General Calculations with Triple Integrals
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- Intro
- Exercise 1 part a
- Exercise 1 part b
- Exercise 1 part c
- Exercise 1 part d
- Exercise 2 part a
- Exercise 2 part b
- Exercise 2 part c
- Exercise 2 part d
- Exercise 3 part a
- Exercise 3 part b
- Exercise 3 part c
- Exercise 3 part d
- Exercise 3 part e
- Exercise 3 part f
- Exercise 4 part 1
- Exercise 4 part 2
- Exercise 5

Triple Integrals, Cylindrical and Spherical
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Triple Integrals, Jacobian
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[{"Name":"General Calculations with Triple Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intro","Duration":"19m 8s","ChapterTopicVideoID":8533,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8533.jpeg","UploadDate":"2020-02-26T06:13:38.2070000","DurationForVideoObject":"PT19M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.525","Text":"In this clip, we\u0027ll be talking about triple integrals."},{"Start":"00:03.525 ","End":"00:07.515","Text":"Now, most of the work is in the exercises."},{"Start":"00:07.515 ","End":"00:12.600","Text":"I\u0027m just going to give an introduction here and some of the main scenarios."},{"Start":"00:12.600 ","End":"00:14.190","Text":"What is a triple integral?"},{"Start":"00:14.190 ","End":"00:18.525","Text":"It\u0027s an extension of a double integral but the 3 dimensions is one way of looking at it."},{"Start":"00:18.525 ","End":"00:21.450","Text":"The notation, as you might expect,"},{"Start":"00:21.450 ","End":"00:24.165","Text":"instead of a double integral, we have two of them."},{"Start":"00:24.165 ","End":"00:25.935","Text":"Triple you have three of them."},{"Start":"00:25.935 ","End":"00:30.194","Text":"It\u0027s over some region or body."},{"Start":"00:30.194 ","End":"00:33.240","Text":"Let\u0027s say R, some use E,"},{"Start":"00:33.240 ","End":"00:35.310","Text":"some use B, it doesn\u0027t matter,"},{"Start":"00:35.310 ","End":"00:39.615","Text":"of some function of 3 variables,"},{"Start":"00:39.615 ","End":"00:42.270","Text":"x, y, and z."},{"Start":"00:42.270 ","End":"00:46.860","Text":"Then just like we had da in 2 dimensions,"},{"Start":"00:46.860 ","End":"00:49.440","Text":"we have dv in 3 dimensions,"},{"Start":"00:49.440 ","End":"00:55.290","Text":"this typically will turn out to be dx dy dz,"},{"Start":"00:55.290 ","End":"00:59.640","Text":"or one of the permutations of these."},{"Start":"00:59.640 ","End":"01:01.410","Text":"Actually, the 6 possibilities,"},{"Start":"01:01.410 ","End":"01:02.880","Text":"dx dy dz, dx dz dy,"},{"Start":"01:02.880 ","End":"01:05.145","Text":"dy dx dz and so on."},{"Start":"01:05.145 ","End":"01:08.070","Text":"Rather than in 2d, We just had dx,"},{"Start":"01:08.070 ","End":"01:09.670","Text":"dy, or dy, dx."},{"Start":"01:09.670 ","End":"01:14.405","Text":"The easiest kind of region to work with is a box."},{"Start":"01:14.405 ","End":"01:20.025","Text":"I\u0027ll change the R to a B, B for box."},{"Start":"01:20.025 ","End":"01:22.430","Text":"This here is the x-direction."},{"Start":"01:22.430 ","End":"01:24.320","Text":"It\u0027s pretty hard to read that."},{"Start":"01:24.320 ","End":"01:34.220","Text":"This here is the y-direction and upwards is the z-direction."},{"Start":"01:34.220 ","End":"01:36.370","Text":"I\u0027ll show where the origin is."},{"Start":"01:36.370 ","End":"01:40.310","Text":"Then this box can be described in more than one way."},{"Start":"01:40.310 ","End":"01:46.250","Text":"We can just use set theory interval notation and say it\u0027s the interval a,"},{"Start":"01:46.250 ","End":"01:50.210","Text":"b Cartesian product with the interval c,"},{"Start":"01:50.210 ","End":"01:53.165","Text":"d Cartesian product with,"},{"Start":"01:53.165 ","End":"02:01.985","Text":"this looks like r and s or we could just say that a is less than or equal to x,"},{"Start":"02:01.985 ","End":"02:04.279","Text":"less than or equal to b,"},{"Start":"02:04.279 ","End":"02:08.150","Text":"that y is between c and"},{"Start":"02:08.150 ","End":"02:14.840","Text":"d and that z is between r and f,"},{"Start":"02:14.840 ","End":"02:17.550","Text":"whichever notation you prefer."},{"Start":"02:17.550 ","End":"02:21.860","Text":"In this case, the integral comes out pretty straightforward."},{"Start":"02:21.860 ","End":"02:27.950","Text":"The integral over such a box B of f, of x,"},{"Start":"02:27.950 ","End":"02:32.120","Text":"y and z, d,"},{"Start":"02:32.120 ","End":"02:35.695","Text":"v is just the iterated integral."},{"Start":"02:35.695 ","End":"02:40.615","Text":"It doesn\u0027t matter in what order but I\u0027ll take it first of all as x"},{"Start":"02:40.615 ","End":"02:47.620","Text":"from a to b. I\u0027ll even write it so it\u0027s clear."},{"Start":"02:47.620 ","End":"02:53.185","Text":"Then y will go from c to d,"},{"Start":"02:53.185 ","End":"02:58.750","Text":"and z will go from r to s. Then"},{"Start":"02:58.750 ","End":"03:06.075","Text":"f of x, y, z."},{"Start":"03:06.075 ","End":"03:10.180","Text":"Then I have to do it in the right order if the innermost is z,"},{"Start":"03:10.180 ","End":"03:16.315","Text":"so this has to be dz and then I need dy and then I need dx"},{"Start":"03:16.315 ","End":"03:23.845","Text":"but you could change the order to any of the six combinations and that would be okay too."},{"Start":"03:23.845 ","End":"03:28.700","Text":"Often, we\u0027re not given any diagram or any particular 3D region,"},{"Start":"03:28.700 ","End":"03:34.050","Text":"we\u0027re just given the integral like this but some particular f to compute."},{"Start":"03:35.480 ","End":"03:39.260","Text":"Let me give an example of one such computation."},{"Start":"03:39.260 ","End":"03:45.180","Text":"If I want to compute the integral from 1 to 2 the integral from 2 to 3,"},{"Start":"03:45.180 ","End":"03:49.360","Text":"the integral from 0 to 1 of 8xyz."},{"Start":"03:51.080 ","End":"03:54.870","Text":"I\u0027ll take it as, let\u0027s say dz,"},{"Start":"03:54.870 ","End":"03:58.535","Text":"dx, dy in this order."},{"Start":"03:58.535 ","End":"04:03.320","Text":"This would correspond to the box where you see this is"},{"Start":"04:03.320 ","End":"04:08.510","Text":"the z. I often do write which variable it isn\u0027t, there\u0027s no confusion."},{"Start":"04:08.510 ","End":"04:15.680","Text":"That\u0027s the innermost one then this would be for x and this would be the outer one for y."},{"Start":"04:15.680 ","End":"04:20.660","Text":"The box would be using the interval notation."},{"Start":"04:20.660 ","End":"04:22.940","Text":"You always write it as the x first."},{"Start":"04:22.940 ","End":"04:32.145","Text":"So x is from 2 to 3 and then y from 1 to 2,"},{"Start":"04:32.145 ","End":"04:35.940","Text":"and z from 0 to 1."},{"Start":"04:35.940 ","End":"04:40.815","Text":"We just happened to choose a set for a box into the order dz, dx, dy."},{"Start":"04:40.815 ","End":"04:44.555","Text":"Now, let\u0027s actually compute what it comes out to be."},{"Start":"04:44.555 ","End":"04:46.490","Text":"We work from the inside out."},{"Start":"04:46.490 ","End":"04:48.935","Text":"First of all I do the dz."},{"Start":"04:48.935 ","End":"04:56.160","Text":"The rest of it I can copy y and x, dx dy."},{"Start":"04:57.290 ","End":"05:00.485","Text":"This integral with respect to z,"},{"Start":"05:00.485 ","End":"05:03.005","Text":"well, it\u0027s a constant times z."},{"Start":"05:03.005 ","End":"05:06.200","Text":"The integral of z is z squared over 2."},{"Start":"05:06.200 ","End":"05:12.160","Text":"What I\u0027m left with is 4xyz squared."},{"Start":"05:12.160 ","End":"05:19.610","Text":"Now, this is a definite integral need to substitute z to be between 0 and 1."},{"Start":"05:19.610 ","End":"05:22.205","Text":"Again, I like to emphasize that this is z."},{"Start":"05:22.205 ","End":"05:24.260","Text":"Well, when z is 0 this is nothing,"},{"Start":"05:24.260 ","End":"05:27.965","Text":"when z is 1 this is just for 4xy."},{"Start":"05:27.965 ","End":"05:33.060","Text":"At the next step, I have the integral from 1 to 2,"},{"Start":"05:33.060 ","End":"05:41.785","Text":"the integral from 2 to 3 and then it\u0027s 4xy, dx, dy."},{"Start":"05:41.785 ","End":"05:47.880","Text":"I think I\u0027ll keep writing the variable x goes from, y goes from."},{"Start":"05:47.930 ","End":"05:52.260","Text":"Now, we need to integrate the 4xy."},{"Start":"05:52.260 ","End":"05:57.600","Text":"We have the integral from 1 to 2 and then 4xy"},{"Start":"05:57.600 ","End":"06:04.145","Text":"with respect to x, 2x squared y."},{"Start":"06:04.145 ","End":"06:09.920","Text":"If I replace x by x squared over 2 this is what I get and I have to evaluate"},{"Start":"06:09.920 ","End":"06:17.490","Text":"this from 2 to 3 and this is with regard to x and then we\u0027ll still have a dy."},{"Start":"06:17.490 ","End":"06:20.290","Text":"I\u0027m going to continue over here."},{"Start":"06:22.280 ","End":"06:26.615","Text":"If we plug in 3, we\u0027re plugging it in for x."},{"Start":"06:26.615 ","End":"06:30.830","Text":"If we plug in 3, x squared is 9 and if we plug in 2,"},{"Start":"06:30.830 ","End":"06:32.165","Text":"x squared is 4."},{"Start":"06:32.165 ","End":"06:34.475","Text":"9 minus 4 is 5."},{"Start":"06:34.475 ","End":"06:39.895","Text":"5 times 2 is 10 so we get 10y."},{"Start":"06:39.895 ","End":"06:43.560","Text":"Then the integral from 1 to 2, dy."},{"Start":"06:43.560 ","End":"06:50.440","Text":"We should be putting equals."},{"Start":"06:52.070 ","End":"07:02.820","Text":"Then we have 10y is y squared over 2 times 10 is 5y squared from 1 to 2,"},{"Start":"07:02.820 ","End":"07:06.675","Text":"and when y is 2 this is 4 otherwise it\u0027s 1,"},{"Start":"07:06.675 ","End":"07:08.865","Text":"4 minus 1 is 3."},{"Start":"07:08.865 ","End":"07:14.340","Text":"The answer is 5 times 3, which is 15."},{"Start":"07:14.340 ","End":"07:17.855","Text":"Now, you might want to check that if you change the order,"},{"Start":"07:17.855 ","End":"07:22.085","Text":"I change the order of these integrals and I correspondingly change these,"},{"Start":"07:22.085 ","End":"07:24.485","Text":"then you\u0027ll get the same answer."},{"Start":"07:24.485 ","End":"07:27.010","Text":"For a box, it doesn\u0027t matter."},{"Start":"07:27.010 ","End":"07:30.320","Text":"Now, let\u0027s get to something more general."},{"Start":"07:30.320 ","End":"07:33.110","Text":"Sometimes we\u0027re not even given the body."},{"Start":"07:33.110 ","End":"07:36.140","Text":"I\u0027ll give you an example of an integral."},{"Start":"07:36.140 ","End":"07:42.715","Text":"We\u0027re just given the integral from 0 to 1,"},{"Start":"07:42.715 ","End":"07:46.705","Text":"integral from 0 to x,"},{"Start":"07:46.705 ","End":"07:51.120","Text":"integral from 0 to x plus y,"},{"Start":"07:51.120 ","End":"07:58.720","Text":"6xy, dz, dy, dx."},{"Start":"07:58.720 ","End":"08:02.450","Text":"We might just be given this without any description of the region."},{"Start":"08:02.450 ","End":"08:05.930","Text":"We could try to make a sketch though it\u0027s 3D of the region,"},{"Start":"08:05.930 ","End":"08:07.940","Text":"but no need to."},{"Start":"08:07.940 ","End":"08:09.935","Text":"It is important."},{"Start":"08:09.935 ","End":"08:14.270","Text":"You can\u0027t just change the order around here if it\u0027s not a box."},{"Start":"08:14.270 ","End":"08:16.960","Text":"If I have an integral dz as the inner 1,"},{"Start":"08:16.960 ","End":"08:19.400","Text":"then in the limit of integration,"},{"Start":"08:19.400 ","End":"08:20.780","Text":"I can only have x and y,"},{"Start":"08:20.780 ","End":"08:22.295","Text":"I can\u0027t have z."},{"Start":"08:22.295 ","End":"08:26.960","Text":"In the next layer when I do it dy, I\u0027ve already done dz,"},{"Start":"08:26.960 ","End":"08:29.380","Text":"then we have to eliminate z and y,"},{"Start":"08:29.380 ","End":"08:33.650","Text":"we can only have x and the last of the outer 1 can only have constant,"},{"Start":"08:33.650 ","End":"08:35.855","Text":"they can\u0027t have z, y, or x."},{"Start":"08:35.855 ","End":"08:38.660","Text":"It won\u0027t make sense to change them around it"},{"Start":"08:38.660 ","End":"08:41.300","Text":"just when you start evaluating you\u0027ll see it doesn\u0027t make sense."},{"Start":"08:41.300 ","End":"08:45.810","Text":"Anyway, let\u0027s just compute this 1 as an exercise and as always,"},{"Start":"08:45.810 ","End":"08:47.475","Text":"we work them from the inside out."},{"Start":"08:47.475 ","End":"08:54.795","Text":"The first integral will be dz and the integral of 6xy it\u0027s a constant as far as z goes."},{"Start":"08:54.795 ","End":"09:00.200","Text":"This is just going to be 6xyz but it\u0027s going to be"},{"Start":"09:00.200 ","End":"09:06.650","Text":"between the limits of 0 to x plus y."},{"Start":"09:06.650 ","End":"09:14.120","Text":"Here, I write dy dx and here are the first 2 integral from 0 to 1,"},{"Start":"09:14.120 ","End":"09:18.690","Text":"integral from 0 to x."},{"Start":"09:18.690 ","End":"09:22.295","Text":"Now if we plug in z equals 0 we get nothing."},{"Start":"09:22.295 ","End":"09:27.350","Text":"If you plug in z equals x plus y then we"},{"Start":"09:27.350 ","End":"09:35.605","Text":"get 6xy and then z is x plus y."},{"Start":"09:35.605 ","End":"09:42.040","Text":"Then this integral, dy dx."},{"Start":"09:42.040 ","End":"09:45.030","Text":"I think I\u0027ll continue up here."},{"Start":"09:45.030 ","End":"09:49.840","Text":"I\u0027m going to just copy pasted it but now I want to expand this."},{"Start":"09:50.030 ","End":"09:52.125","Text":"Let\u0027s see. I\u0027m looking over here."},{"Start":"09:52.125 ","End":"09:56.520","Text":"This would be 6xyx so 6x squared y."},{"Start":"09:56.520 ","End":"10:03.390","Text":"The other 6xy squared and brackets."},{"Start":"10:03.390 ","End":"10:07.885","Text":"Now, this integral dy is not difficult."},{"Start":"10:07.885 ","End":"10:13.290","Text":"The first one integral of y is y squared over 2 so"},{"Start":"10:13.290 ","End":"10:18.360","Text":"I get the over 2 makes it 3 x squared y squared."},{"Start":"10:18.360 ","End":"10:24.090","Text":"The other one, I get y cubed over 3 so it\u0027s"},{"Start":"10:24.090 ","End":"10:31.050","Text":"2xy cubed and all this is from 0 to x."},{"Start":"10:31.050 ","End":"10:35.085","Text":"This, of course, is y because it\u0027s the integral dy."},{"Start":"10:35.085 ","End":"10:39.390","Text":"I still have from 0 to 1 dx."},{"Start":"10:39.390 ","End":"10:43.155","Text":"If y is 0 this whole thing comes out 0."},{"Start":"10:43.155 ","End":"10:45.820","Text":"If y equals x,"},{"Start":"10:45.820 ","End":"10:48.230","Text":"then what we get, y is x,"},{"Start":"10:48.230 ","End":"10:57.590","Text":"this is 3x squared x squared plus 2x x cubed dx."},{"Start":"10:57.590 ","End":"11:00.490","Text":"Should I be writing equals all along."},{"Start":"11:00.490 ","End":"11:03.340","Text":"I guess it\u0027s understood that it doesn\u0027t hurt."},{"Start":"11:03.340 ","End":"11:07.750","Text":"What is this equal? This is 3x^4th plus 2x^4th, is 5x^4th."},{"Start":"11:07.750 ","End":"11:11.390","Text":"I have the integral from 0-1,"},{"Start":"11:11.790 ","End":"11:19.705","Text":"5x^4th, dx, and this is equal to,"},{"Start":"11:19.705 ","End":"11:21.490","Text":"let\u0027s see, 5x^4th,"},{"Start":"11:21.490 ","End":"11:22.795","Text":"x^5th over 5."},{"Start":"11:22.795 ","End":"11:28.705","Text":"It\u0027s just x^5th from 0-1."},{"Start":"11:28.705 ","End":"11:30.085","Text":"That\u0027s just 1."},{"Start":"11:30.085 ","End":"11:33.610","Text":"The answer come out nice and neat, just 1."},{"Start":"11:33.610 ","End":"11:35.590","Text":"Now, we could, as I said,"},{"Start":"11:35.590 ","End":"11:38.800","Text":"describe the region, but often it\u0027s just given like this."},{"Start":"11:38.800 ","End":"11:44.200","Text":"The next thing I want to talk about is volume as a triple integral,"},{"Start":"11:44.200 ","End":"11:47.170","Text":"just like we had area as a double integral."},{"Start":"11:47.170 ","End":"11:49.270","Text":"Go to a new page."},{"Start":"11:49.270 ","End":"11:56.000","Text":"Now, I\u0027m going to talk about volume as a triple integral."},{"Start":"11:57.330 ","End":"12:00.850","Text":"I\u0027ll just give you a formula."},{"Start":"12:00.850 ","End":"12:08.965","Text":"Suppose I want to know the volume of a body and let\u0027s say the body is called the 3D body."},{"Start":"12:08.965 ","End":"12:14.605","Text":"B is given by the triple integral"},{"Start":"12:14.605 ","End":"12:22.165","Text":"over this body B of just dv."},{"Start":"12:22.165 ","End":"12:25.720","Text":"Although I usually like to write it as 1dv."},{"Start":"12:25.720 ","End":"12:33.290","Text":"Just like we had the area of a region was the double integral of 1da."},{"Start":"12:33.900 ","End":"12:37.630","Text":"Let me just give an example of this."},{"Start":"12:37.630 ","End":"12:44.065","Text":"As an example, I\u0027m going to describe B and you\u0027ll find the volume."},{"Start":"12:44.065 ","End":"12:47.980","Text":"Let\u0027s say that B is bounded by,"},{"Start":"12:47.980 ","End":"12:49.480","Text":"and this is what\u0027s commonly done."},{"Start":"12:49.480 ","End":"12:55.285","Text":"You give a set of surfaces that bound it by the following surfaces."},{"Start":"12:55.285 ","End":"12:59.530","Text":"Let\u0027s take x equals naught as a plane,"},{"Start":"12:59.530 ","End":"13:02.185","Text":"y equals naught as a plane."},{"Start":"13:02.185 ","End":"13:07.570","Text":"X plus y equals 2, that\u0027s the plane."},{"Start":"13:07.570 ","End":"13:14.750","Text":"Z equals naught, that\u0027s the plane, z equals 3xy."},{"Start":"13:15.750 ","End":"13:21.550","Text":"We want to figure out the volume of B."},{"Start":"13:21.550 ","End":"13:25.810","Text":"Now, it\u0027s not quite clear without a sketch or anything,"},{"Start":"13:25.810 ","End":"13:28.180","Text":"but here are some things."},{"Start":"13:28.180 ","End":"13:30.490","Text":"If I look at this,"},{"Start":"13:30.490 ","End":"13:36.700","Text":"z depends on x and y. I know that when I describe B,"},{"Start":"13:36.700 ","End":"13:40.030","Text":"I have to do as the innermost integral."},{"Start":"13:40.030 ","End":"13:45.970","Text":"The innermost iteration is going to be z from 0 to 3xy,"},{"Start":"13:45.970 ","End":"13:49.600","Text":"or the other way around depending on which is higher or up."},{"Start":"13:49.600 ","End":"13:54.490","Text":"We\u0027ll see that, how we find that in a moment."},{"Start":"13:54.490 ","End":"14:02.050","Text":"Let me just start off by writing the integral from 0 to 3xy,"},{"Start":"14:02.050 ","End":"14:05.560","Text":"although it could be the other way around, I don\u0027t know,"},{"Start":"14:05.560 ","End":"14:09.535","Text":"we\u0027ll see in a moment of 1."},{"Start":"14:09.535 ","End":"14:12.145","Text":"Then dv is going to be,"},{"Start":"14:12.145 ","End":"14:13.690","Text":"starts off by dz."},{"Start":"14:13.690 ","End":"14:17.420","Text":"Then I have to figure whether it\u0027s dxdy or dydx."},{"Start":"14:17.490 ","End":"14:22.810","Text":"I\u0027m going to need a sketch here. Let\u0027s see."},{"Start":"14:22.810 ","End":"14:31.255","Text":"I just need dxy plane because I want to figure out what are these three. Let\u0027s see."},{"Start":"14:31.255 ","End":"14:33.985","Text":"X equals naught is the y axis."},{"Start":"14:33.985 ","End":"14:36.760","Text":"Y equals naught is the x-axis,"},{"Start":"14:36.760 ","End":"14:38.545","Text":"and x plus y equals 2."},{"Start":"14:38.545 ","End":"14:41.410","Text":"Well, when x is 0, y is 2 minus 0, x is 2."},{"Start":"14:41.410 ","End":"14:42.940","Text":"In the xy plane,"},{"Start":"14:42.940 ","End":"14:45.460","Text":"it would be something like this,"},{"Start":"14:45.460 ","End":"14:48.760","Text":"where this is 2 and this is 2."},{"Start":"14:48.760 ","End":"14:52.030","Text":"Then we said we had the y-axis for x equals 0."},{"Start":"14:52.030 ","End":"14:55.915","Text":"Obviously, I just need this bit and this bit."},{"Start":"14:55.915 ","End":"14:59.665","Text":"This will have to integrate over this 2d region."},{"Start":"14:59.665 ","End":"15:03.250","Text":"We could do it as a type 2 or type 1 integral."},{"Start":"15:03.250 ","End":"15:05.455","Text":"Let\u0027s do it as a type 1 integral,"},{"Start":"15:05.455 ","End":"15:09.260","Text":"which means that we\u0027re taking vertical slices."},{"Start":"15:09.390 ","End":"15:19.030","Text":"We enter at, this is y equals 0, the x-axis."},{"Start":"15:19.030 ","End":"15:22.960","Text":"Here, well, I can just extract y from here."},{"Start":"15:22.960 ","End":"15:26.830","Text":"I can write this as y equals 2 minus x."},{"Start":"15:26.830 ","End":"15:29.465","Text":"If I do it as a vertical slice,"},{"Start":"15:29.465 ","End":"15:38.910","Text":"so I have the integral from 0-2 minus xdy,"},{"Start":"15:38.910 ","End":"15:45.760","Text":"and x goes from 0-2, 0 to 2dx."},{"Start":"15:45.760 ","End":"15:48.100","Text":"Let me return to the point that we don\u0027t know which of these"},{"Start":"15:48.100 ","End":"15:51.285","Text":"2 is higher and which is lower."},{"Start":"15:51.285 ","End":"15:53.070","Text":"Well, just plug in any value."},{"Start":"15:53.070 ","End":"16:00.210","Text":"Let\u0027s say the point 11 would be in here,11 if I plug it in here,"},{"Start":"16:00.210 ","End":"16:01.470","Text":"is going to be something positive."},{"Start":"16:01.470 ","End":"16:03.480","Text":"It\u0027s 3, plug it in here with 0,"},{"Start":"16:03.480 ","End":"16:04.740","Text":"so this is higher or up."},{"Start":"16:04.740 ","End":"16:06.785","Text":"There\u0027s no question mark here."},{"Start":"16:06.785 ","End":"16:09.190","Text":"Now it\u0027s just straightforward computation,"},{"Start":"16:09.190 ","End":"16:11.665","Text":"it\u0027s technical, but you know what, I\u0027ll do it."},{"Start":"16:11.665 ","End":"16:15.055","Text":"Let\u0027s start from the inside out, we have to do that."},{"Start":"16:15.055 ","End":"16:19.165","Text":"We\u0027ve got the integral from 0-2."},{"Start":"16:19.165 ","End":"16:24.445","Text":"The outer 2 go from 0-2 minus x."},{"Start":"16:24.445 ","End":"16:29.680","Text":"This integral of 1dz is just z."},{"Start":"16:29.680 ","End":"16:37.990","Text":"From here to here means that we plugin just z equals 3xy and z equals naught,"},{"Start":"16:37.990 ","End":"16:45.580","Text":"so it\u0027s just 3xy minus 0 and then dydx."},{"Start":"16:45.580 ","End":"16:49.735","Text":"Of course, this is y and this is x."},{"Start":"16:49.735 ","End":"16:56.410","Text":"Now, the next one is the 3xy dy."},{"Start":"16:56.410 ","End":"17:00.700","Text":"The integral of y is y squared over 2."},{"Start":"17:00.700 ","End":"17:08.710","Text":"I\u0027ve got 3xy squared over 2,"},{"Start":"17:08.710 ","End":"17:13.869","Text":"so 3 over 2xy squared from"},{"Start":"17:13.869 ","End":"17:17.500","Text":"0-2 minus x,"},{"Start":"17:17.500 ","End":"17:24.040","Text":"dx from 0-2."},{"Start":"17:24.040 ","End":"17:26.035","Text":"Let\u0027s see what this is,"},{"Start":"17:26.035 ","End":"17:28.150","Text":"y is 0 gives me nothing."},{"Start":"17:28.150 ","End":"17:34.495","Text":"Y is 2 minus x gives me the integral from 0-2,"},{"Start":"17:34.495 ","End":"17:43.370","Text":"3 over 2x times 2x minus squared dx."},{"Start":"17:45.630 ","End":"17:48.370","Text":"Continue over here."},{"Start":"17:48.370 ","End":"17:54.490","Text":"If I multiply this out using special binomial expansion,"},{"Start":"17:54.490 ","End":"17:56.934","Text":"what I\u0027ll have is,"},{"Start":"17:56.934 ","End":"18:02.275","Text":"this will be 4 minus 4x plus x squared."},{"Start":"18:02.275 ","End":"18:06.805","Text":"I\u0027ll keep the 3 over 2,"},{"Start":"18:06.805 ","End":"18:09.445","Text":"and then I\u0027ll have, let\u0027s see,"},{"Start":"18:09.445 ","End":"18:13.300","Text":"4 minus 4x plus x squared,"},{"Start":"18:13.300 ","End":"18:14.725","Text":"but I\u0027ll multiply that by x."},{"Start":"18:14.725 ","End":"18:22.120","Text":"It\u0027s 4x minus 4x squared plus x cubed,"},{"Start":"18:22.120 ","End":"18:26.425","Text":"yeah, because this was 4 minus 4x plus x squared."},{"Start":"18:26.425 ","End":"18:30.430","Text":"Multiply this by x, that\u0027s what I get."},{"Start":"18:30.430 ","End":"18:39.580","Text":"From 0-2, so we get 3 over 2."},{"Start":"18:39.580 ","End":"18:43.390","Text":"Now, this will be 4x squared over 2 is 2x."},{"Start":"18:43.390 ","End":"18:48.520","Text":"Here, 4 over 3x cubed, here,"},{"Start":"18:48.520 ","End":"18:57.640","Text":"x_4th over 4 from 0-2."},{"Start":"18:57.640 ","End":"18:59.410","Text":"I\u0027ll leave you to finish this off."},{"Start":"18:59.410 ","End":"19:02.545","Text":"It\u0027s just straightforward and I make it 2."},{"Start":"19:02.545 ","End":"19:04.525","Text":"That\u0027s enough for this introduction,"},{"Start":"19:04.525 ","End":"19:09.380","Text":"all the rest will be in the solved exercises. That\u0027s it."}],"ID":8723},{"Watched":false,"Name":"Exercise 1 part a","Duration":"5m 42s","ChapterTopicVideoID":8534,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8534.jpeg","UploadDate":"2017-02-13T03:18:00.6300000","DurationForVideoObject":"PT5M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.485","Text":"In this exercise, we have to compute the following triple integral."},{"Start":"00:04.485 ","End":"00:06.675","Text":"I\u0027ll start by copying it."},{"Start":"00:06.675 ","End":"00:09.300","Text":"The integrals are done from the inside out."},{"Start":"00:09.300 ","End":"00:11.565","Text":"In other words we first do this integral dy,"},{"Start":"00:11.565 ","End":"00:13.290","Text":"then this integral dx,"},{"Start":"00:13.290 ","End":"00:14.760","Text":"then this integral dz."},{"Start":"00:14.760 ","End":"00:19.755","Text":"Sometimes I like to emphasize to keep things straight."},{"Start":"00:19.755 ","End":"00:21.945","Text":"Let\u0027s say here the outer integral,"},{"Start":"00:21.945 ","End":"00:24.120","Text":"z goes from 0-1,"},{"Start":"00:24.120 ","End":"00:27.630","Text":"here x goes from 0 to z,"},{"Start":"00:27.630 ","End":"00:30.495","Text":"and here y goes from this,"},{"Start":"00:30.495 ","End":"00:35.170","Text":"and then you really can\u0027t get them mixed up."},{"Start":"00:35.450 ","End":"00:41.560","Text":"The innermost first, that would be this integral dy."},{"Start":"00:42.660 ","End":"00:45.490","Text":"I\u0027d like to do this highlighted bit,"},{"Start":"00:45.490 ","End":"00:46.930","Text":"the inner integral this side."},{"Start":"00:46.930 ","End":"00:49.150","Text":"I\u0027ll call it the asterisk."},{"Start":"00:49.150 ","End":"00:55.370","Text":"What I have is the integral from 0 to x plus z."},{"Start":"00:55.370 ","End":"01:03.270","Text":"The 6, I can actually bring out in front because it\u0027s a constant. Changed my mind,"},{"Start":"01:03.270 ","End":"01:05.955","Text":"we\u0027ll keep the 6xz here,"},{"Start":"01:05.955 ","End":"01:10.915","Text":"because the whole thing is a constant as far as y goes."},{"Start":"01:10.915 ","End":"01:14.815","Text":"This integral is just 6xz,"},{"Start":"01:14.815 ","End":"01:17.335","Text":"the constant times y,"},{"Start":"01:17.335 ","End":"01:24.720","Text":"and this I have to take between 0 and x plus z."},{"Start":"01:24.720 ","End":"01:27.810","Text":"These limits are for y."},{"Start":"01:27.810 ","End":"01:32.420","Text":"If you\u0027re not sure, you can also write here y goes from 0 to x plus z."},{"Start":"01:32.420 ","End":"01:36.530","Text":"If I plug in y equals 0, I\u0027ll get nothing."},{"Start":"01:36.530 ","End":"01:39.155","Text":"I only have the top limit to substitute,"},{"Start":"01:39.155 ","End":"01:47.455","Text":"and so I get 6xz and instead of y, x plus z."},{"Start":"01:47.455 ","End":"01:50.220","Text":"That\u0027s this integral, and"},{"Start":"01:50.220 ","End":"01:55.820","Text":"now I\u0027ll just mark it so we can see that this yellow corresponds to this yellow."},{"Start":"01:55.820 ","End":"02:01.925","Text":"Now back here, we have the integral from 0-1,"},{"Start":"02:01.925 ","End":"02:06.360","Text":"from 0 to z of what\u0027s here,"},{"Start":"02:06.360 ","End":"02:11.865","Text":"which I can write if I expand the bracket as"},{"Start":"02:11.865 ","End":"02:17.580","Text":"6x squared z from"},{"Start":"02:17.580 ","End":"02:24.520","Text":"this times this and then plus 6xz squared."},{"Start":"02:24.950 ","End":"02:27.540","Text":"I\u0027ll need the brackets."},{"Start":"02:27.540 ","End":"02:31.060","Text":"It\u0027s now dx dz."},{"Start":"02:31.060 ","End":"02:35.705","Text":"Once again, we do the innermost integral."},{"Start":"02:35.705 ","End":"02:40.979","Text":"This time it is the dx integral."},{"Start":"02:41.600 ","End":"02:45.410","Text":"As before, I like to do these aside calculations."},{"Start":"02:45.410 ","End":"02:50.460","Text":"Let me call this double asterisk and I\u0027ll do that 1 over here."},{"Start":"02:53.030 ","End":"02:55.710","Text":"Well, I won\u0027t keep and copy it,"},{"Start":"02:55.710 ","End":"03:01.510","Text":"I\u0027ll just straight away go for the integral of this with respect to x."},{"Start":"03:01.670 ","End":"03:11.370","Text":"From this, x squared gives me 1/3x cubed."},{"Start":"03:11.370 ","End":"03:21.870","Text":"Just make a little note that the integral of x squared in shorthand is 1/3x cubed."},{"Start":"03:21.870 ","End":"03:23.850","Text":"If I\u0027m doing that here,"},{"Start":"03:23.850 ","End":"03:26.295","Text":"the 1/3 will cancel with the 6,"},{"Start":"03:26.295 ","End":"03:34.245","Text":"and it will leave me with 2x cubed z."},{"Start":"03:34.245 ","End":"03:37.710","Text":"The second 1 is linear."},{"Start":"03:37.710 ","End":"03:42.145","Text":"The 6z squared is a constant times x."},{"Start":"03:42.145 ","End":"03:44.725","Text":"The integral of x is x squared over 2."},{"Start":"03:44.725 ","End":"03:46.360","Text":"Maybe I\u0027ll write that also."},{"Start":"03:46.360 ","End":"03:52.665","Text":"The integral of x dx is 1/2x squared."},{"Start":"03:52.665 ","End":"03:59.730","Text":"This time I get 3 because 1/2 goes into the 6."},{"Start":"03:59.730 ","End":"04:04.740","Text":"This time I get x squared z squared."},{"Start":"04:04.740 ","End":"04:10.755","Text":"All this taken between 0 and z."},{"Start":"04:10.755 ","End":"04:14.910","Text":"To emphasize, that\u0027s what I substitute for x."},{"Start":"04:17.600 ","End":"04:21.615","Text":"Well, if I put x equals 0, it\u0027s 0."},{"Start":"04:21.615 ","End":"04:25.865","Text":"I only have to put in the top limit z and what it gives me,"},{"Start":"04:25.865 ","End":"04:27.500","Text":"I put instead of xz,"},{"Start":"04:27.500 ","End":"04:35.240","Text":"2z cubed z plus"},{"Start":"04:35.240 ","End":"04:43.945","Text":"3z squared z squared."},{"Start":"04:43.945 ","End":"04:47.760","Text":"Now, this is z to the 4th and this is z to the 4th."},{"Start":"04:47.760 ","End":"04:49.215","Text":"I\u0027ve got 5 of them."},{"Start":"04:49.215 ","End":"04:57.500","Text":"I\u0027ve got 5z to the 4th and this corresponds to this integral."},{"Start":"04:57.500 ","End":"04:59.225","Text":"Back here again."},{"Start":"04:59.225 ","End":"05:01.549","Text":"Since now the final layer,"},{"Start":"05:01.549 ","End":"05:05.320","Text":"I just have the integral from 0-1 of whatever is here,"},{"Start":"05:05.320 ","End":"05:11.190","Text":"5z to the 4th dz."},{"Start":"05:11.420 ","End":"05:16.450","Text":"Now, the integral of 5z to the 4th,"},{"Start":"05:16.610 ","End":"05:21.000","Text":"I raise the power by 1 and get 5 and divide by 5,"},{"Start":"05:21.000 ","End":"05:24.070","Text":"so it\u0027s just z to the 5th."},{"Start":"05:25.870 ","End":"05:29.615","Text":"Between 0 and 1,"},{"Start":"05:29.615 ","End":"05:31.820","Text":"if I plug in 1, I get 1."},{"Start":"05:31.820 ","End":"05:33.725","Text":"If I plug in 0, I get 0."},{"Start":"05:33.725 ","End":"05:36.319","Text":"I just get 1 minus 0."},{"Start":"05:36.319 ","End":"05:42.400","Text":"The final answer is 1 and after the final answer we\u0027re done."}],"ID":8724},{"Watched":false,"Name":"Exercise 1 part b","Duration":"8m 17s","ChapterTopicVideoID":8535,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8535.jpeg","UploadDate":"2017-02-13T03:19:44.6800000","DurationForVideoObject":"PT8M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.974","Text":"In this exercise, we need to compute this triple integral."},{"Start":"00:03.974 ","End":"00:06.345","Text":"I\u0027d like to start by copying it."},{"Start":"00:06.345 ","End":"00:08.984","Text":"Actually, I did more than just copy."},{"Start":"00:08.984 ","End":"00:13.590","Text":"I often like to not get confused as to which variable is being integrated."},{"Start":"00:13.590 ","End":"00:15.390","Text":"We go from the inside out,"},{"Start":"00:15.390 ","End":"00:17.370","Text":"it\u0027s dx, dz, dy."},{"Start":"00:17.370 ","End":"00:19.395","Text":"This integral goes with the dx."},{"Start":"00:19.395 ","End":"00:21.075","Text":"These are the limits for x."},{"Start":"00:21.075 ","End":"00:24.450","Text":"Likewise, these are the limits for z and these are the limits for y."},{"Start":"00:24.450 ","End":"00:29.025","Text":"Sometimes I add these and avoid confusion."},{"Start":"00:29.025 ","End":"00:32.235","Text":"We have to do the innermost first,"},{"Start":"00:32.235 ","End":"00:34.590","Text":"and that would be this one,"},{"Start":"00:34.590 ","End":"00:41.150","Text":"that\u0027s dx, and I\u0027d like to do this one separately at the side,"},{"Start":"00:41.150 ","End":"00:43.025","Text":"I\u0027ll call it asterisk."},{"Start":"00:43.025 ","End":"00:48.800","Text":"What I get from here is because all this doesn\u0027t"},{"Start":"00:48.800 ","End":"00:55.655","Text":"depend on x. I get after integrating it,"},{"Start":"00:55.655 ","End":"01:03.755","Text":"ze to the y times just x because that was a constant,"},{"Start":"01:03.755 ","End":"01:14.265","Text":"and this taken from 0 to square root of 1 minus z squared."},{"Start":"01:14.265 ","End":"01:16.220","Text":"This equals, first of all,"},{"Start":"01:16.220 ","End":"01:19.040","Text":"I substitute x equals the upper limit."},{"Start":"01:19.040 ","End":"01:28.314","Text":"I get ze to the y square root of 1 minus z squared,"},{"Start":"01:28.314 ","End":"01:31.579","Text":"and that\u0027s it because when I put x equals 0,"},{"Start":"01:31.579 ","End":"01:35.790","Text":"it\u0027s just 0 though you could write minus 0."},{"Start":"01:37.340 ","End":"01:44.225","Text":"I\u0027ll highlight this to show that this is the answer to this and now we can continue."},{"Start":"01:44.225 ","End":"01:53.250","Text":"Get the integral from 0-3, integral from 0-1."},{"Start":"01:53.250 ","End":"01:57.740","Text":"This bit I replace by ze to"},{"Start":"01:57.740 ","End":"02:06.210","Text":"the y square root of 1 minus z squared dz dy."},{"Start":"02:08.770 ","End":"02:14.825","Text":"Next in line is this integral, the dz integral."},{"Start":"02:14.825 ","End":"02:17.480","Text":"Again, I\u0027ll do it at the side,"},{"Start":"02:17.480 ","End":"02:19.740","Text":"call it double asterisk,"},{"Start":"02:19.740 ","End":"02:21.730","Text":"and I\u0027ll do it over here."},{"Start":"02:21.730 ","End":"02:26.690","Text":"I\u0027ll start out by noticing that e to the y is a constant."},{"Start":"02:26.690 ","End":"02:32.095","Text":"It will be easier for me to take e to the y outside the integral,"},{"Start":"02:32.095 ","End":"02:35.770","Text":"and then it goes from 0-1,"},{"Start":"02:36.470 ","End":"02:43.060","Text":"z root 1 minus z squared dz."},{"Start":"02:44.040 ","End":"02:47.170","Text":"You might wonder, how do we do such an integral,"},{"Start":"02:47.170 ","End":"02:49.505","Text":"and the answer is substitution."},{"Start":"02:49.505 ","End":"02:54.910","Text":"There\u0027s actually 2 reasonable possibilities for substitution."},{"Start":"02:54.910 ","End":"03:00.530","Text":"I could let t be 1 minus z squared."},{"Start":"03:00.530 ","End":"03:04.960","Text":"Or I could let t be the whole square root of 1 minus z squared."},{"Start":"03:04.960 ","End":"03:07.105","Text":"Both of them will work."},{"Start":"03:07.105 ","End":"03:09.280","Text":"Just to choose 1,"},{"Start":"03:09.280 ","End":"03:17.250","Text":"I\u0027ll go with the substitution t equals the square root of 1 minus z squared."},{"Start":"03:17.250 ","End":"03:26.510","Text":"Then what I\u0027ll get is that dt is equal to the derivative."},{"Start":"03:26.510 ","End":"03:29.295","Text":"Oops, went too fast."},{"Start":"03:29.295 ","End":"03:34.205","Text":"Best thing to do here is to square both sides and get rid of that square root,"},{"Start":"03:34.205 ","End":"03:39.445","Text":"so t squared equals 1 minus z squared."},{"Start":"03:39.445 ","End":"03:46.160","Text":"Now from here, I get that if I take d of both sides"},{"Start":"03:46.160 ","End":"03:54.360","Text":"2tdt would equal minus 2zdz."},{"Start":"03:58.630 ","End":"04:05.630","Text":"I can cancel the 2 and then I\u0027d like to extract dz, which is equal to,"},{"Start":"04:05.630 ","End":"04:08.015","Text":"I\u0027ll just divide both sides by minus z,"},{"Start":"04:08.015 ","End":"04:14.645","Text":"is tdt over minus z."},{"Start":"04:14.645 ","End":"04:18.185","Text":"Now, I\u0027ll go and make the substitution."},{"Start":"04:18.185 ","End":"04:22.340","Text":"We get e to the y, just stays there."},{"Start":"04:22.340 ","End":"04:27.590","Text":"The integral, we have to substitute the limits of integration also."},{"Start":"04:27.590 ","End":"04:30.075","Text":"We can do this in our heads."},{"Start":"04:30.075 ","End":"04:32.105","Text":"When z is 0,"},{"Start":"04:32.105 ","End":"04:37.670","Text":"t is the square root of 1 minus 0, so that\u0027s 1."},{"Start":"04:37.670 ","End":"04:44.525","Text":"These are limits for t. I\u0027ll even maybe emphasize that this is z equals 0-1."},{"Start":"04:44.525 ","End":"04:47.225","Text":"Here t goes from 1 to what?"},{"Start":"04:47.225 ","End":"04:50.425","Text":"When we put in z equals 1,"},{"Start":"04:50.425 ","End":"04:53.040","Text":"we\u0027ll get 1 minus 1 squared is 0,"},{"Start":"04:53.040 ","End":"04:56.955","Text":"square root of 0 is 0, so t goes from 1-0."},{"Start":"04:56.955 ","End":"05:02.445","Text":"Next, we get, if I put dz"},{"Start":"05:02.445 ","End":"05:10.690","Text":"here as tdt over minus z,"},{"Start":"05:11.000 ","End":"05:15.450","Text":"and I replace this thing by t, here\u0027s what we get."},{"Start":"05:15.450 ","End":"05:19.395","Text":"We get z, this thing is t,"},{"Start":"05:19.395 ","End":"05:22.625","Text":"and dz from here is,"},{"Start":"05:22.625 ","End":"05:24.005","Text":"I\u0027ll put the minus on the top,"},{"Start":"05:24.005 ","End":"05:29.525","Text":"minus tdt over z."},{"Start":"05:29.525 ","End":"05:33.340","Text":"Z cancels with z,"},{"Start":"05:33.340 ","End":"05:40.745","Text":"so what we get is e to the y stays."},{"Start":"05:40.745 ","End":"05:48.435","Text":"We\u0027ve got the integral from 1-0 of,"},{"Start":"05:48.435 ","End":"05:50.984","Text":"minus I can put in front,"},{"Start":"05:50.984 ","End":"05:58.020","Text":"and then t with t is t squared dt."},{"Start":"05:58.020 ","End":"06:01.200","Text":"Now, this integral we can do in our heads."},{"Start":"06:01.200 ","End":"06:06.900","Text":"The integral of t squared is 1/3t cubed,"},{"Start":"06:06.900 ","End":"06:10.725","Text":"and if I take this 1/3t cubed,"},{"Start":"06:10.725 ","End":"06:12.945","Text":"maybe I better write it."},{"Start":"06:12.945 ","End":"06:19.550","Text":"This is equal to minus e to the y times"},{"Start":"06:19.550 ","End":"06:28.905","Text":"1/3t cubed between the limits of 1 and 0."},{"Start":"06:28.905 ","End":"06:34.890","Text":"That\u0027s a 3, and this equals,"},{"Start":"06:34.890 ","End":"06:37.670","Text":"let\u0027s see, what does it equal?"},{"Start":"06:37.670 ","End":"06:42.900","Text":"We\u0027ve got minus e to the y,"},{"Start":"06:42.900 ","End":"06:47.990","Text":"and then I plug in first the 0 and I get 0,"},{"Start":"06:47.990 ","End":"06:53.155","Text":"then I plug in the 1 and I get 1/3."},{"Start":"06:53.155 ","End":"06:57.845","Text":"It\u0027s minus e to the y times minus 1/3,"},{"Start":"06:57.845 ","End":"07:02.150","Text":"so it\u0027s 1/3e to the y,"},{"Start":"07:02.150 ","End":"07:07.920","Text":"and that\u0027s the answer to this middle integral."},{"Start":"07:08.600 ","End":"07:11.570","Text":"Now we\u0027re on the last layer."},{"Start":"07:11.570 ","End":"07:14.885","Text":"We have the integral from 0-3."},{"Start":"07:14.885 ","End":"07:22.620","Text":"This I just copy from here as 1/3e to the y dy."},{"Start":"07:26.930 ","End":"07:31.610","Text":"The integral of e to the y is just e to the y and the constant stays."},{"Start":"07:31.610 ","End":"07:39.170","Text":"It\u0027s 1/3e to the y, taken from 0-3."},{"Start":"07:39.170 ","End":"07:42.799","Text":"If I plug in 3,"},{"Start":"07:42.799 ","End":"07:48.245","Text":"I get 1/3e cubed,"},{"Start":"07:48.245 ","End":"07:52.720","Text":"and I could say minus"},{"Start":"07:52.720 ","End":"08:00.940","Text":"1/3e to the 0,"},{"Start":"08:00.940 ","End":"08:03.415","Text":"and this is equal to,"},{"Start":"08:03.415 ","End":"08:06.225","Text":"I\u0027ll take the 1/3 outside the brackets,"},{"Start":"08:06.225 ","End":"08:09.365","Text":"and this just gives me e cubed minus 1."},{"Start":"08:09.365 ","End":"08:11.960","Text":"Sorry, yeah, e to the 0 is 1."},{"Start":"08:11.960 ","End":"08:14.045","Text":"This is the answer,"},{"Start":"08:14.045 ","End":"08:18.300","Text":"and I\u0027ll just highlight it and declare that we are done."}],"ID":8725},{"Watched":false,"Name":"Exercise 1 part c","Duration":"5m 17s","ChapterTopicVideoID":8536,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8536.jpeg","UploadDate":"2017-02-13T03:21:06.4470000","DurationForVideoObject":"PT5M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.530","Text":"Here in this exercise,"},{"Start":"00:01.530 ","End":"00:08.040","Text":"we have to compute the triple integral of this function of x, y, z."},{"Start":"00:08.040 ","End":"00:12.270","Text":"dV is the element of volume."},{"Start":"00:12.270 ","End":"00:15.195","Text":"It\u0027s either dx, dy, dz,"},{"Start":"00:15.195 ","End":"00:19.650","Text":"but in any order you choose or that\u0027s convenient."},{"Start":"00:19.650 ","End":"00:22.710","Text":"The region B is a box."},{"Start":"00:22.710 ","End":"00:25.065","Text":"That\u0027s, actually, B stands for box"},{"Start":"00:25.065 ","End":"00:30.690","Text":"because it\u0027s just an interval times an interval times an interval x,"},{"Start":"00:30.690 ","End":"00:32.535","Text":"all the x\u0027s from 0 to 1,"},{"Start":"00:32.535 ","End":"00:34.575","Text":"all the y\u0027s from minus 1 to 2,"},{"Start":"00:34.575 ","End":"00:37.275","Text":"all the z\u0027s from 0 to 3,"},{"Start":"00:37.275 ","End":"00:39.910","Text":"so we just have to decide on an order."},{"Start":"00:39.910 ","End":"00:44.270","Text":"Let\u0027s go dx, dy, dz."},{"Start":"00:44.270 ","End":"00:45.860","Text":"Change your mind, do it in reverse."},{"Start":"00:45.860 ","End":"00:48.250","Text":"I mean, we\u0027ll have the x integral here,"},{"Start":"00:48.250 ","End":"00:50.405","Text":"we\u0027ll have the y integral here,"},{"Start":"00:50.405 ","End":"00:52.370","Text":"and we\u0027ll have the z integral here."},{"Start":"00:52.370 ","End":"00:54.185","Text":"It\u0027s not dx, dy, dz,"},{"Start":"00:54.185 ","End":"01:01.155","Text":"it\u0027s actually going to be dz, dy, dx"},{"Start":"01:01.155 ","End":"01:08.160","Text":"and what we have here is the x, y, z^2 I just copy."},{"Start":"01:10.130 ","End":"01:15.470","Text":"I replace the B,"},{"Start":"01:15.470 ","End":"01:21.575","Text":"the box, by the limit z goes from 0 to 3,"},{"Start":"01:21.575 ","End":"01:28.175","Text":"y goes from minus 1 up to 2,"},{"Start":"01:28.175 ","End":"01:32.840","Text":"and x goes from 0 up to 1."},{"Start":"01:32.840 ","End":"01:35.795","Text":"We start with the innermost integral,"},{"Start":"01:35.795 ","End":"01:39.155","Text":"and that would be the dz integral,"},{"Start":"01:39.155 ","End":"01:42.005","Text":"and I\u0027d like to do that at the side."},{"Start":"01:42.005 ","End":"01:44.720","Text":"I\u0027ll do it over there somewhere."},{"Start":"01:44.720 ","End":"01:49.950","Text":"What I have is an integral with respect to z."},{"Start":"01:49.950 ","End":"01:55.025","Text":"I can already take constants in front of the integral sign x, y,"},{"Start":"01:55.025 ","End":"01:59.630","Text":"for example, a constant so I can say that this is xy times the"},{"Start":"01:59.630 ","End":"02:05.730","Text":"integral from 0 to 3 of z^2 dz,"},{"Start":"02:05.930 ","End":"02:10.670","Text":"and this gives me what xy,"},{"Start":"02:10.670 ","End":"02:15.453","Text":"the integral of z^2 is just"},{"Start":"02:15.453 ","End":"02:21.930","Text":"1/3 z^3, taken from 0 to 3."},{"Start":"02:22.190 ","End":"02:25.670","Text":"When I put in z equals 0, I get nothing."},{"Start":"02:25.670 ","End":"02:28.060","Text":"When I put in z equals 3,"},{"Start":"02:28.060 ","End":"02:31.050","Text":"I get 3 cubed over 3,"},{"Start":"02:31.050 ","End":"02:35.350","Text":"which is 9 so it comes out to be 9xy."},{"Start":"02:35.710 ","End":"02:39.970","Text":"I\u0027ll highlight this in the same color as that."},{"Start":"02:39.970 ","End":"02:43.079","Text":"Now I\u0027m going to put this back here,"},{"Start":"02:43.079 ","End":"02:47.460","Text":"and so I got 1 less layer."},{"Start":"02:47.460 ","End":"02:52.365","Text":"I got from 0 to 1, integral from minus 1 to 2."},{"Start":"02:52.365 ","End":"03:03.770","Text":"Here I copy the 9xy and dy, dx."},{"Start":"03:03.770 ","End":"03:07.860","Text":"Next layer from the inside is going to be this one,"},{"Start":"03:07.860 ","End":"03:14.015","Text":"the integral dy, and I\u0027d like to do this one at the side also over here."},{"Start":"03:14.015 ","End":"03:19.100","Text":"I can take the 9 outside and get the integral from minus 1 to 2."},{"Start":"03:19.100 ","End":"03:21.680","Text":"Now I can do better than taking the 9 out."},{"Start":"03:21.680 ","End":"03:24.270","Text":"I can take the 9x out."},{"Start":"03:24.610 ","End":"03:29.220","Text":"9x and then all I\u0027m left with is a y, dy,"},{"Start":"03:29.740 ","End":"03:41.090","Text":"and this is equal to 9x, integral of y is 1/2 y^2 from minus 1 to 2,"},{"Start":"03:41.090 ","End":"03:43.624","Text":"so that gives me 9x."},{"Start":"03:43.624 ","End":"03:46.875","Text":"Now, when y is 2,"},{"Start":"03:46.875 ","End":"03:53.090","Text":"2 squared over 2 is 2 less,"},{"Start":"03:53.090 ","End":"03:55.775","Text":"when y is minus 1,"},{"Start":"03:55.775 ","End":"04:00.215","Text":"minus 1 squared is 1 over 2 is 1/2."},{"Start":"04:00.215 ","End":"04:06.410","Text":"I\u0027ve got 9x times 1/2,"},{"Start":"04:06.410 ","End":"04:08.795","Text":"9 times 3 over 2,"},{"Start":"04:08.795 ","End":"04:15.175","Text":"27 over 2 or 13 1/2,"},{"Start":"04:15.175 ","End":"04:18.210","Text":"I\u0027ll leave it at 27 over 2 x."},{"Start":"04:19.610 ","End":"04:25.835","Text":"Put this back over here and we\u0027ve got the integral from 0 to 1."},{"Start":"04:25.835 ","End":"04:30.555","Text":"This came out to be 27 over 2 x."},{"Start":"04:30.555 ","End":"04:36.590","Text":"Well, I could put the 27 over 2 in front,"},{"Start":"04:36.590 ","End":"04:42.280","Text":"and then all I\u0027m missing is the x, dx."},{"Start":"04:42.280 ","End":"04:48.296","Text":"Now, this is equal to,"},{"Start":"04:48.296 ","End":"04:49.017","Text":"I should be writing equal signs,"},{"Start":"04:49.017 ","End":"04:59.135","Text":"27 over 2 times the integral of x is 1/2 x^2 between 0 and 1."},{"Start":"04:59.135 ","End":"05:01.680","Text":"At 0, I get nothing,"},{"Start":"05:01.680 ","End":"05:03.930","Text":"so all I have to do is plug in 1."},{"Start":"05:03.930 ","End":"05:07.345","Text":"I plug in 1, I get 1 squared over 2 is 1/2,"},{"Start":"05:07.345 ","End":"05:13.454","Text":"so 27 over 2 times 1/2 is 27 over 4,"},{"Start":"05:13.454 ","End":"05:18.000","Text":"and I\u0027ll highlight this and that\u0027s our answer."}],"ID":8726},{"Watched":false,"Name":"Exercise 1 part d","Duration":"8m 13s","ChapterTopicVideoID":8537,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8537.jpeg","UploadDate":"2017-02-13T03:23:00.1630000","DurationForVideoObject":"PT8M13S","Description":null,"VideoComments":[],"Subtitles":[],"ID":8727},{"Watched":false,"Name":"Exercise 2 part a","Duration":"12m 16s","ChapterTopicVideoID":8538,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8538.jpeg","UploadDate":"2017-02-13T03:25:57.8930000","DurationForVideoObject":"PT12M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.530","Text":"In this exercise, we need to compute the following triple integral."},{"Start":"00:05.780 ","End":"00:08.550","Text":"It\u0027s dx, dy, dz,"},{"Start":"00:08.550 ","End":"00:11.175","Text":"which means that the outermost is dz."},{"Start":"00:11.175 ","End":"00:13.410","Text":"Normally we would start, first of all,"},{"Start":"00:13.410 ","End":"00:18.075","Text":"with the dx, then do the integral dy, and lastly dz."},{"Start":"00:18.075 ","End":"00:22.560","Text":"We\u0027re given a hint that we\u0027re going to have to change the order of"},{"Start":"00:22.560 ","End":"00:26.970","Text":"integration because we\u0027re going to have a difficult time otherwise."},{"Start":"00:26.970 ","End":"00:30.195","Text":"But let\u0027s assume that I didn\u0027t notice this."},{"Start":"00:30.195 ","End":"00:32.400","Text":"Let me start naively."},{"Start":"00:32.400 ","End":"00:36.075","Text":"I copy this with a slight change."},{"Start":"00:36.075 ","End":"00:40.830","Text":"I\u0027ll say this is the integral from 0-4."},{"Start":"00:40.830 ","End":"00:43.545","Text":"Maybe I\u0027ll emphasize this is z."},{"Start":"00:43.545 ","End":"00:52.400","Text":"Since this denominator 2 square root of z doesn\u0027t depend on x and y,"},{"Start":"00:52.400 ","End":"00:57.275","Text":"I can actually take this out and put this 1 here,"},{"Start":"00:57.275 ","End":"01:00.445","Text":"1 over twice square root of z."},{"Start":"01:00.445 ","End":"01:04.150","Text":"Then I have the integral,"},{"Start":"01:04.820 ","End":"01:11.025","Text":"this 1 is y, I\u0027ll emphasize that y goes from 0-1."},{"Start":"01:11.025 ","End":"01:19.275","Text":"Then the integral, the inner most is x goes from 2y to 2."},{"Start":"01:19.275 ","End":"01:26.565","Text":"I have 4 cosine of x squared dx,"},{"Start":"01:26.565 ","End":"01:30.760","Text":"dy, and lastly, dz."},{"Start":"01:32.210 ","End":"01:35.025","Text":"By the way, deliberately,"},{"Start":"01:35.025 ","End":"01:36.560","Text":"I\u0027m keeping the 4 here."},{"Start":"01:36.560 ","End":"01:38.270","Text":"It\u0027s a constant. I could take it out,"},{"Start":"01:38.270 ","End":"01:40.745","Text":"but it suits me to leave it there."},{"Start":"01:40.745 ","End":"01:43.910","Text":"Now, normally we start with the innermost"},{"Start":"01:43.910 ","End":"01:47.900","Text":"integral and we would try to do this integral dx."},{"Start":"01:47.900 ","End":"01:52.130","Text":"The thing is that it\u0027s 1 of those impossible integrals."},{"Start":"01:52.130 ","End":"01:55.879","Text":"There is no closed formula for the cosine of x squared,"},{"Start":"01:55.879 ","End":"01:57.920","Text":"not in any of the formula sheets."},{"Start":"01:57.920 ","End":"02:02.270","Text":"Nobody knows how to do that and so we seem to be stuck."},{"Start":"02:02.270 ","End":"02:09.620","Text":"But we now notice the hint to change the order of integration and what we do is try to,"},{"Start":"02:09.620 ","End":"02:18.285","Text":"first of all, compute the whole of the double integral up to the dy."},{"Start":"02:18.285 ","End":"02:20.565","Text":"I\u0027ll do this at the side."},{"Start":"02:20.565 ","End":"02:24.320","Text":"Here we\u0027ll do a change of the order of integration because often when"},{"Start":"02:24.320 ","End":"02:28.175","Text":"we get stuck changing the order of integration will help."},{"Start":"02:28.175 ","End":"02:29.990","Text":"I can tell you that in this case,"},{"Start":"02:29.990 ","End":"02:33.840","Text":"it will help. I\u0027ll just copy it."},{"Start":"02:33.840 ","End":"02:39.660","Text":"At the side, I want to do the integral from 0-1,"},{"Start":"02:39.660 ","End":"02:44.055","Text":"the integral from 2y to 2."},{"Start":"02:44.055 ","End":"02:51.275","Text":"Better keep the variables here, it\u0027s less confusing."},{"Start":"02:51.275 ","End":"02:53.450","Text":"Now what\u0027s going from what to where,"},{"Start":"02:53.450 ","End":"03:02.545","Text":"of 4 cosine x squared dx, dy."},{"Start":"03:02.545 ","End":"03:07.110","Text":"Here\u0027s where I want to do a change of the order of integration."},{"Start":"03:07.110 ","End":"03:14.790","Text":"We want to sketch of what this region is that we\u0027re integrating over in 2-dimensions."},{"Start":"03:14.870 ","End":"03:17.495","Text":"Here are some axes."},{"Start":"03:17.495 ","End":"03:19.370","Text":"Now on the outside,"},{"Start":"03:19.370 ","End":"03:22.325","Text":"y goes from 0-1,"},{"Start":"03:22.325 ","End":"03:27.300","Text":"so I need 0 and I need 1."},{"Start":"03:27.350 ","End":"03:35.430","Text":"For any typical y between 0 and 1,"},{"Start":"03:35.430 ","End":"03:39.060","Text":"the x goes from 2y to 2."},{"Start":"03:39.060 ","End":"03:45.495","Text":"I need to sketch x equals 2y and x equals 2."},{"Start":"03:45.495 ","End":"03:49.115","Text":"X equals 2 would be"},{"Start":"03:49.115 ","End":"03:58.155","Text":"a vertical line going through 2 and x equals 2y,"},{"Start":"03:58.155 ","End":"04:00.555","Text":"the line through the origin,"},{"Start":"04:00.555 ","End":"04:04.650","Text":"when y is 0,"},{"Start":"04:04.650 ","End":"04:13.560","Text":"x is 0, and when y equals 1,"},{"Start":"04:13.560 ","End":"04:18.374","Text":"then x equals 2y is 2,"},{"Start":"04:18.374 ","End":"04:21.430","Text":"so we\u0027ll get a point here."},{"Start":"04:25.240 ","End":"04:29.970","Text":"Then a straight line through here."},{"Start":"04:31.160 ","End":"04:33.690","Text":"I\u0027ll just label them."},{"Start":"04:33.690 ","End":"04:36.630","Text":"Here\u0027s x equals 2."},{"Start":"04:36.630 ","End":"04:42.455","Text":"This line here is where x equals 2y."},{"Start":"04:42.455 ","End":"04:48.890","Text":"We see that we have this triangular region."},{"Start":"04:48.890 ","End":"04:50.540","Text":"Let me just shade it."},{"Start":"04:50.540 ","End":"04:53.600","Text":"But taking it as a type 2 region,"},{"Start":"04:53.600 ","End":"04:55.295","Text":"meaning that the outer loop,"},{"Start":"04:55.295 ","End":"04:59.570","Text":"as we said, is y goes from 0-1 and for each y,"},{"Start":"04:59.570 ","End":"05:08.460","Text":"we enter the region here and leave the region here,"},{"Start":"05:08.460 ","End":"05:13.925","Text":"x goes from 2y up to 2."},{"Start":"05:13.925 ","End":"05:17.270","Text":"To reverse the order of integration,"},{"Start":"05:17.270 ","End":"05:19.610","Text":"instead of horizontal slices,"},{"Start":"05:19.610 ","End":"05:22.620","Text":"I want to do vertical slices."},{"Start":"05:22.810 ","End":"05:27.000","Text":"I need the inverse function of this."},{"Start":"05:27.270 ","End":"05:29.920","Text":"To put y in terms of x,"},{"Start":"05:29.920 ","End":"05:35.300","Text":"this would be y equals x over 2."},{"Start":"05:35.720 ","End":"05:46.030","Text":"I need the equation of this line also because if I take a typical x here,"},{"Start":"05:46.030 ","End":"05:48.129","Text":"I want to know where it cuts,"},{"Start":"05:48.129 ","End":"05:51.220","Text":"like what is this and what is this?"},{"Start":"05:51.220 ","End":"05:56.360","Text":"This is obviously the line y equals 0."},{"Start":"05:56.360 ","End":"06:02.205","Text":"Here as we said, y is x over 2 and we see that x goes from 0-2."},{"Start":"06:02.205 ","End":"06:05.935","Text":"After changing the order of integration,"},{"Start":"06:05.935 ","End":"06:08.110","Text":"and I\u0027ll continue over here,"},{"Start":"06:08.110 ","End":"06:10.150","Text":"we\u0027ll get the integral,"},{"Start":"06:10.150 ","End":"06:15.740","Text":"the outer integral is x runs from 0-2,"},{"Start":"06:16.010 ","End":"06:19.995","Text":"and that would be dx,"},{"Start":"06:19.995 ","End":"06:27.810","Text":"then inside that, y runs from 0 to x over 2."},{"Start":"06:32.210 ","End":"06:35.410","Text":"That\u0027s y."},{"Start":"06:35.630 ","End":"06:41.110","Text":"We still have this 4 cosine of x."},{"Start":"06:45.790 ","End":"06:49.070","Text":"Now that we\u0027ve changed the order of integration,"},{"Start":"06:49.070 ","End":"06:50.510","Text":"we\u0027re not stuck anymore."},{"Start":"06:50.510 ","End":"06:52.880","Text":"If the first integral is dy,"},{"Start":"06:52.880 ","End":"06:57.110","Text":"I can actually take this in front of the integral and say"},{"Start":"06:57.110 ","End":"07:04.630","Text":"that I have the integral from 0-2."},{"Start":"07:04.630 ","End":"07:08.850","Text":"Let me just pull the cosine x in front."},{"Start":"07:11.080 ","End":"07:20.115","Text":"Then I have the integral y goes from 0 to x over"},{"Start":"07:20.115 ","End":"07:28.950","Text":"2 of just 4 dy, dx."},{"Start":"07:28.950 ","End":"07:32.865","Text":"Now this is my inner integral."},{"Start":"07:32.865 ","End":"07:35.525","Text":"That\u0027s pretty straightforward."},{"Start":"07:35.525 ","End":"07:38.905","Text":"In fact, we can do this mentally,"},{"Start":"07:38.905 ","End":"07:43.185","Text":"well, or we could do it at the side."},{"Start":"07:43.185 ","End":"07:46.240","Text":"I\u0027ll just do this over here."},{"Start":"07:48.950 ","End":"07:53.820","Text":"The integral of 4 is just 4y."},{"Start":"07:53.820 ","End":"08:00.870","Text":"This I have to take from 0 to x over 2."},{"Start":"08:01.280 ","End":"08:03.890","Text":"When y is x over 2,"},{"Start":"08:03.890 ","End":"08:07.055","Text":"I get 4 times x over 2 is 2x."},{"Start":"08:07.055 ","End":"08:10.295","Text":"When y is 0, it gives me 0."},{"Start":"08:10.295 ","End":"08:15.180","Text":"This whole thing just comes out to be 2x."},{"Start":"08:15.670 ","End":"08:20.750","Text":"What we have is the integral,"},{"Start":"08:20.750 ","End":"08:23.860","Text":"I think I should have been writing equals signs here,"},{"Start":"08:24.800 ","End":"08:31.540","Text":"from 0-2 of 2x,"},{"Start":"08:31.540 ","End":"08:33.905","Text":"I\u0027ll put the x in front of the cosine,"},{"Start":"08:33.905 ","End":"08:41.005","Text":"2x cosine of x dx."},{"Start":"08:41.005 ","End":"08:47.560","Text":"Sorry, I think I lost an x squared somewhere"},{"Start":"08:47.560 ","End":"08:59.280","Text":"here, here, here, here."},{"Start":"08:59.280 ","End":"09:02.090","Text":"This time, we can do this integral."},{"Start":"09:02.090 ","End":"09:04.340","Text":"If it was just cosine x squared,"},{"Start":"09:04.340 ","End":"09:05.370","Text":"we couldn\u0027t do it,"},{"Start":"09:05.370 ","End":"09:09.515","Text":"but the fact that there\u0027s an x here makes all the difference."},{"Start":"09:09.515 ","End":"09:12.445","Text":"We could do it with a substitution,"},{"Start":"09:12.445 ","End":"09:15.010","Text":"but that\u0027s a bit of an overkill."},{"Start":"09:15.010 ","End":"09:19.470","Text":"What I\u0027d like to just point out is that this is of the form,"},{"Start":"09:19.570 ","End":"09:25.580","Text":"the integral, let\u0027s just make it an indefinite integral for the moment."},{"Start":"09:25.580 ","End":"09:29.930","Text":"If I have cosine of something,"},{"Start":"09:29.930 ","End":"09:32.515","Text":"some expression involving x,"},{"Start":"09:32.515 ","End":"09:38.360","Text":"and then I have the derivative of that something dx,"},{"Start":"09:38.360 ","End":"09:43.865","Text":"then the integral of this is just sine"},{"Start":"09:43.865 ","End":"09:50.090","Text":"of whatever this was plus the constant if it\u0027s an indefinite integral."},{"Start":"09:50.090 ","End":"09:54.200","Text":"Easiest way to see this is just to differentiate this."},{"Start":"09:54.200 ","End":"09:56.735","Text":"If I differentiate sine of something,"},{"Start":"09:56.735 ","End":"09:58.360","Text":"I get cosine of something,"},{"Start":"09:58.360 ","End":"10:01.045","Text":"and by the chain rule, the anti-derivative."},{"Start":"10:01.045 ","End":"10:04.535","Text":"I\u0027m claiming this is exactly what we have here,"},{"Start":"10:04.535 ","End":"10:09.559","Text":"except that the box prime could be before."},{"Start":"10:09.559 ","End":"10:14.430","Text":"This is the box prime."},{"Start":"10:14.430 ","End":"10:18.455","Text":"This is cosine of box,"},{"Start":"10:18.455 ","End":"10:21.485","Text":"where box is x squared."},{"Start":"10:21.485 ","End":"10:27.350","Text":"This integral becomes sine of box,"},{"Start":"10:27.350 ","End":"10:33.510","Text":"which is x squared between 0 and 2."},{"Start":"10:33.940 ","End":"10:36.680","Text":"When I plug in 2,"},{"Start":"10:36.680 ","End":"10:40.875","Text":"I get sine of 4,"},{"Start":"10:40.875 ","End":"10:46.740","Text":"then minus sine of 0 squared, sine of 0."},{"Start":"10:46.740 ","End":"10:50.010","Text":"But sine of 0 is just 0."},{"Start":"10:50.010 ","End":"10:52.685","Text":"So this is just sine 4."},{"Start":"10:52.685 ","End":"11:01.680","Text":"Now we\u0027ve figured out this whole bit is sine of 4."},{"Start":"11:02.480 ","End":"11:08.820","Text":"Continuing, I can take the sine 4 in front of the integral sign."},{"Start":"11:08.820 ","End":"11:15.425","Text":"I have sine of 4 times the integral from"},{"Start":"11:15.425 ","End":"11:19.195","Text":"0-4 of 1 over"},{"Start":"11:19.195 ","End":"11:28.140","Text":"2 root z dz."},{"Start":"11:28.140 ","End":"11:30.425","Text":"Now this is 1 of those immediate integrals."},{"Start":"11:30.425 ","End":"11:38.060","Text":"The sine of 4 stays and the integral of 1 over 2 root z is just root z."},{"Start":"11:38.060 ","End":"11:41.990","Text":"That\u0027s 1 of the basic ones."},{"Start":"11:41.990 ","End":"11:46.790","Text":"I have to evaluate this between 0 and 4."},{"Start":"11:46.790 ","End":"11:53.340","Text":"What we get is sine 4."},{"Start":"11:53.340 ","End":"11:56.270","Text":"Then if I plug in 4,"},{"Start":"11:56.270 ","End":"12:01.355","Text":"I\u0027ve got root 4 minus root 0."},{"Start":"12:01.355 ","End":"12:11.420","Text":"Altogether, this is just 2 minus 0 is 2 and so our final answer is 2 sine of 4."},{"Start":"12:11.420 ","End":"12:15.900","Text":"I\u0027ll just highlight it and we\u0027re done."}],"ID":8728},{"Watched":false,"Name":"Exercise 2 part b","Duration":"11m 9s","ChapterTopicVideoID":8539,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8539.jpeg","UploadDate":"2017-02-13T03:28:33.4170000","DurationForVideoObject":"PT11M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this exercise, we have to compute a triple integral."},{"Start":"00:04.335 ","End":"00:07.680","Text":"Let me first copy it over here."},{"Start":"00:07.680 ","End":"00:16.710","Text":"I emphasized that this is an integral with respect to y. Y goes from this to this."},{"Start":"00:16.710 ","End":"00:22.620","Text":"Then next we have the dx and the limits for x and then dz with the limits for z."},{"Start":"00:22.620 ","End":"00:25.140","Text":"We work from the inside out."},{"Start":"00:25.140 ","End":"00:30.630","Text":"Now, let\u0027s pretend you didn\u0027t see this hint here and go about it naively."},{"Start":"00:30.630 ","End":"00:34.410","Text":"What I would normally do is do the inner integral first."},{"Start":"00:34.410 ","End":"00:43.549","Text":"I would normally start to do this and then I would come across a snag."},{"Start":"00:43.549 ","End":"00:48.140","Text":"Essentially this integral the main part of it is this part here,"},{"Start":"00:48.140 ","End":"00:51.710","Text":"because the 12 xz are just constants as far as y goes."},{"Start":"00:51.710 ","End":"00:55.100","Text":"This is a variation on e to the y squared,"},{"Start":"00:55.100 ","End":"00:56.630","Text":"and this is an unknown integral."},{"Start":"00:56.630 ","End":"01:01.225","Text":"There\u0027s no formula for the integral of e to the y squared."},{"Start":"01:01.225 ","End":"01:03.405","Text":"We appear to be stuck."},{"Start":"01:03.405 ","End":"01:07.150","Text":"Now we notice the hint change the order of integration."},{"Start":"01:07.150 ","End":"01:12.110","Text":"What we do is we extend the first bit we do to"},{"Start":"01:12.110 ","End":"01:19.400","Text":"the double integral dy/dx."},{"Start":"01:19.400 ","End":"01:23.840","Text":"Now let me take this double integral and put it over side here."},{"Start":"01:23.840 ","End":"01:30.165","Text":"What I have is the integral from 0 to 1,"},{"Start":"01:30.165 ","End":"01:34.180","Text":"the integral from x squared to 1."},{"Start":"01:34.180 ","End":"01:38.600","Text":"I\u0027ll keep the letters here so we\u0027ll know what\u0027s going from what to"},{"Start":"01:38.600 ","End":"01:43.790","Text":"wear of 12 xz, e"},{"Start":"01:43.790 ","End":"01:49.040","Text":"to the z y squared dy/dx."},{"Start":"01:49.040 ","End":"01:51.560","Text":"Now, this as a double integral,"},{"Start":"01:51.560 ","End":"01:54.380","Text":"we still have the same problem as we had here."},{"Start":"01:54.380 ","End":"01:57.365","Text":"We can\u0027t do this integral dy and that\u0027s why"},{"Start":"01:57.365 ","End":"02:01.280","Text":"we look at the hint and we want to change the order of integration."},{"Start":"02:01.280 ","End":"02:05.150","Text":"What we do first is to sketch the region that is being"},{"Start":"02:05.150 ","End":"02:09.725","Text":"integrated here using these limits y goes from x squared to 1,"},{"Start":"02:09.725 ","End":"02:12.485","Text":"and x goes from 0 to 1."},{"Start":"02:12.485 ","End":"02:14.330","Text":"I make a little sketch."},{"Start":"02:14.330 ","End":"02:21.095","Text":"I see that the outer loop is x goes from 0 to 1."},{"Start":"02:21.095 ","End":"02:25.575","Text":"Then for each particular x say this is my typical"},{"Start":"02:25.575 ","End":"02:30.165","Text":"x. I have that y goes from x squared to 1."},{"Start":"02:30.165 ","End":"02:31.930","Text":"I need to draw 2 functions,"},{"Start":"02:31.930 ","End":"02:35.030","Text":"y equals x squared and y equals 1."},{"Start":"02:35.030 ","End":"02:43.510","Text":"Now, y equals 1 is just a horizontal line and y equals x squared is a parabola."},{"Start":"02:43.510 ","End":"02:47.630","Text":"It\u0027s easy to check that they meet at the point 1,"},{"Start":"02:47.630 ","End":"02:54.710","Text":"1 because here y equals 1 and here y equals x squared also."},{"Start":"02:54.710 ","End":"02:57.649","Text":"This is the region."},{"Start":"02:57.649 ","End":"03:00.020","Text":"I\u0027ve shaded the region,"},{"Start":"03:00.020 ","End":"03:01.160","Text":"and let\u0027s label them."},{"Start":"03:01.160 ","End":"03:04.219","Text":"This is y equals x squared."},{"Start":"03:04.219 ","End":"03:08.000","Text":"This line is y equals 1."},{"Start":"03:08.000 ","End":"03:16.580","Text":"We see that for each x if I just take a vertical slice,"},{"Start":"03:16.580 ","End":"03:21.215","Text":"we go from y equals x squared to y equals 1."},{"Start":"03:21.215 ","End":"03:24.800","Text":"It\u0027s a type 1 region vertical slices."},{"Start":"03:24.800 ","End":"03:30.330","Text":"Now we want to convert it into a type 2 region and take horizontal slices."},{"Start":"03:30.350 ","End":"03:40.430","Text":"For a typical y we see that x goes from this point to this point."},{"Start":"03:40.430 ","End":"03:47.480","Text":"What we need to know the equation of this line and this curve."},{"Start":"03:47.480 ","End":"03:52.460","Text":"But as x as a function of y,"},{"Start":"03:52.460 ","End":"03:58.530","Text":"this line here would be the y-axis."},{"Start":"03:58.570 ","End":"04:04.400","Text":"I could label the y-axis as x equals 0."},{"Start":"04:04.400 ","End":"04:09.545","Text":"The parabola here, I just have to reverse y and x."},{"Start":"04:09.545 ","End":"04:11.405","Text":"If y is x squared,"},{"Start":"04:11.405 ","End":"04:16.639","Text":"then x equals the square root of y."},{"Start":"04:16.639 ","End":"04:21.260","Text":"Took the positive square root because we\u0027re in the first quadrant,"},{"Start":"04:21.260 ","End":"04:24.175","Text":"the minus square root of y would be on the other side."},{"Start":"04:24.175 ","End":"04:29.835","Text":"Now I know that y goes from 0 to 1 on the output loop."},{"Start":"04:29.835 ","End":"04:39.165","Text":"I can start rewriting this as outwardly y from 0 to 1, and that\u0027s dy."},{"Start":"04:39.165 ","End":"04:46.180","Text":"Then inward loop x goes from 0 to this here;"},{"Start":"04:46.180 ","End":"04:50.750","Text":"x equals 0 to x equals square root of y. I go"},{"Start":"04:50.750 ","End":"04:57.680","Text":"from 0 to square root of y, and that\u0027s dx."},{"Start":"04:57.680 ","End":"05:07.320","Text":"Here as before, 12 x ze to the zy squared."},{"Start":"05:07.450 ","End":"05:12.350","Text":"This time we\u0027re not going to get stuck because it\u0027s dx."},{"Start":"05:12.350 ","End":"05:13.850","Text":"This is not a difficult integral."},{"Start":"05:13.850 ","End":"05:15.170","Text":"Let me just highlight it."},{"Start":"05:15.170 ","End":"05:19.440","Text":"The inner integral this time is dx."},{"Start":"05:19.640 ","End":"05:23.030","Text":"I\u0027m going to do this 1 also as a side integral,"},{"Start":"05:23.030 ","End":"05:25.280","Text":"let\u0027s say over here."},{"Start":"05:25.280 ","End":"05:31.610","Text":"What I get is the integral from 0 to root y of"},{"Start":"05:31.610 ","End":"05:37.995","Text":"12 xz e to the zy squared dx."},{"Start":"05:37.995 ","End":"05:44.525","Text":"Since it\u0027s dx, I can take some constants in front of the integration sign."},{"Start":"05:44.525 ","End":"05:50.370","Text":"For example, the 12 and all this ze to the zy squared."},{"Start":"05:50.370 ","End":"06:00.065","Text":"I\u0027ve got 12 ze to the zy squared integral from 0 to root y."},{"Start":"06:00.065 ","End":"06:09.110","Text":"All that\u0027s left here is the dx12 ze to the zy squared."},{"Start":"06:09.110 ","End":"06:18.330","Text":"The integral of this is 1/2 x squared taken from 0 to square root of y."},{"Start":"06:18.330 ","End":"06:23.060","Text":"Zero gives me nothing but when I plug in the square root of y,"},{"Start":"06:23.060 ","End":"06:24.890","Text":"then x squared is just y,"},{"Start":"06:24.890 ","End":"06:27.095","Text":"so I get 1/2 y."},{"Start":"06:27.095 ","End":"06:35.600","Text":"It becomes 12 ze to the zy squared times"},{"Start":"06:35.600 ","End":"06:46.545","Text":"1/2 y. I can just write this more simply the 1/2 with the 12 gives me 6."},{"Start":"06:46.545 ","End":"06:52.590","Text":"I\u0027ve got 6y from here, z from there,"},{"Start":"06:52.590 ","End":"07:02.110","Text":"e to the zy squared should have used another color here."},{"Start":"07:02.110 ","End":"07:07.490","Text":"Then this answer here corresponds to the answer here."},{"Start":"07:07.490 ","End":"07:09.665","Text":"Now I can plug it back in."},{"Start":"07:09.665 ","End":"07:12.260","Text":"Putting this instead of what was shaded here,"},{"Start":"07:12.260 ","End":"07:17.165","Text":"I\u0027ve got now the integral from 0-1,"},{"Start":"07:17.165 ","End":"07:23.565","Text":"copying this 6y ze"},{"Start":"07:23.565 ","End":"07:30.180","Text":"to the zy squared dy."},{"Start":"07:30.180 ","End":"07:33.155","Text":"The way I\u0027m going to tackle this is as follows."},{"Start":"07:33.155 ","End":"07:35.780","Text":"I have e to the power of something."},{"Start":"07:35.780 ","End":"07:38.120","Text":"In general, if I have the integral,"},{"Start":"07:38.120 ","End":"07:41.555","Text":"There\u0027s 1 of these templates formulas."},{"Start":"07:41.555 ","End":"07:47.810","Text":"If I have e to the power of something times that something derivative,"},{"Start":"07:47.810 ","End":"07:52.790","Text":"let\u0027s say these are functions of an x or in this case y,"},{"Start":"07:52.790 ","End":"07:54.545","Text":"if these are functions of y."},{"Start":"07:54.545 ","End":"08:02.420","Text":"This integral is just going to be e to the power of box plus C if you\u0027re being precise,"},{"Start":"08:02.420 ","End":"08:04.040","Text":"if it\u0027s an indefinite integral."},{"Start":"08:04.040 ","End":"08:06.170","Text":"You can see this by differentiating this,"},{"Start":"08:06.170 ","End":"08:08.060","Text":"if I differentiate e to the something,"},{"Start":"08:08.060 ","End":"08:11.060","Text":"is e to that something times the derivative."},{"Start":"08:11.060 ","End":"08:16.085","Text":"Now in this case I want my box to be this."},{"Start":"08:16.085 ","End":"08:23.100","Text":"Now box prime would be 2zy,"},{"Start":"08:23.100 ","End":"08:24.450","Text":"I don\u0027t have to 2yz."},{"Start":"08:24.450 ","End":"08:25.680","Text":"I have 6yz."},{"Start":"08:25.680 ","End":"08:35.390","Text":"The trick is to rewrite this by taking the 3 outside the integral."},{"Start":"08:35.390 ","End":"08:45.820","Text":"Then I have the integral from 0 to 1 of 2 yz e to the zy squared dy."},{"Start":"08:46.370 ","End":"08:51.425","Text":"Now this bit here is like this is the box prime,"},{"Start":"08:51.425 ","End":"08:55.210","Text":"and this is the box from this formula here."},{"Start":"08:55.210 ","End":"09:00.320","Text":"I get 3 times and the integral I don\u0027t need the constant is just e"},{"Start":"09:00.320 ","End":"09:07.620","Text":"to the z y squared evaluated between 0 and 1,"},{"Start":"09:07.620 ","End":"09:10.980","Text":"that\u0027s y equals 0 to y equals 1."},{"Start":"09:10.980 ","End":"09:15.590","Text":"Just to get some space."},{"Start":"09:15.590 ","End":"09:18.575","Text":"This would be equal to 3 times."},{"Start":"09:18.575 ","End":"09:26.510","Text":"If I put in 1, I\u0027ve got z1 squared is just e to the z."},{"Start":"09:26.510 ","End":"09:28.564","Text":"If I put in y equals 0,"},{"Start":"09:28.564 ","End":"09:32.275","Text":"this is 0, so e to the 0 is 1."},{"Start":"09:32.275 ","End":"09:38.605","Text":"This is now the answer to this part that I\u0027ve shaded here."},{"Start":"09:38.605 ","End":"09:40.955","Text":"I\u0027ll shade it in the same color."},{"Start":"09:40.955 ","End":"09:42.890","Text":"So I\u0027m going to know this is that."},{"Start":"09:42.890 ","End":"09:50.560","Text":"Now we go back here and we get the integral from 0 to 1."},{"Start":"09:50.560 ","End":"09:53.445","Text":"This bit I just copy from here."},{"Start":"09:53.445 ","End":"09:56.645","Text":"Suppose I could put the 3 in front."},{"Start":"09:56.645 ","End":"09:58.765","Text":"Say I\u0027ll put the 3 here,"},{"Start":"09:58.765 ","End":"10:05.850","Text":"and then I\u0027ve got e to the z minus 1 dz."},{"Start":"10:05.850 ","End":"10:09.145","Text":"I don\u0027t need this anymore."},{"Start":"10:09.145 ","End":"10:17.915","Text":"What I get is 3 times the integral of e to the z is e to the z."},{"Start":"10:17.915 ","End":"10:22.530","Text":"The integral of 1 minus 1 is just minus z,"},{"Start":"10:22.530 ","End":"10:26.910","Text":"and all this has to be taken from 0 to 1."},{"Start":"10:27.800 ","End":"10:31.260","Text":"If I plug in 1,"},{"Start":"10:31.260 ","End":"10:34.110","Text":"that you put a 3 with a square bracket if I plug in 1,"},{"Start":"10:34.110 ","End":"10:36.240","Text":"I\u0027ve got e to the 1 minus 1,"},{"Start":"10:36.240 ","End":"10:38.580","Text":"which is e minus 1."},{"Start":"10:38.580 ","End":"10:46.315","Text":"If I plug in 0. I get here e to the 0 is 1 minus 0."},{"Start":"10:46.315 ","End":"10:50.030","Text":"This equals, if I evaluate this bit,"},{"Start":"10:50.030 ","End":"10:51.995","Text":"it\u0027s just e minus 2."},{"Start":"10:51.995 ","End":"10:56.500","Text":"I have 3 times e minus 2."},{"Start":"10:56.500 ","End":"11:04.500","Text":"You can leave the answer is that or we could multiply out and say 3e minus 6."},{"Start":"11:04.500 ","End":"11:06.570","Text":"I\u0027ll highlight that."},{"Start":"11:06.570 ","End":"11:09.520","Text":"Finally, that\u0027s our answer."}],"ID":8729},{"Watched":false,"Name":"Exercise 2 part c","Duration":"13m 54s","ChapterTopicVideoID":8522,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8522.jpeg","UploadDate":"2017-02-13T02:51:34.9170000","DurationForVideoObject":"PT13M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"In this exercise, we have to compute the following triple integral."},{"Start":"00:04.065 ","End":"00:09.150","Text":"It\u0027s dxdydz means first with respect to x and these are the limits for x,"},{"Start":"00:09.150 ","End":"00:10.365","Text":"for y and for z."},{"Start":"00:10.365 ","End":"00:12.880","Text":"In fact, I\u0027m going to rewrite it."},{"Start":"00:13.130 ","End":"00:17.070","Text":"Here I\u0027ve emphasized that this is x that goes from"},{"Start":"00:17.070 ","End":"00:23.625","Text":"0 to this and then here it\u0027s y and here it\u0027s z and there\u0027s less confusion."},{"Start":"00:23.625 ","End":"00:27.240","Text":"Ignore for a moment this hint, if we get stuck,"},{"Start":"00:27.240 ","End":"00:30.270","Text":"then we maybe will use this idea."},{"Start":"00:30.270 ","End":"00:34.695","Text":"Meanwhile, let\u0027s just start naively and see if we can do the inner integral."},{"Start":"00:34.695 ","End":"00:39.135","Text":"I mean this one with respect to x, sorry,"},{"Start":"00:39.135 ","End":"00:44.675","Text":"there and there and I\u0027ll do that at the side."},{"Start":"00:44.675 ","End":"00:48.400","Text":"I\u0027ll just put an arrow to say I\u0027m going to continue over here."},{"Start":"00:48.400 ","End":"00:54.889","Text":"We just get the integral from 0 to natural log of 3,"},{"Start":"00:54.889 ","End":"00:56.990","Text":"and we\u0027re going to do it dx."},{"Start":"00:56.990 ","End":"00:59.585","Text":"Now since this thing is dx,"},{"Start":"00:59.585 ","End":"01:02.890","Text":"the stuff that doesn\u0027t belong with x,"},{"Start":"01:02.890 ","End":"01:04.370","Text":"the others are constant."},{"Start":"01:04.370 ","End":"01:06.920","Text":"I can actually take them outside."},{"Start":"01:06.920 ","End":"01:08.820","Text":"But you know what? I\u0027ll copy it first."},{"Start":"01:08.820 ","End":"01:18.585","Text":"Pi e^2x sine of Pi y squared over y squared dx."},{"Start":"01:18.585 ","End":"01:25.640","Text":"Now I\u0027ll take outside the brackets the stuff that\u0027s constant with respect to x,"},{"Start":"01:25.640 ","End":"01:27.845","Text":"which is everything except to e^2x."},{"Start":"01:27.845 ","End":"01:36.195","Text":"I have Pi sine of Pi y squared."},{"Start":"01:36.195 ","End":"01:38.670","Text":"Maybe I better put brackets here,"},{"Start":"01:38.670 ","End":"01:43.860","Text":"over y squared integral from"},{"Start":"01:43.860 ","End":"01:50.875","Text":"0 to natural log of 3 of e^2x dx."},{"Start":"01:50.875 ","End":"01:53.465","Text":"Now this is a straightforward integral."},{"Start":"01:53.465 ","End":"01:59.185","Text":"The integral of this is just 1.5 e^2x."},{"Start":"01:59.185 ","End":"02:02.900","Text":"In fact I also do this as a side exercise."},{"Start":"02:02.900 ","End":"02:06.950","Text":"In fact I\u0027ll do just this bit here at the side over here."},{"Start":"02:06.950 ","End":"02:11.960","Text":"What I would get would be the integral of e^2x is"},{"Start":"02:11.960 ","End":"02:21.220","Text":"1/2e^2x and I would take this from 0 to natural log of 3."},{"Start":"02:21.220 ","End":"02:22.970","Text":"This would come out to,"},{"Start":"02:22.970 ","End":"02:29.370","Text":"if I substitute natural log of 3, I get 1.5."},{"Start":"02:29.890 ","End":"02:37.000","Text":"I can write this as e^,"},{"Start":"02:37.000 ","End":"02:41.590","Text":"well, I\u0027ll just write it as twice natural log of 3 and then we\u0027ll simplify it."},{"Start":"02:41.590 ","End":"02:45.100","Text":"Minus, and when I put in x equals 0,"},{"Start":"02:45.100 ","End":"02:48.415","Text":"it\u0027s just one that\u0027s minus 1/2."},{"Start":"02:48.415 ","End":"02:53.420","Text":"But e^2 log 3 is e to the log 3,"},{"Start":"02:53.420 ","End":"02:56.575","Text":"natural log that is to the power of 2."},{"Start":"02:56.575 ","End":"02:59.630","Text":"It comes out to be 3 squared."},{"Start":"03:00.270 ","End":"03:04.100","Text":"1/2 3 squared minus 1/2."},{"Start":"03:04.100 ","End":"03:06.025","Text":"If you\u0027re not sure about that,"},{"Start":"03:06.025 ","End":"03:07.870","Text":"I\u0027ll go even more slowly."},{"Start":"03:07.870 ","End":"03:10.570","Text":"This is, I can change the order of integration."},{"Start":"03:10.570 ","End":"03:15.730","Text":"It\u0027s e to the power of natural log 3 times 2 and this"},{"Start":"03:15.730 ","End":"03:21.950","Text":"is e^natural log 3,"},{"Start":"03:21.950 ","End":"03:24.590","Text":"all this to the power of 2."},{"Start":"03:24.590 ","End":"03:28.370","Text":"This is e^log of something,"},{"Start":"03:28.370 ","End":"03:29.750","Text":"is that something itself."},{"Start":"03:29.750 ","End":"03:31.025","Text":"To the power of 2."},{"Start":"03:31.025 ","End":"03:33.420","Text":"That brings me here."},{"Start":"03:33.710 ","End":"03:38.595","Text":"This is 9, so it\u0027s 9 over 2 is 4.5,"},{"Start":"03:38.595 ","End":"03:42.360","Text":"4.5 minus 1/2 is 4."},{"Start":"03:42.360 ","End":"03:48.825","Text":"All this bit is 4 and now if we just put the 4 up here,"},{"Start":"03:48.825 ","End":"03:54.870","Text":"I have 4Pi sine"},{"Start":"03:54.870 ","End":"04:00.990","Text":"of Pi y squared over y squared."},{"Start":"04:00.990 ","End":"04:04.715","Text":"All this was the inner integral and there was no problem at all,"},{"Start":"04:04.715 ","End":"04:08.465","Text":"as in the previous exercises when we had to do a change of order."},{"Start":"04:08.465 ","End":"04:10.280","Text":"But wait and see."},{"Start":"04:10.280 ","End":"04:18.575","Text":"Now I put this back up here and so I get a double integral,"},{"Start":"04:18.575 ","End":"04:22.945","Text":"integral z from 0-1,"},{"Start":"04:22.945 ","End":"04:29.690","Text":"y from cube root of z-1."},{"Start":"04:29.690 ","End":"04:34.730","Text":"This whole thing is 4Pi sine of"},{"Start":"04:34.730 ","End":"04:41.640","Text":"Pi y squared over"},{"Start":"04:41.640 ","End":"04:46.965","Text":"y squared and this is dy, dz."},{"Start":"04:46.965 ","End":"04:50.795","Text":"This is the point at which we get stuck."},{"Start":"04:50.795 ","End":"04:58.094","Text":"In general, even without the Pi sine of y squared over y squared,"},{"Start":"04:58.094 ","End":"05:00.455","Text":"it\u0027s one of these impossible integrals."},{"Start":"05:00.455 ","End":"05:04.145","Text":"You can look in the integral tables, you can try doing it."},{"Start":"05:04.145 ","End":"05:05.720","Text":"You\u0027ll get stuck here."},{"Start":"05:05.720 ","End":"05:07.865","Text":"This is the point at which we\u0027re stuck,"},{"Start":"05:07.865 ","End":"05:09.860","Text":"and this is the point at which we\u0027re going to use"},{"Start":"05:09.860 ","End":"05:12.635","Text":"the hint to change the order of integrating."},{"Start":"05:12.635 ","End":"05:14.570","Text":"Unlike in previous exercises,"},{"Start":"05:14.570 ","End":"05:16.115","Text":"we change these two."},{"Start":"05:16.115 ","End":"05:18.020","Text":"This time the inner one was okay,"},{"Start":"05:18.020 ","End":"05:20.495","Text":"and that\u0027s the outer two that we\u0027re going to change."},{"Start":"05:20.495 ","End":"05:25.615","Text":"Let\u0027s see what is the region that is described by these limits."},{"Start":"05:25.615 ","End":"05:29.100","Text":"I had some room over here to put a couple of axes,"},{"Start":"05:29.100 ","End":"05:31.985","Text":"but notices the y and the z-axis."},{"Start":"05:31.985 ","End":"05:35.915","Text":"The outer one is z, goes from 0-1."},{"Start":"05:35.915 ","End":"05:39.925","Text":"Let\u0027s say this is 0 and let\u0027s say this is 1."},{"Start":"05:39.925 ","End":"05:43.820","Text":"Then y for any particular z,"},{"Start":"05:43.820 ","End":"05:46.500","Text":"if this is a typical z,"},{"Start":"05:47.470 ","End":"05:50.455","Text":"this is cube root,"},{"Start":"05:50.455 ","End":"05:55.460","Text":"y will go from the cube root of z to 1."},{"Start":"05:55.460 ","End":"05:57.590","Text":"One is easier to sketch,"},{"Start":"05:57.590 ","End":"06:01.620","Text":"if this is 1, that\u0027s just a horizontal line."},{"Start":"06:01.620 ","End":"06:04.605","Text":"The cube root of z,"},{"Start":"06:04.605 ","End":"06:09.005","Text":"the general shape is something like this."},{"Start":"06:09.005 ","End":"06:14.370","Text":"But you can see that when z is equal to 1,"},{"Start":"06:14.370 ","End":"06:19.590","Text":"the cube root of z is also 1 so we"},{"Start":"06:19.590 ","End":"06:27.420","Text":"actually go through the point 1,1 and it looks roughly like this."},{"Start":"06:27.420 ","End":"06:29.265","Text":"It doesn\u0027t have to be exact."},{"Start":"06:29.265 ","End":"06:32.045","Text":"This would be our region and I\u0027ll shade it."},{"Start":"06:32.045 ","End":"06:34.580","Text":"I\u0027ll just go over quickly again."},{"Start":"06:34.580 ","End":"06:39.940","Text":"For each z, we take the limit,"},{"Start":"06:39.940 ","End":"06:42.785","Text":"so we take the slice,"},{"Start":"06:42.785 ","End":"06:47.780","Text":"it cuts into the region here where y equals"},{"Start":"06:47.780 ","End":"06:53.910","Text":"cube root of z and here where y equals 1."},{"Start":"06:53.910 ","End":"06:58.220","Text":"This is like a type one region because we\u0027re slicing it vertically."},{"Start":"06:58.220 ","End":"07:01.940","Text":"Since we\u0027re stuck, we want to try reversing the order."},{"Start":"07:01.940 ","End":"07:03.920","Text":"Let\u0027s horizontal slices."},{"Start":"07:03.920 ","End":"07:08.620","Text":"We see that y also goes from 0-1."},{"Start":"07:08.620 ","End":"07:14.255","Text":"So I can already start by saying y goes from 0-1."},{"Start":"07:14.255 ","End":"07:17.270","Text":"Here I\u0027ll put the dy."},{"Start":"07:17.270 ","End":"07:23.775","Text":"Then I need to see where z goes from and to, and then put a dz."},{"Start":"07:23.775 ","End":"07:26.280","Text":"Let\u0027s see a horizontal slice,"},{"Start":"07:26.280 ","End":"07:28.405","Text":"if this is a typical y,"},{"Start":"07:28.405 ","End":"07:32.570","Text":"this horizontal slice will cut into the region here,"},{"Start":"07:32.570 ","End":"07:34.400","Text":"out to the region here."},{"Start":"07:34.400 ","End":"07:40.350","Text":"This vertical line is where z equals 0."},{"Start":"07:40.350 ","End":"07:47.135","Text":"It\u0027s the y-axis and where it exits is the inverse of y equals cube root of z."},{"Start":"07:47.135 ","End":"07:48.860","Text":"Well, if you cube both sides,"},{"Start":"07:48.860 ","End":"07:53.330","Text":"you see that the inverse of this is just z equals y cubed."},{"Start":"07:53.330 ","End":"07:57.285","Text":"X goes from along here from,"},{"Start":"07:57.285 ","End":"07:59.340","Text":"not x, sorry, z,"},{"Start":"07:59.340 ","End":"08:07.920","Text":"z goes from 0-y cubed and the rest of it\u0027s the same."},{"Start":"08:07.920 ","End":"08:18.310","Text":"4Pi sine of Pi y squared over y squared."},{"Start":"08:19.700 ","End":"08:22.230","Text":"Now this is the integral,"},{"Start":"08:22.230 ","End":"08:26.200","Text":"the 1dz. I went earlier."},{"Start":"08:26.200 ","End":"08:31.315","Text":"I should have shaded this just to show that this is what correlates with this."},{"Start":"08:31.315 ","End":"08:38.525","Text":"Now we\u0027re going to compute this integral and we\u0027ll do that at the side also."},{"Start":"08:38.525 ","End":"08:40.750","Text":"I\u0027ll just make a bit of room."},{"Start":"08:40.750 ","End":"08:48.100","Text":"It\u0027s okay, so I\u0027ll take this color and do it over here and what we have now is the"},{"Start":"08:48.100 ","End":"08:55.360","Text":"integral from 0-y cubed of"},{"Start":"08:55.360 ","End":"09:05.615","Text":"4Pi sine of Pi y squared over y squared dz."},{"Start":"09:05.615 ","End":"09:10.355","Text":"But look, all of this doesn\u0027t contain z,"},{"Start":"09:10.355 ","End":"09:13.140","Text":"this is just a constant."},{"Start":"09:14.270 ","End":"09:19.025","Text":"Well, I could just take the constant in front of the integral sign."},{"Start":"09:19.025 ","End":"09:22.115","Text":"I just copy pasted the constant,"},{"Start":"09:22.115 ","End":"09:25.880","Text":"the integral from 0-y cubed."},{"Start":"09:25.880 ","End":"09:27.590","Text":"All we\u0027re left with is dz,"},{"Start":"09:27.590 ","End":"09:29.180","Text":"or if you like 1dz."},{"Start":"09:29.180 ","End":"09:33.370","Text":"The integral of 1,"},{"Start":"09:33.370 ","End":"09:37.145","Text":"we said is always the upper limit minus the lower limit."},{"Start":"09:37.145 ","End":"09:47.130","Text":"What we get is this constant just copied it times y cubed minus 0,"},{"Start":"09:47.130 ","End":"09:49.530","Text":"which is just y cubed."},{"Start":"09:49.530 ","End":"09:53.975","Text":"If we just simplify this y squared into y cubed goes y times,"},{"Start":"09:53.975 ","End":"10:00.290","Text":"I\u0027ll just write it over here as 4Pi and then I\u0027ll put"},{"Start":"10:00.290 ","End":"10:10.145","Text":"a y and then the sine of Pi y squared."},{"Start":"10:10.145 ","End":"10:17.630","Text":"That\u0027s this bit, which I\u0027ll highlight in this color,"},{"Start":"10:17.630 ","End":"10:19.580","Text":"because that\u0027s the answer to this."},{"Start":"10:19.580 ","End":"10:22.540","Text":"Now I\u0027m going back here,"},{"Start":"10:22.540 ","End":"10:30.410","Text":"over here and then we just have the integral from 0-1 of this,"},{"Start":"10:30.410 ","End":"10:39.480","Text":"4Pi y sine of pi y squared dy."},{"Start":"10:39.550 ","End":"10:42.575","Text":"Now how do we do this integral?"},{"Start":"10:42.575 ","End":"10:47.705","Text":"Now, we could do this as integration by parts,"},{"Start":"10:47.705 ","End":"10:49.430","Text":"but that might be an overkill."},{"Start":"10:49.430 ","End":"10:54.600","Text":"What I\u0027d like you to notice is that the derivative of Pi y"},{"Start":"10:54.600 ","End":"11:01.265","Text":"squared is 2Pi y and we almost have that."},{"Start":"11:01.265 ","End":"11:03.740","Text":"If we had 2Pi y here,"},{"Start":"11:03.740 ","End":"11:08.180","Text":"then I could look at Pi y squared like some box,"},{"Start":"11:08.180 ","End":"11:12.629","Text":"it would look like the template box,"},{"Start":"11:12.629 ","End":"11:20.675","Text":"prime sine of box where box is some function in this case of y."},{"Start":"11:20.675 ","End":"11:23.510","Text":"If you take the integral of this,"},{"Start":"11:23.510 ","End":"11:25.550","Text":"I\u0027m claiming that this is equal to,"},{"Start":"11:25.550 ","End":"11:27.905","Text":"what it it says, the function of y dy,"},{"Start":"11:27.905 ","End":"11:32.390","Text":"then this is just equal to the anti-derivative of sine,"},{"Start":"11:32.390 ","End":"11:39.865","Text":"which is minus cosine of that same box plus a constant theoretically."},{"Start":"11:39.865 ","End":"11:43.020","Text":"You can see this by differentiating this."},{"Start":"11:43.020 ","End":"11:47.030","Text":"The derivative of minus cosine is sine because it\u0027s not y,"},{"Start":"11:47.030 ","End":"11:48.230","Text":"it\u0027s a function of y."},{"Start":"11:48.230 ","End":"11:51.800","Text":"We take also the inner derivative which you could put in front."},{"Start":"11:51.800 ","End":"11:56.825","Text":"If I use this and I take this as my box,"},{"Start":"11:56.825 ","End":"12:05.640","Text":"then box prime is just 2Pi y. I rewrite this as 2 and I can put the 2 in front and then"},{"Start":"12:05.640 ","End":"12:15.970","Text":"I have the integral of 2Pi y sine of Pi y squared dy."},{"Start":"12:16.150 ","End":"12:19.415","Text":"This bit here is the box,"},{"Start":"12:19.415 ","End":"12:22.835","Text":"and this bit here is the box prime."},{"Start":"12:22.835 ","End":"12:29.540","Text":"I can use this template and the answer to this just comes out to be like this,"},{"Start":"12:29.540 ","End":"12:30.860","Text":"but the 2 sticks,"},{"Start":"12:30.860 ","End":"12:33.635","Text":"so it\u0027s minus and there\u0027s a 2 here,"},{"Start":"12:33.635 ","End":"12:37.070","Text":"there is cosine of whatever the box was,"},{"Start":"12:37.070 ","End":"12:40.960","Text":"which in this case is Pi y squared."},{"Start":"12:40.960 ","End":"12:44.795","Text":"I don\u0027t need the plus C because I\u0027m going to take this,"},{"Start":"12:44.795 ","End":"12:48.140","Text":"it\u0027s an indefinite integral from 0-1."},{"Start":"12:48.140 ","End":"12:51.485","Text":"I\u0027ll put here from 0-1."},{"Start":"12:51.485 ","End":"12:54.665","Text":"Let\u0027s see what we get."},{"Start":"12:54.665 ","End":"12:58.445","Text":"If we plug in 1."},{"Start":"12:58.445 ","End":"13:02.075","Text":"Let me put the minus 2 separately."},{"Start":"13:02.075 ","End":"13:08.720","Text":"Now, I need cosine sine of this thing with y equals 1."},{"Start":"13:08.720 ","End":"13:17.855","Text":"It\u0027s cosine of Pi 1 squared is Pi minus cosine of what I get if I put y equals 0,"},{"Start":"13:17.855 ","End":"13:22.000","Text":"which is cosign of 0."},{"Start":"13:22.000 ","End":"13:23.930","Text":"Let\u0027s see what this equals."},{"Start":"13:23.930 ","End":"13:30.785","Text":"Now, cosine pi is cosine of 180 degrees is minus 1."},{"Start":"13:30.785 ","End":"13:33.385","Text":"Cosine of 0 is 1."},{"Start":"13:33.385 ","End":"13:43.560","Text":"Minus 1, minus 1 so we get minus 2 times minus 2 from minus 1,"},{"Start":"13:43.560 ","End":"13:50.055","Text":"minus 1 so altogether we get plus 4 and that would be the answer."},{"Start":"13:50.055 ","End":"13:54.360","Text":"I\u0027ll just highlight this and then declare that we\u0027re done."}],"ID":8730},{"Watched":false,"Name":"Exercise 2 part d","Duration":"10m 6s","ChapterTopicVideoID":8523,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8523.jpeg","UploadDate":"2017-02-13T02:54:22.9970000","DurationForVideoObject":"PT10M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this exercise, we have to compute the following triple integral."},{"Start":"00:04.770 ","End":"00:07.965","Text":"It\u0027s first dy in the innermost,"},{"Start":"00:07.965 ","End":"00:10.545","Text":"then dz, then dx."},{"Start":"00:10.545 ","End":"00:13.545","Text":"I\u0027m going to copy it here."},{"Start":"00:13.545 ","End":"00:20.580","Text":"I copied it so I could emphasize which variable goes from where to where,"},{"Start":"00:20.580 ","End":"00:24.700","Text":"the innermost is y from 0-x and so on."},{"Start":"00:25.460 ","End":"00:30.510","Text":"This Hinton brackets just basically tells us if we get stuck,"},{"Start":"00:30.510 ","End":"00:35.040","Text":"we could always try changing the order of integration and maybe get unstuck."},{"Start":"00:35.040 ","End":"00:39.660","Text":"Let\u0027s just start simply at the middle."},{"Start":"00:39.660 ","End":"00:41.935","Text":"As usual, we take the innermost,"},{"Start":"00:41.935 ","End":"00:46.870","Text":"and this is the dy integral, this one."},{"Start":"00:47.450 ","End":"00:52.270","Text":"If I do this inner one at the side over here,"},{"Start":"00:52.270 ","End":"00:57.710","Text":"what I see is that there is no y in here at all,"},{"Start":"00:57.710 ","End":"00:59.015","Text":"there\u0027s no x either,"},{"Start":"00:59.015 ","End":"01:01.490","Text":"so I could take it in front of the integral,"},{"Start":"01:01.490 ","End":"01:08.280","Text":"and get sine 2z over 4 minus z."},{"Start":"01:08.280 ","End":"01:13.710","Text":"Then just the integral from 0-x of dy,"},{"Start":"01:13.710 ","End":"01:16.500","Text":"which I prefer to write it as just 1dy."},{"Start":"01:16.500 ","End":"01:19.005","Text":"We know that an integral like this,"},{"Start":"01:19.005 ","End":"01:22.640","Text":"you just take the top minus the bottom,"},{"Start":"01:22.640 ","End":"01:24.200","Text":"it\u0027s x minus 0,"},{"Start":"01:24.200 ","End":"01:27.160","Text":"so this thing is just x."},{"Start":"01:27.160 ","End":"01:33.290","Text":"Basically this whole thing comes out to be x times"},{"Start":"01:33.290 ","End":"01:39.775","Text":"sine 2z over 4 minus z,"},{"Start":"01:39.775 ","End":"01:41.360","Text":"and I\u0027ll highlight this,"},{"Start":"01:41.360 ","End":"01:45.485","Text":"and then we\u0027ll substitute back in here where I took it from."},{"Start":"01:45.485 ","End":"01:51.710","Text":"Continuing, we get the integral x goes from 0-2,"},{"Start":"01:51.710 ","End":"01:58.710","Text":"and z goes from 0-4 minus x"},{"Start":"01:58.710 ","End":"02:06.820","Text":"squared of x sine z, no it\u0027s 2z."},{"Start":"02:06.890 ","End":"02:10.485","Text":"Yeah, I corrected that in here also,"},{"Start":"02:10.485 ","End":"02:18.700","Text":"over 4 minus z and this time dz dx."},{"Start":"02:18.830 ","End":"02:24.260","Text":"Normally I would start with the inner one, I mean."},{"Start":"02:24.260 ","End":"02:28.495","Text":"We would try to do this integral dz,"},{"Start":"02:28.495 ","End":"02:31.769","Text":"and here we would get stuck."},{"Start":"02:31.769 ","End":"02:35.300","Text":"Because basically ignoring a constant like x,"},{"Start":"02:35.300 ","End":"02:38.910","Text":"it\u0027s roughly which of"},{"Start":"02:38.910 ","End":"02:45.550","Text":"the same type as sine z over z with a variation substitutions, constants."},{"Start":"02:45.550 ","End":"02:48.855","Text":"This integral that\u0027s based on this,"},{"Start":"02:48.855 ","End":"02:52.530","Text":"is not known, we don\u0027t know how to do this integral."},{"Start":"02:52.530 ","End":"02:54.930","Text":"This is the point where we\u0027re stuck,"},{"Start":"02:54.930 ","End":"02:59.675","Text":"this is the point where we\u0027re going to use the hint to change the order of integration."},{"Start":"02:59.675 ","End":"03:02.955","Text":"I won\u0027t be working on this inner integral,"},{"Start":"03:02.955 ","End":"03:04.970","Text":"and we\u0027re going to do the change of order,"},{"Start":"03:04.970 ","End":"03:07.045","Text":"so we need a diagram."},{"Start":"03:07.045 ","End":"03:13.430","Text":"I see that the outer integral is x from 0-2."},{"Start":"03:13.430 ","End":"03:17.820","Text":"Let\u0027s say this is 0 and this is 2,"},{"Start":"03:17.820 ","End":"03:23.209","Text":"and the inner integral for each x in this range,"},{"Start":"03:23.209 ","End":"03:25.910","Text":"when x is between 0 and 2,"},{"Start":"03:25.910 ","End":"03:33.030","Text":"z is going to go from 0 to 4 minus x squared."},{"Start":"03:33.030 ","End":"03:37.760","Text":"Well, z equals 0 is the x-axis."},{"Start":"03:37.760 ","End":"03:40.140","Text":"I could color it,"},{"Start":"03:40.250 ","End":"03:42.510","Text":"just to emphasize it,"},{"Start":"03:42.510 ","End":"03:46.590","Text":"x from 0-2, that\u0027s the 0,"},{"Start":"03:46.590 ","End":"03:49.100","Text":"and 4 minus x squared is the parabola."},{"Start":"03:49.100 ","End":"03:52.890","Text":"When x is 0, y is 4,"},{"Start":"03:52.890 ","End":"03:54.780","Text":"so this might be 4,"},{"Start":"03:54.780 ","End":"03:58.530","Text":"and as x goes from 0-2, when we get to 2,"},{"Start":"03:58.530 ","End":"04:00.450","Text":"it\u0027s 4 minus 2 squared is 0,"},{"Start":"04:00.450 ","End":"04:02.760","Text":"we go down to here,"},{"Start":"04:02.760 ","End":"04:05.115","Text":"with something like this."},{"Start":"04:05.115 ","End":"04:07.205","Text":"Now we have these 2 graphs."},{"Start":"04:07.205 ","End":"04:11.900","Text":"This x-axis is the place where z equals 0,"},{"Start":"04:11.900 ","End":"04:17.535","Text":"and this is where z equals 4 minus x squared."},{"Start":"04:17.535 ","End":"04:22.100","Text":"We take vertical slices through each x,"},{"Start":"04:22.100 ","End":"04:28.485","Text":"that z goes from here to here."},{"Start":"04:28.485 ","End":"04:31.260","Text":"We have the formula for these two,"},{"Start":"04:31.660 ","End":"04:38.680","Text":"and now we want to change the direction of integration or the order,"},{"Start":"04:38.680 ","End":"04:41.330","Text":"instead of a type 1 with vertical slices,"},{"Start":"04:41.330 ","End":"04:44.090","Text":"you want a type 2 with horizontal slices."},{"Start":"04:44.090 ","End":"04:49.025","Text":"Well, we can see already that z will go from 0-4 on the outer loop."},{"Start":"04:49.025 ","End":"04:53.550","Text":"I import from 0-4 and that\u0027s z."},{"Start":"04:53.550 ","End":"04:58.460","Text":"For each z, this is a typical z from 0-4,"},{"Start":"04:58.460 ","End":"05:05.155","Text":"the horizontal slice will cut at 2 places here and here,"},{"Start":"05:05.155 ","End":"05:08.685","Text":"and this one is just the line,"},{"Start":"05:08.685 ","End":"05:10.220","Text":"it\u0027s the z axis,"},{"Start":"05:10.220 ","End":"05:15.300","Text":"which means that x equals 0, up to here."},{"Start":"05:16.040 ","End":"05:21.900","Text":"Well, we have to just reverse the function here."},{"Start":"05:21.900 ","End":"05:24.220","Text":"Let me just do some computation at the side,"},{"Start":"05:24.220 ","End":"05:26.210","Text":"z is 4 minus x squared."},{"Start":"05:26.210 ","End":"05:28.080","Text":"I\u0027ll write that here,"},{"Start":"05:28.080 ","End":"05:30.225","Text":"z is 4 minus x squared,"},{"Start":"05:30.225 ","End":"05:34.415","Text":"that means that x squared is 4 minus z."},{"Start":"05:34.415 ","End":"05:37.010","Text":"Since we want the positive x,"},{"Start":"05:37.010 ","End":"05:38.765","Text":"we\u0027re going to the right,"},{"Start":"05:38.765 ","End":"05:43.540","Text":"then x is plus the square root of 4 minus z."},{"Start":"05:43.540 ","End":"05:47.075","Text":"That\u0027s this function, square root of 4 minus z."},{"Start":"05:47.075 ","End":"05:49.535","Text":"That\u0027s what also I can write here,"},{"Start":"05:49.535 ","End":"05:53.175","Text":"that given a z,"},{"Start":"05:53.175 ","End":"06:00.315","Text":"the x goes from 0 to root of 4 minus z,"},{"Start":"06:00.315 ","End":"06:02.250","Text":"and the rest of it,"},{"Start":"06:02.250 ","End":"06:05.905","Text":"will be dx dz."},{"Start":"06:05.905 ","End":"06:10.205","Text":"Because this inner loop is for x and the outer loop was dz,"},{"Start":"06:10.205 ","End":"06:12.995","Text":"from 0-4, and this is the same,"},{"Start":"06:12.995 ","End":"06:20.290","Text":"x sine 2z over 4 minus z."},{"Start":"06:20.290 ","End":"06:23.420","Text":"Now we can work on this in an integral,"},{"Start":"06:23.420 ","End":"06:26.490","Text":"which would be the dx integral."},{"Start":"06:26.500 ","End":"06:30.740","Text":"I\u0027ll do this one at the side here."},{"Start":"06:30.740 ","End":"06:33.300","Text":"But I\u0027ll rewrite it a little bit,"},{"Start":"06:33.300 ","End":"06:35.940","Text":"because look, x only appears here."},{"Start":"06:35.940 ","End":"06:45.084","Text":"The whole of the sine 2z over 4 minus z can come out in front of the integral sign,"},{"Start":"06:45.084 ","End":"06:53.150","Text":"so I\u0027ve gone from 0 to square root of 4 minus z of just the x dx."},{"Start":"06:54.590 ","End":"07:00.430","Text":"This equals, the integral of x is a 1/2 x squared."},{"Start":"07:01.550 ","End":"07:10.825","Text":"This goes from 0 to root 4 minus z and I have just have to just copy this piece,"},{"Start":"07:10.825 ","End":"07:18.950","Text":"multiply each sine of 2z over 4 minus z."},{"Start":"07:18.980 ","End":"07:21.330","Text":"Now, let\u0027s see what happens."},{"Start":"07:21.330 ","End":"07:23.550","Text":"When we put in 0, we get 0,"},{"Start":"07:23.550 ","End":"07:26.245","Text":"so we just have to substitute the upper limit."},{"Start":"07:26.245 ","End":"07:29.440","Text":"I put x equals 4 minus z,"},{"Start":"07:29.440 ","End":"07:34.620","Text":"x square root, x squared is just 4 minus z."},{"Start":"07:34.620 ","End":"07:37.485","Text":"What we end up with is,"},{"Start":"07:37.485 ","End":"07:45.285","Text":"sine 2z over 4 minus z."},{"Start":"07:45.285 ","End":"07:48.165","Text":"The half I\u0027ll put in front,"},{"Start":"07:48.165 ","End":"07:54.189","Text":"and the x squared we said was 4 minus z."},{"Start":"07:54.290 ","End":"07:58.785","Text":"We have luck, this cancels with this,"},{"Start":"07:58.785 ","End":"08:02.765","Text":"and so the answer to this integral is just this bit,"},{"Start":"08:02.765 ","End":"08:05.315","Text":"just the 1/2 sine 2z,"},{"Start":"08:05.315 ","End":"08:08.490","Text":"and I\u0027ll put that back here,"},{"Start":"08:08.530 ","End":"08:14.840","Text":"and we\u0027ll get the integral from 0-4,"},{"Start":"08:14.840 ","End":"08:17.750","Text":"of 1/2 sine 2z,"},{"Start":"08:17.750 ","End":"08:21.100","Text":"I put the half in front."},{"Start":"08:21.100 ","End":"08:23.160","Text":"Actually it doesn\u0027t matter,"},{"Start":"08:23.160 ","End":"08:29.530","Text":"but okay. Sine 2z, dz."},{"Start":"08:32.500 ","End":"08:35.495","Text":"If it was just sine,"},{"Start":"08:35.495 ","End":"08:41.135","Text":"then I would say that this would be minus cosine 2z."},{"Start":"08:41.135 ","End":"08:43.660","Text":"But it\u0027s not sine,"},{"Start":"08:43.660 ","End":"08:50.180","Text":"it\u0027s sine 2z, so we have to divide by the inner derivative."},{"Start":"08:50.180 ","End":"08:52.100","Text":"It\u0027s a linear function of z, so we can do that."},{"Start":"08:52.100 ","End":"08:54.610","Text":"We have to divide by 2,"},{"Start":"08:54.610 ","End":"08:56.445","Text":"like multiplying by 1/2,"},{"Start":"08:56.445 ","End":"08:59.350","Text":"and we also have the 1/2 from here."},{"Start":"08:59.510 ","End":"09:06.320","Text":"All this has to go from 0-4,"},{"Start":"09:06.320 ","End":"09:07.760","Text":"let\u0027s see what we get."},{"Start":"09:07.760 ","End":"09:15.070","Text":"The 1/2 with the 1/2 with the minus, is minus 1/4."},{"Start":"09:15.140 ","End":"09:18.645","Text":"Now we just have to plug in,"},{"Start":"09:18.645 ","End":"09:26.760","Text":"we get cosine of twice z is 8,"},{"Start":"09:26.760 ","End":"09:31.785","Text":"minus is here, so it\u0027s cosine 2z,"},{"Start":"09:31.785 ","End":"09:37.630","Text":"plug in 4, plug-in 0, get cosine 0."},{"Start":"09:37.820 ","End":"09:41.385","Text":"Let\u0027s see what this equals."},{"Start":"09:41.385 ","End":"09:44.040","Text":"I\u0027m not going to compute cosine 8,"},{"Start":"09:44.040 ","End":"09:45.350","Text":"it\u0027s not some simple number."},{"Start":"09:45.350 ","End":"09:47.465","Text":"Cosine 0 is 1."},{"Start":"09:47.465 ","End":"09:49.790","Text":"What I can do is get rid of this minus,"},{"Start":"09:49.790 ","End":"09:52.010","Text":"and reverse the order of the subtraction."},{"Start":"09:52.010 ","End":"09:53.525","Text":"Maybe it\u0027s a bit simpler,"},{"Start":"09:53.525 ","End":"09:56.990","Text":"variety does 1/4 this in front,"},{"Start":"09:56.990 ","End":"10:00.920","Text":"minus this, cosine 8."},{"Start":"10:00.920 ","End":"10:06.630","Text":"I\u0027ll highlight this, and declare that this is the answer and we\u0027re done."}],"ID":8731},{"Watched":false,"Name":"Exercise 3 part a","Duration":"10m 35s","ChapterTopicVideoID":8524,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8524.jpeg","UploadDate":"2017-02-13T02:57:02.8530000","DurationForVideoObject":"PT10M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.480","Text":"In this exercise, you have to compute the volume of the solid bounded by these surfaces."},{"Start":"00:06.480 ","End":"00:08.955","Text":"There\u0027s 5 of them."},{"Start":"00:08.955 ","End":"00:13.830","Text":"Now, the easy part is to quote the theorem"},{"Start":"00:13.830 ","End":"00:19.770","Text":"that the volume is given by the triple integral over the body."},{"Start":"00:19.770 ","End":"00:21.930","Text":"Solid is also called the body,"},{"Start":"00:21.930 ","End":"00:30.690","Text":"and so I\u0027ll call it B for the body of 1 dv."},{"Start":"00:30.690 ","End":"00:32.325","Text":"Perhaps I\u0027ll spell it out."},{"Start":"00:32.325 ","End":"00:41.300","Text":"The hard part is to try and describe this as an iterated integral to set limits for x,"},{"Start":"00:41.300 ","End":"00:45.130","Text":"y, and z even to decide which order."},{"Start":"00:45.130 ","End":"00:49.800","Text":"In this case, z is the odd one out."},{"Start":"00:49.800 ","End":"00:54.650","Text":"Here we have some equations that define a domain in the xy plane,"},{"Start":"00:54.650 ","End":"01:00.120","Text":"which will be the projection of our body onto the xy plane."},{"Start":"01:00.120 ","End":"01:04.875","Text":"Here we have 2 functions of z in terms of x and y,"},{"Start":"01:04.875 ","End":"01:07.400","Text":"and this would be the upper surface and the lowest surface."},{"Start":"01:07.400 ","End":"01:10.650","Text":"I know the inner integral"},{"Start":"01:14.620 ","End":"01:19.580","Text":"will be z from some lower surface to an upper surface"},{"Start":"01:19.580 ","End":"01:23.930","Text":"and the outer integrals will be x and y doesn\u0027t really matter,"},{"Start":"01:23.930 ","End":"01:30.215","Text":"but we\u0027ll say estimate in order like outermost x, then y."},{"Start":"01:30.215 ","End":"01:32.780","Text":"Here\u0027s a picture I found on the Internet."},{"Start":"01:32.780 ","End":"01:40.450","Text":"I guess here I\u0027m calling it b not v and its projection onto the xy plane."},{"Start":"01:40.450 ","End":"01:43.340","Text":"One of these surfaces will be the upper,"},{"Start":"01:43.340 ","End":"01:44.360","Text":"one will be the lower,"},{"Start":"01:44.360 ","End":"01:45.680","Text":"and that\u0027s these 2."},{"Start":"01:45.680 ","End":"01:48.515","Text":"Z equals something, z equals something."},{"Start":"01:48.515 ","End":"01:52.595","Text":"The other thing I\u0027ll need to do is find out what this d is,"},{"Start":"01:52.595 ","End":"01:58.595","Text":"and I\u0027ll be using these 3 equations to see what is the projection."},{"Start":"01:58.595 ","End":"02:02.005","Text":"In our case, it\u0027s going to turn out to be a triangle."},{"Start":"02:02.005 ","End":"02:08.780","Text":"Here we have a pair of axes for the xy plane and I want these 3 surfaces,"},{"Start":"02:08.780 ","End":"02:11.540","Text":"which are actually lines."},{"Start":"02:11.540 ","End":"02:14.485","Text":"If you project them down to the plane."},{"Start":"02:14.485 ","End":"02:25.110","Text":"Y equals 0 is just the x-axis and x equals 0 is just the y-axis,"},{"Start":"02:25.110 ","End":"02:28.055","Text":"and x plus y equals 1."},{"Start":"02:28.055 ","End":"02:31.610","Text":"We can mentally figure out what this is."},{"Start":"02:31.610 ","End":"02:33.650","Text":"If x is 0, y is 1."},{"Start":"02:33.650 ","End":"02:35.780","Text":"If y is 0, x is 1."},{"Start":"02:35.780 ","End":"02:37.850","Text":"If this is where y is 1,"},{"Start":"02:37.850 ","End":"02:39.530","Text":"and this is where x is 1,"},{"Start":"02:39.530 ","End":"02:43.880","Text":"then we have this line here or at least part of it."},{"Start":"02:43.880 ","End":"02:49.000","Text":"This will be our region D,"},{"Start":"02:49.000 ","End":"02:51.000","Text":"and I shaded it."},{"Start":"02:51.000 ","End":"02:54.140","Text":"I\u0027ll just write the equation of this one because these 2 are obvious."},{"Start":"02:54.140 ","End":"03:00.820","Text":"This one is the x plus y equals 1 equation."},{"Start":"03:00.820 ","End":"03:05.690","Text":"That\u0027s this D. Now I need to identify which of"},{"Start":"03:05.690 ","End":"03:11.105","Text":"these 2 is the upper and which is the lower."},{"Start":"03:11.105 ","End":"03:14.915","Text":"Which one is the higher up?"},{"Start":"03:14.915 ","End":"03:20.030","Text":"Z equals 0 or z equals 1 plus x plus y?"},{"Start":"03:20.030 ","End":"03:22.840","Text":"Now in principle, there could be a situation."},{"Start":"03:22.840 ","End":"03:24.440","Text":"Sometimes one was higher,"},{"Start":"03:24.440 ","End":"03:26.345","Text":"sometimes the other is higher,"},{"Start":"03:26.345 ","End":"03:28.235","Text":"this is not the case here."},{"Start":"03:28.235 ","End":"03:29.780","Text":"The thing to do is to check,"},{"Start":"03:29.780 ","End":"03:31.655","Text":"can they both be equal?"},{"Start":"03:31.655 ","End":"03:32.960","Text":"If they were both equal,"},{"Start":"03:32.960 ","End":"03:37.315","Text":"I would get 1 plus x plus y equals 0."},{"Start":"03:37.315 ","End":"03:38.960","Text":"If you solve that,"},{"Start":"03:38.960 ","End":"03:42.530","Text":"you get that x plus y is minus 1,"},{"Start":"03:42.530 ","End":"03:44.120","Text":"and if you sketch this,"},{"Start":"03:44.120 ","End":"03:48.679","Text":"you get some kind of a line here which is completely outside the region"},{"Start":"03:48.679 ","End":"03:54.050","Text":"D. If it\u0027s never 0 in the region because of continuity,"},{"Start":"03:54.050 ","End":"03:56.855","Text":"It\u0027s always one above the other."},{"Start":"03:56.855 ","End":"03:58.840","Text":"Just the question is which way round?"},{"Start":"03:58.840 ","End":"04:01.070","Text":"We just take a sample point. I don\u0027t know."},{"Start":"04:01.070 ","End":"04:04.980","Text":"Let\u0027s say I\u0027ll take this point, x is 0,"},{"Start":"04:04.980 ","End":"04:11.375","Text":"y is 1, and then I substitute in the right-hand side of each of these. Here I get 0."},{"Start":"04:11.375 ","End":"04:16.905","Text":"Here I get 1 plus x plus y is 2,"},{"Start":"04:16.905 ","End":"04:20.710","Text":"and so this one is the bigger one."},{"Start":"04:20.720 ","End":"04:25.980","Text":"This is the upper,"},{"Start":"04:25.980 ","End":"04:28.185","Text":"and the upper we place here,"},{"Start":"04:28.185 ","End":"04:30.900","Text":"1 plus x plus y,"},{"Start":"04:30.900 ","End":"04:34.295","Text":"and the lower, the 0 is here."},{"Start":"04:34.295 ","End":"04:40.250","Text":"That\u0027s the z upper surface, lower surface."},{"Start":"04:40.250 ","End":"04:45.080","Text":"The rest of it, we just have to slice up the region."},{"Start":"04:45.080 ","End":"04:47.570","Text":"Well, when I wrote y here and x here,"},{"Start":"04:47.570 ","End":"04:52.190","Text":"really what I\u0027m saying is that x is the outer loop and y is the inner loop."},{"Start":"04:52.190 ","End":"04:53.660","Text":"It\u0027s a type one."},{"Start":"04:53.660 ","End":"04:56.120","Text":"If I take a typical x,"},{"Start":"04:56.120 ","End":"05:05.275","Text":"somewhere between 0 and 1 and I take a vertical slice,"},{"Start":"05:05.275 ","End":"05:11.625","Text":"then y goes from here up to here,"},{"Start":"05:11.625 ","End":"05:17.720","Text":"and on this equation of this is where y equals 0,"},{"Start":"05:17.720 ","End":"05:19.655","Text":"that\u0027s the x axis."},{"Start":"05:19.655 ","End":"05:23.860","Text":"This one, I can easily write as y equals something,"},{"Start":"05:23.860 ","End":"05:27.690","Text":"y equals 1 minus x."},{"Start":"05:27.690 ","End":"05:30.975","Text":"As x goes from 0 to 1,"},{"Start":"05:30.975 ","End":"05:35.220","Text":"y goes from 0-1 minus x,"},{"Start":"05:35.220 ","End":"05:37.440","Text":"and I can write that here."},{"Start":"05:37.440 ","End":"05:42.485","Text":"X goes from 0 to 1 and for each such x,"},{"Start":"05:42.485 ","End":"05:51.020","Text":"y goes from 0-1 minus x. I can fill in the rest."},{"Start":"05:51.020 ","End":"05:52.835","Text":"1 is just the 1,"},{"Start":"05:52.835 ","End":"05:56.450","Text":"but for the dv, I have to put them in the right order."},{"Start":"05:56.450 ","End":"05:58.520","Text":"Now the inner one is dz,"},{"Start":"05:58.520 ","End":"06:04.485","Text":"next one dy and the outer one dx."},{"Start":"06:04.485 ","End":"06:06.545","Text":"Now I have the expression,"},{"Start":"06:06.545 ","End":"06:09.650","Text":"and that\u0027s all computational from now on,"},{"Start":"06:09.650 ","End":"06:11.960","Text":"no need anymore for the pictures."},{"Start":"06:11.960 ","End":"06:15.975","Text":"We do the integral first."},{"Start":"06:15.975 ","End":"06:20.585","Text":"The integral of 1 is always just the upper minus the lower,"},{"Start":"06:20.585 ","End":"06:23.165","Text":"and that\u0027s an easy one."},{"Start":"06:23.165 ","End":"06:25.085","Text":"You just copy this bit."},{"Start":"06:25.085 ","End":"06:30.425","Text":"X goes from 0-1 and y goes from 0-1 minus x."},{"Start":"06:30.425 ","End":"06:32.330","Text":"Here it\u0027s just this minus this,"},{"Start":"06:32.330 ","End":"06:35.734","Text":"which is 1 plus x plus y,"},{"Start":"06:35.734 ","End":"06:39.800","Text":"and then dy, dx."},{"Start":"06:39.800 ","End":"06:47.070","Text":"The next inner one is this, the dy integral."},{"Start":"06:47.290 ","End":"06:51.200","Text":"I find it convenient to compute this at the side."},{"Start":"06:51.200 ","End":"06:55.370","Text":"Let me just put it an asterisk and I may compute the asterisk,"},{"Start":"06:55.370 ","End":"06:57.485","Text":"which if I look at it,"},{"Start":"06:57.485 ","End":"06:59.605","Text":"the integral of this dy,"},{"Start":"06:59.605 ","End":"07:01.680","Text":"remember x is a constant,"},{"Start":"07:01.680 ","End":"07:08.370","Text":"so we get y for this plus xy,"},{"Start":"07:08.370 ","End":"07:14.280","Text":"and the y gives me 1.5y squared and all"},{"Start":"07:14.280 ","End":"07:21.015","Text":"this between 0 and 1 minus x."},{"Start":"07:21.015 ","End":"07:25.430","Text":"This of course, is the limits for y."},{"Start":"07:25.430 ","End":"07:28.760","Text":"Now, when y is 0, I\u0027ll get nothing."},{"Start":"07:28.760 ","End":"07:38.220","Text":"All I have to do is substitute the 1 minus x. I\u0027ve got 1 minus x plus x,"},{"Start":"07:38.220 ","End":"07:46.000","Text":"1 minus x plus 1.5 of 1 minus x squared."},{"Start":"07:47.000 ","End":"07:51.215","Text":"This is equal to, let\u0027s see,"},{"Start":"07:51.215 ","End":"07:53.290","Text":"I guess write it all out,"},{"Start":"07:53.290 ","End":"07:58.610","Text":"one minus x plus x minus x squared."},{"Start":"07:58.680 ","End":"08:03.310","Text":"Using the formula for one minus x squared,"},{"Start":"08:03.310 ","End":"08:06.030","Text":"that would be, I just write it at the side."},{"Start":"08:06.030 ","End":"08:12.300","Text":"It\u0027s 1 minus 2x plus x squared is this bit here."},{"Start":"08:12.300 ","End":"08:14.440","Text":"Half of that would be,"},{"Start":"08:14.440 ","End":"08:20.275","Text":"so plus 1.5 minus x"},{"Start":"08:20.275 ","End":"08:26.270","Text":"plus 1.5x squared, and simplify."},{"Start":"08:26.270 ","End":"08:29.985","Text":"Well, this cancels with this."},{"Start":"08:29.985 ","End":"08:34.050","Text":"Other than that it\u0027s collecting like terms."},{"Start":"08:34.050 ","End":"08:37.805","Text":"If I do them in order, let\u0027s see,"},{"Start":"08:37.805 ","End":"08:41.600","Text":"minus x squared plus a 1/2 x squared,"},{"Start":"08:41.600 ","End":"08:46.405","Text":"that\u0027s minus a 1/2x squared."},{"Start":"08:46.405 ","End":"08:52.955","Text":"Now for x\u0027s, all I have left is the minus x here,"},{"Start":"08:52.955 ","End":"08:57.920","Text":"constants plus 1 plus 1/2."},{"Start":"08:57.920 ","End":"09:02.395","Text":"That\u0027s plus 3 over 2."},{"Start":"09:02.395 ","End":"09:05.370","Text":"That concludes the asterisk parts."},{"Start":"09:05.370 ","End":"09:14.150","Text":"I can go back here and just substitute this bit and get the integral from 0 to"},{"Start":"09:14.150 ","End":"09:23.685","Text":"1 of minus 1/2x squared minus x plus 3 over 2 dx."},{"Start":"09:23.685 ","End":"09:26.510","Text":"This is equal to, let\u0027s see."},{"Start":"09:26.510 ","End":"09:30.200","Text":"The integral of this would be x cubed and I have to divide by 3,"},{"Start":"09:30.200 ","End":"09:34.595","Text":"so minus 1/6th x cubed here,"},{"Start":"09:34.595 ","End":"09:40.565","Text":"minus 1/2x squared, and here plus 3 over 2x."},{"Start":"09:40.565 ","End":"09:44.345","Text":"All this between 0 and 1."},{"Start":"09:44.345 ","End":"09:47.030","Text":"Once again, I\u0027m going to put in 0,"},{"Start":"09:47.030 ","End":"09:51.295","Text":"I get nothing, so all I have to do is plug in one."},{"Start":"09:51.295 ","End":"09:54.930","Text":"Just get this fraction exercise,"},{"Start":"09:54.930 ","End":"09:58.245","Text":"minus 1/6th, minus 1/2,"},{"Start":"09:58.245 ","End":"10:01.360","Text":"plus 3 over 2,"},{"Start":"10:01.400 ","End":"10:07.005","Text":"let\u0027s see, let\u0027s put it all over 6,"},{"Start":"10:07.005 ","End":"10:10.500","Text":"and then I\u0027ve got minus 1,"},{"Start":"10:10.500 ","End":"10:14.190","Text":"this is minus 3/6ths,"},{"Start":"10:14.190 ","End":"10:19.640","Text":"this is plus 9/6ths."},{"Start":"10:19.640 ","End":"10:23.855","Text":"I guess I should have really said minus 1/2 plus 1.5 is 1, never mind."},{"Start":"10:23.855 ","End":"10:30.300","Text":"In any event, it comes out to be equal to 5/6ths,"},{"Start":"10:30.980 ","End":"10:35.890","Text":"and this is the final answer. We\u0027re done."}],"ID":8732},{"Watched":false,"Name":"Exercise 3 part b","Duration":"12m 21s","ChapterTopicVideoID":8525,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8525.jpeg","UploadDate":"2017-02-13T02:59:54.4170000","DurationForVideoObject":"PT12M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this exercise, we have to compute the volume of a"},{"Start":"00:03.090 ","End":"00:06.195","Text":"solid bounded by the following surfaces."},{"Start":"00:06.195 ","End":"00:09.250","Text":"I brought in a sketch."},{"Start":"00:09.410 ","End":"00:13.110","Text":"Here in this whole set of exercises,"},{"Start":"00:13.110 ","End":"00:16.800","Text":"the body will be such that we can project it down onto"},{"Start":"00:16.800 ","End":"00:24.510","Text":"the xy-plane and compute this shape region of projection."},{"Start":"00:24.510 ","End":"00:29.310","Text":"There\u0027ll be 2 functions where z is a function of x and y."},{"Start":"00:29.310 ","End":"00:31.440","Text":"There\u0027ll be these 2 and 1 of them is going to be"},{"Start":"00:31.440 ","End":"00:33.890","Text":"an upper surface and 1 of them is going to be a lower surface,"},{"Start":"00:33.890 ","End":"00:35.775","Text":"and that\u0027s the general setup."},{"Start":"00:35.775 ","End":"00:44.370","Text":"Now we\u0027re going to use the formula that the volume of a body, let say B,"},{"Start":"00:44.370 ","End":"00:47.305","Text":"I don\u0027t want to use the letter V I\u0027ll use B,"},{"Start":"00:47.305 ","End":"00:54.735","Text":"is equal to the triple integral over B of just the function 1."},{"Start":"00:54.735 ","End":"00:59.015","Text":"Then it\u0027s going to be dx dy dz in some particular order."},{"Start":"00:59.015 ","End":"01:01.040","Text":"In our case, well,"},{"Start":"01:01.040 ","End":"01:03.835","Text":"I\u0027ll just leave it as dV for now."},{"Start":"01:03.835 ","End":"01:07.760","Text":"Then the idea is to convert this integral"},{"Start":"01:07.760 ","End":"01:11.905","Text":"because of the shape of B to an iterated integral,"},{"Start":"01:11.905 ","End":"01:20.069","Text":"and we\u0027ll always have the z in the innermost integral,"},{"Start":"01:20.069 ","End":"01:21.645","Text":"and it\u0027ll be something,"},{"Start":"01:21.645 ","End":"01:23.850","Text":"and it\u0027ll be dz."},{"Start":"01:23.850 ","End":"01:26.730","Text":"Then x and y can be in either order,"},{"Start":"01:26.730 ","End":"01:30.515","Text":"we\u0027ll see if we want to do it at a type 1 or type 2 region."},{"Start":"01:30.515 ","End":"01:32.780","Text":"But let me just assume that we\u0027re going to have it in"},{"Start":"01:32.780 ","End":"01:36.560","Text":"the order x equals y equals z equals something to something."},{"Start":"01:36.560 ","End":"01:38.120","Text":"Then we\u0027re going to have"},{"Start":"01:38.120 ","End":"01:48.370","Text":"dz dy dx."},{"Start":"01:48.370 ","End":"01:52.070","Text":"In this section, we have to compute the volume of a solid bounded"},{"Start":"01:52.070 ","End":"01:56.560","Text":"by the following surfaces brought in a sketch."},{"Start":"01:56.560 ","End":"01:58.639","Text":"At least in this problem,"},{"Start":"01:58.639 ","End":"02:06.630","Text":"the shape is such that we can project it down onto some region or domain in the xy-plane."},{"Start":"02:06.980 ","End":"02:11.780","Text":"That\u0027s probably how it\u0027s going to be in the rest of the exercises in this set."},{"Start":"02:11.780 ","End":"02:16.165","Text":"The body I want to call by the name B,"},{"Start":"02:16.165 ","End":"02:18.750","Text":"solid is also a body."},{"Start":"02:18.750 ","End":"02:21.035","Text":"We\u0027re going to use the theorem that"},{"Start":"02:21.035 ","End":"02:27.050","Text":"the volume of a body B in this case is just the triple"},{"Start":"02:27.050 ","End":"02:37.590","Text":"integral over the solid body B of 1 dV."},{"Start":"02:37.590 ","End":"02:42.559","Text":"The difficult part is to convert this triple integral"},{"Start":"02:42.559 ","End":"02:47.580","Text":"into an iterated integral, 3 separate ones."},{"Start":"02:47.580 ","End":"02:52.065","Text":"It\u0027s always going to be dz here, and typically,"},{"Start":"02:52.065 ","End":"02:57.445","Text":"sorry, I meant to say z is going to be the middle integral."},{"Start":"02:57.445 ","End":"02:59.130","Text":"It\u0027s going to be 1."},{"Start":"02:59.130 ","End":"03:03.150","Text":"We\u0027re going to start out with dz and the other 2 will be x and y."},{"Start":"03:03.150 ","End":"03:07.110","Text":"Not sure in what order but we will always try x, y,"},{"Start":"03:07.110 ","End":"03:09.710","Text":"and if it doesn\u0027t work we\u0027ll reverse y and x,"},{"Start":"03:09.710 ","End":"03:12.850","Text":"and then it\u0027ll be dy dx."},{"Start":"03:12.850 ","End":"03:15.950","Text":"But the thing is where the z go from and to,"},{"Start":"03:15.950 ","End":"03:19.020","Text":"and the same for y and same for x."},{"Start":"03:19.360 ","End":"03:21.979","Text":"For the z part,"},{"Start":"03:21.979 ","End":"03:24.830","Text":"we will take these 2 functions,"},{"Start":"03:24.830 ","End":"03:26.180","Text":"the upper and the lower."},{"Start":"03:26.180 ","End":"03:33.135","Text":"We have to figure out which of these 2 is above the other."},{"Start":"03:33.135 ","End":"03:37.290","Text":"Sometimes we use a technique of taking a point to sample,"},{"Start":"03:37.290 ","End":"03:40.860","Text":"but here it\u0027s fairly clear that when I have,"},{"Start":"03:40.860 ","End":"03:42.460","Text":"say the upper one,"},{"Start":"03:42.460 ","End":"03:48.670","Text":"I\u0027m claiming is the x squared plus y squared."},{"Start":"03:48.670 ","End":"03:51.750","Text":"The lower one is z equals 0,"},{"Start":"03:51.750 ","End":"03:53.985","Text":"so I\u0027m saying this is upper,"},{"Start":"03:53.985 ","End":"03:56.189","Text":"that\u0027s the f in the picture,"},{"Start":"03:56.189 ","End":"03:58.485","Text":"and this one is the lower one."},{"Start":"03:58.485 ","End":"04:01.420","Text":"It\u0027s immediately obvious because x squared plus y"},{"Start":"04:01.420 ","End":"04:05.604","Text":"squared is a quantity that is always bigger or equal to 0,"},{"Start":"04:05.604 ","End":"04:07.300","Text":"so this will be the upper."},{"Start":"04:07.300 ","End":"04:11.170","Text":"Lets me right away write what z is,"},{"Start":"04:11.170 ","End":"04:12.985","Text":"limits for z I mean,"},{"Start":"04:12.985 ","End":"04:21.305","Text":"will be from 0-x squared plus y squared."},{"Start":"04:21.305 ","End":"04:23.555","Text":"Now what about x and y?"},{"Start":"04:23.555 ","End":"04:28.820","Text":"Well, I have to figure out what this region is in the xy-plane."},{"Start":"04:28.820 ","End":"04:33.110","Text":"To get this d, if I sketch these 2 functions,"},{"Start":"04:33.110 ","End":"04:37.145","Text":"y equals x squared will be a parabola,"},{"Start":"04:37.145 ","End":"04:44.445","Text":"something like this, not the greatest."},{"Start":"04:44.445 ","End":"04:48.644","Text":"I\u0027m going to label it y equals x squared,"},{"Start":"04:48.644 ","End":"04:53.565","Text":"and then y equals 1 would be something like this."},{"Start":"04:53.565 ","End":"04:57.615","Text":"This would be the point 1 for y."},{"Start":"04:57.615 ","End":"05:02.225","Text":"If I intersect them and I want to find out what these 2 points are,"},{"Start":"05:02.225 ","End":"05:06.690","Text":"we can compare and see that x squared equals 1,"},{"Start":"05:07.090 ","End":"05:10.430","Text":"gives me x equals plus or minus 1,"},{"Start":"05:10.430 ","End":"05:13.415","Text":"so here\u0027s 1, here\u0027s minus 1,"},{"Start":"05:13.415 ","End":"05:15.560","Text":"and so our region D,"},{"Start":"05:15.560 ","End":"05:20.480","Text":"I\u0027ll shade it and also label this y equals 1."},{"Start":"05:20.480 ","End":"05:22.760","Text":"Now we have the sketch."},{"Start":"05:22.760 ","End":"05:27.730","Text":"Now, we can decide that we want to go dy dx,"},{"Start":"05:27.730 ","End":"05:29.770","Text":"meaning is a type 1 region."},{"Start":"05:29.770 ","End":"05:31.215","Text":"Could also do it the other way."},{"Start":"05:31.215 ","End":"05:32.660","Text":"I think this is better,"},{"Start":"05:32.660 ","End":"05:35.390","Text":"because y is already extracted as a function of x."},{"Start":"05:35.390 ","End":"05:40.300","Text":"We would take x going from minus 1-1."},{"Start":"05:40.300 ","End":"05:43.715","Text":"I can already put that in here, minus 1-1,"},{"Start":"05:43.715 ","End":"05:49.175","Text":"and y goes from the lower one here, the x squared."},{"Start":"05:49.175 ","End":"05:52.835","Text":"In other words, a vertical slice through a typical"},{"Start":"05:52.835 ","End":"05:57.905","Text":"x would be where y goes from the parabola,"},{"Start":"05:57.905 ","End":"06:01.295","Text":"x squared up to the line y equals 1."},{"Start":"06:01.295 ","End":"06:03.649","Text":"Here it would be x squared,"},{"Start":"06:03.649 ","End":"06:05.405","Text":"and here it\u0027s 1."},{"Start":"06:05.405 ","End":"06:08.430","Text":"From here on it\u0027s all technical,"},{"Start":"06:08.430 ","End":"06:10.685","Text":"we don\u0027t need any diagrams anymore."},{"Start":"06:10.685 ","End":"06:13.535","Text":"We start with the inner integral,"},{"Start":"06:13.535 ","End":"06:16.800","Text":"and that will be the dz."},{"Start":"06:19.400 ","End":"06:23.659","Text":"This we can do in our heads"},{"Start":"06:23.659 ","End":"06:28.720","Text":"because the integral of 1 is just the upper limit minus the lower limit."},{"Start":"06:28.720 ","End":"06:30.535","Text":"In the middle here,"},{"Start":"06:30.535 ","End":"06:33.715","Text":"I just get x squared plus y squared minus 0,"},{"Start":"06:33.715 ","End":"06:36.890","Text":"which is x squared plus y squared."},{"Start":"06:36.890 ","End":"06:43.575","Text":"Then the rest of it it\u0027s the same integral as x goes from"},{"Start":"06:43.575 ","End":"06:51.855","Text":"minus 1-1 and y goes from x squared to 1,"},{"Start":"06:51.855 ","End":"06:54.960","Text":"and here dy dx."},{"Start":"06:54.960 ","End":"06:58.035","Text":"Now we do the middle one here."},{"Start":"06:58.035 ","End":"07:03.550","Text":"That\u0027s the dy integral."},{"Start":"07:03.590 ","End":"07:07.490","Text":"I\u0027d like to do this one just as a side exercise."},{"Start":"07:07.490 ","End":"07:09.500","Text":"By the way I have space here."},{"Start":"07:09.500 ","End":"07:14.794","Text":"The integral from x squared"},{"Start":"07:14.794 ","End":"07:20.465","Text":"to 1 of x squared plus y squared dy."},{"Start":"07:20.465 ","End":"07:23.675","Text":"That is equal to, with respect to y,"},{"Start":"07:23.675 ","End":"07:29.780","Text":"we have x squared gives me x squared y because x squared is a constant,"},{"Start":"07:29.780 ","End":"07:36.275","Text":"y squared integrated gives me 1/3y cubed."},{"Start":"07:36.275 ","End":"07:42.065","Text":"All this has to be taken between x squared and 1."},{"Start":"07:42.065 ","End":"07:46.070","Text":"This is of course, the limits for y."},{"Start":"07:46.070 ","End":"07:48.575","Text":"When y is 1,"},{"Start":"07:48.575 ","End":"07:52.310","Text":"I get 1x squared plus 1/3,"},{"Start":"07:52.310 ","End":"07:56.360","Text":"so I get x squared plus a 1/3."},{"Start":"07:56.360 ","End":"08:00.065","Text":"Then, when y is x squared,"},{"Start":"08:00.065 ","End":"08:07.580","Text":"I get x squared x squared is x^4 plus 1/3."},{"Start":"08:07.580 ","End":"08:15.120","Text":"Then when y is x squared, x squared cubed is x^6,"},{"Start":"08:15.120 ","End":"08:18.815","Text":"and so we just get,"},{"Start":"08:18.815 ","End":"08:24.100","Text":"let\u0027s see if I put them in order, it\u0027s minus"},{"Start":"08:24.100 ","End":"08:33.805","Text":"a 1/3x^6 minus x^4 plus x squared plus 1/3."},{"Start":"08:33.805 ","End":"08:38.705","Text":"This I can now put back over here,"},{"Start":"08:38.705 ","End":"08:47.515","Text":"and so we get the integral from minus 1-1 of minus"},{"Start":"08:47.515 ","End":"08:53.740","Text":"a 1/3x^6 minus x^4 plus x"},{"Start":"08:53.740 ","End":"09:02.065","Text":"squared plus 1/3 all this dx."},{"Start":"09:02.065 ","End":"09:12.360","Text":"Then, this is equal to the integral of this if I raise the power by 1,"},{"Start":"09:12.360 ","End":"09:17.200","Text":"it x^7, so I\u0027ve got minus"},{"Start":"09:19.010 ","End":"09:28.470","Text":"1/21x^7 minus 1/5x^5 here,"},{"Start":"09:28.470 ","End":"09:32.925","Text":"plus a 1/3x cubed,"},{"Start":"09:32.925 ","End":"09:36.945","Text":"and here plus 1/3x,"},{"Start":"09:36.945 ","End":"09:41.835","Text":"all this from minus 1-1."},{"Start":"09:41.835 ","End":"09:46.445","Text":"Let\u0027s see what we get."},{"Start":"09:46.445 ","End":"09:49.385","Text":"If I plug in 1,"},{"Start":"09:49.385 ","End":"09:57.020","Text":"I\u0027ve got minus 1/21 minus a 1/5 plus"},{"Start":"09:57.020 ","End":"10:07.800","Text":"a 1/3 plus 1/3."},{"Start":"10:07.800 ","End":"10:11.160","Text":"The other parts with the minus 1,"},{"Start":"10:11.160 ","End":"10:13.180","Text":"actually all the powers are odd,"},{"Start":"10:13.180 ","End":"10:16.085","Text":"they\u0027ll all come out just the opposite."},{"Start":"10:16.085 ","End":"10:23.820","Text":"I will get plus 1/21 plus"},{"Start":"10:23.820 ","End":"10:32.160","Text":"1/5 minus a 1/3 minus a 1/3, continuing."},{"Start":"10:32.160 ","End":"10:35.340","Text":"Well, I can just say it\u0027s twice this."},{"Start":"10:35.340 ","End":"10:39.010","Text":"This part here comes out to be,"},{"Start":"10:39.010 ","End":"10:41.970","Text":"let see if I can just do this one,"},{"Start":"10:41.970 ","End":"10:45.430","Text":"common denominator for 3, 5,"},{"Start":"10:45.430 ","End":"10:50.200","Text":"and 21 would be 105."},{"Start":"10:50.200 ","End":"10:55.675","Text":"I could take 21 times 5 and the 3 is already included."},{"Start":"10:55.675 ","End":"10:57.970","Text":"That would be, let see,"},{"Start":"10:57.970 ","End":"11:02.890","Text":"over a 105, this"},{"Start":"11:02.890 ","End":"11:08.150","Text":"goes into it minus 5 times."},{"Start":"11:08.150 ","End":"11:11.075","Text":"Here I have minus 21,"},{"Start":"11:11.075 ","End":"11:19.560","Text":"here 3 goes into this 35 times and another 35 times."},{"Start":"11:20.810 ","End":"11:25.650","Text":"The other one will just be exactly the negative of that."},{"Start":"11:25.660 ","End":"11:29.510","Text":"I\u0027m just subtracting the negative of it,"},{"Start":"11:29.510 ","End":"11:33.200","Text":"so I can just put twice here."},{"Start":"11:33.200 ","End":"11:38.975","Text":"What I\u0027m saying is that a minus minus a is twice a in general,"},{"Start":"11:38.975 ","End":"11:41.960","Text":"so I don\u0027t have to do this one also."},{"Start":"11:41.960 ","End":"11:44.765","Text":"Here I can get,"},{"Start":"11:44.765 ","End":"11:48.680","Text":"let see, 35 and 35 is 70."},{"Start":"11:48.680 ","End":"11:51.215","Text":"The negatives are 26,"},{"Start":"11:51.215 ","End":"11:58.165","Text":"70 minus 26 is 44."},{"Start":"11:58.165 ","End":"12:08.174","Text":"It\u0027s 44/105 times 2,"},{"Start":"12:08.174 ","End":"12:11.205","Text":"I forgot the times 2."},{"Start":"12:11.205 ","End":"12:16.000","Text":"I guess that makes it 88/105,"},{"Start":"12:17.000 ","End":"12:21.520","Text":"and that would be the answer, so we\u0027re done."}],"ID":8733},{"Watched":false,"Name":"Exercise 3 part c","Duration":"18m 53s","ChapterTopicVideoID":8526,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8526.jpeg","UploadDate":"2017-02-13T03:04:34.1570000","DurationForVideoObject":"PT18M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:08.680","Text":"In this exercise, we need to compute the volume of the solid bounded by the surfaces."},{"Start":"00:08.870 ","End":"00:11.595","Text":"In this whole set of exercise,"},{"Start":"00:11.595 ","End":"00:18.420","Text":"it\u0027s always so that we work with the projection onto the x, y plane."},{"Start":"00:18.420 ","End":"00:21.735","Text":"There\u0027s always an upper surface and the lowest surface,"},{"Start":"00:21.735 ","End":"00:23.385","Text":"one of these two."},{"Start":"00:23.385 ","End":"00:27.815","Text":"Typically, we have one of them as z equals 0,"},{"Start":"00:27.815 ","End":"00:30.890","Text":"but in general, it could be an upper and a lower."},{"Start":"00:30.890 ","End":"00:33.590","Text":"Let me just change the letter."},{"Start":"00:33.590 ","End":"00:36.694","Text":"I prefer to use the letter B for body."},{"Start":"00:36.694 ","End":"00:41.330","Text":"Then we\u0027re going to use the standard formula that the volume of"},{"Start":"00:41.330 ","End":"00:50.915","Text":"a body B is just equal to the triple integral over that body B of the function 1,"},{"Start":"00:50.915 ","End":"00:54.845","Text":"constant function, and dv."},{"Start":"00:54.845 ","End":"01:04.770","Text":"Our difficult task is to break this up to iterated integrals like dx, dy, dz."},{"Start":"01:07.130 ","End":"01:10.645","Text":"What I\u0027d like to start with is computing,"},{"Start":"01:10.645 ","End":"01:17.810","Text":"and I\u0027ll use these three equations to figure out what our domain D is in the x, y-plane."},{"Start":"01:17.810 ","End":"01:22.135","Text":"I just need a rough sketch of these three equations."},{"Start":"01:22.135 ","End":"01:25.315","Text":"I\u0027ll start with the easier ones."},{"Start":"01:25.315 ","End":"01:28.480","Text":"This is y equals 1/2x and y equals 2x."},{"Start":"01:28.480 ","End":"01:30.790","Text":"They\u0027re both lines through the origin,"},{"Start":"01:30.790 ","End":"01:34.809","Text":"one with slope 1/2, something like this."},{"Start":"01:34.809 ","End":"01:38.845","Text":"Slope less than 45 degrees because the slope is less than 1."},{"Start":"01:38.845 ","End":"01:41.290","Text":"Here it\u0027ll be more than 45 degrees,"},{"Start":"01:41.290 ","End":"01:43.595","Text":"slope will be 2."},{"Start":"01:43.595 ","End":"01:54.650","Text":"Also, I stopped at the origin because of the restriction that x bigger or equal to 0,"},{"Start":"01:54.650 ","End":"01:55.865","Text":"I just label them."},{"Start":"01:55.865 ","End":"01:58.250","Text":"This is y equals 2x."},{"Start":"01:58.250 ","End":"02:02.810","Text":"This is y equals 1/2x or 0.5x."},{"Start":"02:02.810 ","End":"02:06.770","Text":"The third one, y equals 2 over x."},{"Start":"02:06.770 ","End":"02:11.210","Text":"In general, each one of these hyperbolas normally would have two branches,"},{"Start":"02:11.210 ","End":"02:14.135","Text":"but in the first quadrant we only have one branch."},{"Start":"02:14.135 ","End":"02:17.180","Text":"Something like this with asymptotes here and here."},{"Start":"02:17.180 ","End":"02:22.095","Text":"This will be y equals 2 over x."},{"Start":"02:22.095 ","End":"02:27.255","Text":"Our region D, I just highlighted it."},{"Start":"02:27.255 ","End":"02:32.915","Text":"This is our region D. In order to figure out better what it is,"},{"Start":"02:32.915 ","End":"02:36.250","Text":"I\u0027d like to know what these points are."},{"Start":"02:36.250 ","End":"02:38.790","Text":"I know this one is the origin."},{"Start":"02:38.790 ","End":"02:40.455","Text":"But what are these two?"},{"Start":"02:40.455 ","End":"02:43.584","Text":"We just intersect them in pairs."},{"Start":"02:43.584 ","End":"02:47.920","Text":"Let\u0027s take first of all, this pair."},{"Start":"02:47.920 ","End":"02:56.520","Text":"That would be y equals 2 over x and y equals 2x."},{"Start":"02:56.520 ","End":"02:58.804","Text":"This point is where they intersect."},{"Start":"02:58.804 ","End":"03:05.415","Text":"In other words, I get that 2 over x equals 2x by comparing the y\u0027s."},{"Start":"03:05.415 ","End":"03:08.240","Text":"Then I divide both sides by 2,"},{"Start":"03:08.240 ","End":"03:13.470","Text":"I get that x squared equals 1."},{"Start":"03:13.850 ","End":"03:19.029","Text":"Because it\u0027s in the non-negative area, x equals 1."},{"Start":"03:19.029 ","End":"03:21.185","Text":"If x equals 1,"},{"Start":"03:21.185 ","End":"03:23.900","Text":"I don\u0027t know that we need the y,"},{"Start":"03:23.900 ","End":"03:25.610","Text":"but in case we do,"},{"Start":"03:25.610 ","End":"03:30.620","Text":"we just substitute it into either one of these and y equals 2."},{"Start":"03:30.620 ","End":"03:34.350","Text":"This is the point 1, 2."},{"Start":"03:34.630 ","End":"03:37.430","Text":"The other point here,"},{"Start":"03:37.430 ","End":"03:39.725","Text":"again, I really just need the x,"},{"Start":"03:39.725 ","End":"03:44.690","Text":"so I compare y equals 2 over"},{"Start":"03:44.690 ","End":"03:51.720","Text":"x and see where it intersects with y equals 1/2x."},{"Start":"03:52.220 ","End":"03:54.650","Text":"In this case from these two,"},{"Start":"03:54.650 ","End":"03:57.050","Text":"if I compare the right-hand side,"},{"Start":"03:57.050 ","End":"04:01.220","Text":"so I get that 2 over x equals 1/2x,"},{"Start":"04:01.220 ","End":"04:03.515","Text":"so I\u0027ll write it as x over 2."},{"Start":"04:03.515 ","End":"04:07.520","Text":"Cross multiply, we get that x squared equals 4."},{"Start":"04:07.520 ","End":"04:09.620","Text":"Because x is positive,"},{"Start":"04:09.620 ","End":"04:11.705","Text":"then we get that x equals 2."},{"Start":"04:11.705 ","End":"04:14.375","Text":"This is the point 2."},{"Start":"04:14.375 ","End":"04:22.160","Text":"This region will be naturally divided into two pieces."},{"Start":"04:22.160 ","End":"04:28.025","Text":"One part, maybe I\u0027ll call this one D_1 and this one D_2."},{"Start":"04:28.025 ","End":"04:31.610","Text":"Anyway, I don\u0027t see any reason to label them."},{"Start":"04:31.610 ","End":"04:34.970","Text":"We just are aware that there\u0027s two separate bits."},{"Start":"04:34.970 ","End":"04:41.000","Text":"That settles the x, y which we\u0027re"},{"Start":"04:41.000 ","End":"04:47.380","Text":"doing as a type 1 region with vertical slices."},{"Start":"04:47.380 ","End":"04:54.330","Text":"Notice that if x is between 0 and 1, I\u0027ll get one thing."},{"Start":"04:54.330 ","End":"04:56.660","Text":"If x is between here and here, I\u0027ll get something else."},{"Start":"04:56.660 ","End":"05:04.580","Text":"In this case, a vertical slice will enter the region here and exit here."},{"Start":"05:04.580 ","End":"05:07.640","Text":"We have the formula for each of these curves."},{"Start":"05:07.640 ","End":"05:09.800","Text":"Then the other part, if x is here,"},{"Start":"05:09.800 ","End":"05:14.840","Text":"the vertical slice will enter the region here and exit here."},{"Start":"05:14.840 ","End":"05:17.135","Text":"Again, we have the formula for these two,"},{"Start":"05:17.135 ","End":"05:20.430","Text":"so we have to split the whole thing up."},{"Start":"05:20.600 ","End":"05:23.970","Text":"Let\u0027s start with the left part."},{"Start":"05:23.970 ","End":"05:29.290","Text":"We get the integral x goes from 0-1,"},{"Start":"05:29.420 ","End":"05:36.810","Text":"and then y goes from this function,"},{"Start":"05:36.810 ","End":"05:43.900","Text":"which is 1/2x to this function which is 2x."},{"Start":"05:43.990 ","End":"05:48.265","Text":"We have to still discuss the z."},{"Start":"05:48.265 ","End":"05:56.570","Text":"Well, z goes and we have a case like this from the lower surface to the upper surface."},{"Start":"05:56.570 ","End":"05:58.460","Text":"Now we have two of them."},{"Start":"05:58.460 ","End":"06:01.325","Text":"One of these two is the lower."},{"Start":"06:01.325 ","End":"06:05.780","Text":"I have y equals x squared plus y, sorry,"},{"Start":"06:05.780 ","End":"06:09.135","Text":"this is z I meant,"},{"Start":"06:09.135 ","End":"06:12.990","Text":"and z equals 0."},{"Start":"06:12.990 ","End":"06:16.035","Text":"Now, we\u0027re all in the first quadrant,"},{"Start":"06:16.035 ","End":"06:20.300","Text":"so this is always bigger or equal to 0."},{"Start":"06:20.300 ","End":"06:25.280","Text":"This is going to be the upper surface like f in the picture,"},{"Start":"06:25.280 ","End":"06:30.230","Text":"and this one\u0027s going to be the lower surface like g in this picture."},{"Start":"06:30.230 ","End":"06:33.575","Text":"We go from the lower, which is 0,"},{"Start":"06:33.575 ","End":"06:38.454","Text":"to the upper, which is x squared plus y."},{"Start":"06:38.454 ","End":"06:41.130","Text":"Then it\u0027s just the 1,"},{"Start":"06:41.130 ","End":"06:44.890","Text":"and the inner integral is dz."},{"Start":"06:45.770 ","End":"06:48.230","Text":"But basically in reverse order,"},{"Start":"06:48.230 ","End":"06:52.910","Text":"then we need a dy, and then we need a dx."},{"Start":"06:52.910 ","End":"06:56.735","Text":"But that\u0027s just the first part."},{"Start":"06:56.735 ","End":"06:59.719","Text":"We\u0027re going to have to add another part,"},{"Start":"06:59.719 ","End":"07:01.735","Text":"which is the right one."},{"Start":"07:01.735 ","End":"07:05.310","Text":"Let me just do a copy-paste."},{"Start":"07:05.310 ","End":"07:07.160","Text":"I copied this here,"},{"Start":"07:07.160 ","End":"07:09.725","Text":"but I erased the limits for x and y,"},{"Start":"07:09.725 ","End":"07:12.085","Text":"z is going to be the same."},{"Start":"07:12.085 ","End":"07:14.360","Text":"Everywhere in the region we\u0027re always going to go from"},{"Start":"07:14.360 ","End":"07:18.575","Text":"the lower surface to the upper surface from 0 to x squared plus y."},{"Start":"07:18.575 ","End":"07:23.905","Text":"But the x and y now changed because now x is going from 1-2."},{"Start":"07:23.905 ","End":"07:25.640","Text":"For x between 1 and 2,"},{"Start":"07:25.640 ","End":"07:28.775","Text":"y goes from this line to this curve."},{"Start":"07:28.775 ","End":"07:36.805","Text":"It goes from 1/2x up to 2 over x,"},{"Start":"07:36.805 ","End":"07:39.170","Text":"where you broke it up into two parts."},{"Start":"07:39.170 ","End":"07:43.385","Text":"But luckily, the inner integral is the same for both."},{"Start":"07:43.385 ","End":"07:50.140","Text":"I\u0027m talking about this part here is exactly the same as this part here."},{"Start":"07:50.140 ","End":"07:53.390","Text":"We know that the integral of 1 between an upper and"},{"Start":"07:53.390 ","End":"07:57.860","Text":"a lower limit is just the upper minus the lower, and the lower is 0."},{"Start":"07:57.860 ","End":"08:03.485","Text":"This whole thing just comes out to be x squared plus y. I can write this"},{"Start":"08:03.485 ","End":"08:11.810","Text":"as the integral of x squared plus y, dy dx."},{"Start":"08:11.810 ","End":"08:14.730","Text":"I\u0027ll write the limits in a moment."},{"Start":"08:14.960 ","End":"08:19.755","Text":"I just wanted to copy-paste first to save myself some trouble."},{"Start":"08:19.755 ","End":"08:24.495","Text":"Now we can write that here we go from 0-1 with"},{"Start":"08:24.495 ","End":"08:34.020","Text":"x and from 1/2x to 2x with y."},{"Start":"08:34.020 ","End":"08:35.265","Text":"Here it\u0027s different."},{"Start":"08:35.265 ","End":"08:39.630","Text":"Here x goes from 1-2,"},{"Start":"08:39.630 ","End":"08:46.049","Text":"and y goes from 1/2x to 2 over x."},{"Start":"08:46.049 ","End":"08:50.145","Text":"Here we diverge different paths,"},{"Start":"08:50.145 ","End":"08:51.755","Text":"let\u0027s just get some space."},{"Start":"08:51.755 ","End":"08:54.830","Text":"In any event we\u0027re now purely technical,"},{"Start":"08:54.830 ","End":"08:57.245","Text":"we don\u0027t need any diagrams."},{"Start":"08:57.245 ","End":"08:59.930","Text":"Let\u0027s start with this integral,"},{"Start":"08:59.930 ","End":"09:03.090","Text":"the inner integral of the first bit."},{"Start":"09:04.550 ","End":"09:07.339","Text":"I want to do that at the side."},{"Start":"09:07.339 ","End":"09:11.495","Text":"Just make a bit of space for it."},{"Start":"09:11.495 ","End":"09:15.634","Text":"I\u0027ll do this one over here."},{"Start":"09:15.634 ","End":"09:20.565","Text":"We get, I\u0027ll just copy it first,"},{"Start":"09:20.565 ","End":"09:25.860","Text":"the integral from 1/2x to"},{"Start":"09:25.860 ","End":"09:31.780","Text":"2x of x squared plus y dy."},{"Start":"09:31.780 ","End":"09:37.240","Text":"The y is the variable and x is a constant,"},{"Start":"09:37.240 ","End":"09:46.730","Text":"so this is equal to x squared y plus 1/2y squared."},{"Start":"09:46.730 ","End":"09:52.420","Text":"All this from 1/2x to 2x."},{"Start":"09:52.420 ","End":"10:02.190","Text":"This will equal, if I put in y equals 2x here,"},{"Start":"10:02.190 ","End":"10:05.175","Text":"I get 2x cubed."},{"Start":"10:05.175 ","End":"10:09.240","Text":"If I put y equals 2x here,"},{"Start":"10:09.240 ","End":"10:16.480","Text":"then we get 4x squared over 2 is 2x squared."},{"Start":"10:16.480 ","End":"10:19.594","Text":"Then let\u0027s see."},{"Start":"10:19.594 ","End":"10:21.485","Text":"We put in 1/2x,"},{"Start":"10:21.485 ","End":"10:23.945","Text":"we have to subtract."},{"Start":"10:23.945 ","End":"10:32.550","Text":"We put in 1/2x here we get 1/2x cubed."},{"Start":"10:33.160 ","End":"10:36.290","Text":"If I put in 1/2x here,"},{"Start":"10:36.290 ","End":"10:44.130","Text":"I get 1/4x squared times 1/2 is 1/8x squared."},{"Start":"10:44.320 ","End":"10:47.060","Text":"What does that give me altogether?"},{"Start":"10:47.060 ","End":"10:48.874","Text":"Collecting the x cubed,"},{"Start":"10:48.874 ","End":"10:52.385","Text":"2 minus 1/2 is 1.5,"},{"Start":"10:52.385 ","End":"10:56.975","Text":"or let\u0027s write it as 3 over 2x cubed."},{"Start":"10:56.975 ","End":"11:01.240","Text":"An x squared to have 2 minus 1/8,"},{"Start":"11:01.240 ","End":"11:05.060","Text":"so I\u0027ll write that as an improper fraction,"},{"Start":"11:05.060 ","End":"11:09.720","Text":"that\u0027s 15 over 8x squared."},{"Start":"11:09.720 ","End":"11:14.415","Text":"Now, all this is what goes here."},{"Start":"11:14.415 ","End":"11:22.170","Text":"Let me just write this bit as the"},{"Start":"11:22.170 ","End":"11:31.060","Text":"integral from 0-1 of 3 over 2x cubed,"},{"Start":"11:31.060 ","End":"11:32.555","Text":"I\u0027m copying from here,"},{"Start":"11:32.555 ","End":"11:39.980","Text":"plus 15 over 8x squared dx."},{"Start":"11:39.980 ","End":"11:43.340","Text":"Now I want to do the other inner one,"},{"Start":"11:43.340 ","End":"11:47.610","Text":"that\u0027s this one here."},{"Start":"11:49.500 ","End":"11:51.700","Text":"Perhaps I label them."},{"Start":"11:51.700 ","End":"11:57.205","Text":"Let\u0027s say this was asterisk and this will be double asterisk."},{"Start":"11:57.205 ","End":"12:01.645","Text":"And I\u0027ll do that also at the side and say this was the asterisk."},{"Start":"12:01.645 ","End":"12:08.605","Text":"Here I\u0027ll do double asterisk and get some more space."},{"Start":"12:08.605 ","End":"12:17.635","Text":"What we have this time is the integral from 1/2x to 2 over x,"},{"Start":"12:17.635 ","End":"12:26.290","Text":"also x squared plus y dy."},{"Start":"12:26.290 ","End":"12:28.225","Text":"Well, this part\u0027s the same."},{"Start":"12:28.225 ","End":"12:33.835","Text":"It\u0027s still x squared y plus 1/2y squared."},{"Start":"12:33.835 ","End":"12:36.400","Text":"But the upper and lower limits are different,"},{"Start":"12:36.400 ","End":"12:38.410","Text":"while even the lower one is the same."},{"Start":"12:38.410 ","End":"12:42.415","Text":"That one\u0027s also a half x from here."},{"Start":"12:42.415 ","End":"12:47.980","Text":"The upper one is 2 over x and so this"},{"Start":"12:47.980 ","End":"12:56.965","Text":"equals the first one is different the second one is the same if,"},{"Start":"12:56.965 ","End":"13:03.145","Text":"well, yeah, if I put in 2 over x for y,"},{"Start":"13:03.145 ","End":"13:07.900","Text":"we just stress that this was y equals a should have stressed it here."},{"Start":"13:07.900 ","End":"13:11.050","Text":"If I put y equals 2 over x,"},{"Start":"13:11.050 ","End":"13:14.245","Text":"what I\u0027ll get is one of the x\u0027s will cancel,"},{"Start":"13:14.245 ","End":"13:20.830","Text":"and I\u0027ll just get 2x because x squared over x is x."},{"Start":"13:20.830 ","End":"13:24.490","Text":"If I put in here y equals 2 over x,"},{"Start":"13:24.490 ","End":"13:27.430","Text":"I\u0027ll get 4 over x squared."},{"Start":"13:27.430 ","End":"13:31.869","Text":"But with the 2, it\u0027ll just be 2 over x squared."},{"Start":"13:31.869 ","End":"13:38.575","Text":"That\u0027s the upper one minus the lower one will be the same as here."},{"Start":"13:38.575 ","End":"13:46.060","Text":"A half x cubed plus 1/8x squared."},{"Start":"13:46.060 ","End":"13:51.655","Text":"This time there\u0027s nothing to collect and just put them in order though."},{"Start":"13:51.655 ","End":"14:01.645","Text":"I can say that I have minus a half x cubed."},{"Start":"14:01.645 ","End":"14:03.280","Text":"Next, I have"},{"Start":"14:03.280 ","End":"14:12.805","Text":"1/8 minus an 1/8x squared plus 2x plus"},{"Start":"14:12.805 ","End":"14:17.120","Text":"2 over x squared."},{"Start":"14:17.850 ","End":"14:24.550","Text":"Then all I will have to do is plug this back in here."},{"Start":"14:24.550 ","End":"14:31.645","Text":"We get plus the integral from 1-2 of this thing."},{"Start":"14:31.645 ","End":"14:36.940","Text":"Let\u0027s write it as minus a half x cubed"},{"Start":"14:36.940 ","End":"14:42.310","Text":"minus 1/8x squared plus"},{"Start":"14:42.310 ","End":"14:50.720","Text":"2x plus 2 over x squared dx."},{"Start":"14:50.760 ","End":"14:54.220","Text":"I could work on each one separately or I can do them"},{"Start":"14:54.220 ","End":"14:58.555","Text":"simultaneously maybe I\u0027ll work on them simultaneously."},{"Start":"14:58.555 ","End":"15:04.480","Text":"The integral of this raise the power by one and divide by the new power."},{"Start":"15:04.480 ","End":"15:13.045","Text":"So we get 3 over 2 times 4, 3/8x^4."},{"Start":"15:13.045 ","End":"15:15.160","Text":"Here I\u0027ll get an x cubed,"},{"Start":"15:15.160 ","End":"15:19.225","Text":"but 3 will cancel into 15, 5 times."},{"Start":"15:19.225 ","End":"15:24.100","Text":"So I\u0027ll get 5 over 8 x cubed."},{"Start":"15:24.100 ","End":"15:28.675","Text":"This bit will be taken between 0 and 1."},{"Start":"15:28.675 ","End":"15:34.030","Text":"The second bit will be raised the power and divide."},{"Start":"15:34.030 ","End":"15:39.025","Text":"I got minus 1/8x^4 here,"},{"Start":"15:39.025 ","End":"15:40.960","Text":"3, and then it will go with the 8,"},{"Start":"15:40.960 ","End":"15:45.520","Text":"make it 1 over 24x cubed, here,"},{"Start":"15:45.520 ","End":"15:51.040","Text":"just x squared and here minus 2 over"},{"Start":"15:51.040 ","End":"15:57.625","Text":"x and this between 1 and 2."},{"Start":"15:57.625 ","End":"15:59.740","Text":"So this equals."},{"Start":"15:59.740 ","End":"16:03.670","Text":"Now, let\u0027s see this bit here."},{"Start":"16:03.670 ","End":"16:06.850","Text":"If I plug in 0, I\u0027ll get nothing."},{"Start":"16:06.850 ","End":"16:08.890","Text":"So I just have to plug in the 1."},{"Start":"16:08.890 ","End":"16:13.045","Text":"This part gives me 3/8 plus"},{"Start":"16:13.045 ","End":"16:22.930","Text":"5/8 and this bit comes out to be just 1."},{"Start":"16:22.930 ","End":"16:25.940","Text":"Let\u0027s see what we get."},{"Start":"16:28.950 ","End":"16:33.700","Text":"If I plug in the upper limit 2,"},{"Start":"16:33.700 ","End":"16:41.715","Text":"I plug-in 2^4 is 16,"},{"Start":"16:41.715 ","End":"16:46.529","Text":"16 over 8 is 2, it\u0027s minus 2."},{"Start":"16:46.529 ","End":"16:54.455","Text":"Here I get 8 over 24 minus a 1/3."},{"Start":"16:54.455 ","End":"17:02.450","Text":"Here I get 2 squared is 4 and here 2 over 2 is minus 1."},{"Start":"17:04.230 ","End":"17:07.570","Text":"Should have really do it this minus 0,"},{"Start":"17:07.570 ","End":"17:09.535","Text":"just to emphasize that I\u0027ve taken the 1,"},{"Start":"17:09.535 ","End":"17:11.755","Text":"I get 0 here, I\u0027m taking 2."},{"Start":"17:11.755 ","End":"17:15.970","Text":"I\u0027m going to subtract what happens when I plug in 1 and a plug-in one,"},{"Start":"17:15.970 ","End":"17:19.240","Text":"I\u0027ve got minus an 1/8 minus 1 over"},{"Start":"17:19.240 ","End":"17:27.505","Text":"24 plus 1 minus 2."},{"Start":"17:27.505 ","End":"17:32.780","Text":"So altogether, let\u0027s see what we get so we go, 1 from here."},{"Start":"17:35.520 ","End":"17:38.470","Text":"These brackets, 4 minus 2,"},{"Start":"17:38.470 ","End":"17:41.455","Text":"minus 1 is 1 minus a 1/3."},{"Start":"17:41.455 ","End":"17:47.330","Text":"This bit here is 2/3 and let\u0027s see what this bit is."},{"Start":"17:48.270 ","End":"17:51.760","Text":"An 1/8 and a 1/24 gives me a 1/6."},{"Start":"17:51.760 ","End":"17:53.500","Text":"You do a common denominator, 24,"},{"Start":"17:53.500 ","End":"17:54.865","Text":"you\u0027ll get 3 plus 1,"},{"Start":"17:54.865 ","End":"17:56.470","Text":"which is 4 over 24."},{"Start":"17:56.470 ","End":"17:58.870","Text":"This bit is minus a 1/6."},{"Start":"17:58.870 ","End":"18:04.045","Text":"Here I\u0027ve got minus 1 minus 1/6."},{"Start":"18:04.045 ","End":"18:08.535","Text":"This will be minus 7/6,"},{"Start":"18:08.535 ","End":"18:15.430","Text":"so it\u0027s minus, minus 7/6."},{"Start":"18:15.430 ","End":"18:20.935","Text":"So it\u0027s 1 plus this, minus, minus this."},{"Start":"18:20.935 ","End":"18:31.970","Text":"What I basically get is 1 plus 2/3 plus 7 over 6."},{"Start":"18:32.100 ","End":"18:35.650","Text":"If I put a common denominator, 6,"},{"Start":"18:35.650 ","End":"18:38.260","Text":"I\u0027ve got 6 over 6 here,"},{"Start":"18:38.260 ","End":"18:43.015","Text":"4 over 6 here, 7 over 6."},{"Start":"18:43.015 ","End":"18:48.519","Text":"Altogether the answer would be 17 over 6."},{"Start":"18:48.519 ","End":"18:50.410","Text":"So that\u0027s the answer."},{"Start":"18:50.410 ","End":"18:53.240","Text":"And we are done."}],"ID":8734},{"Watched":false,"Name":"Exercise 3 part d","Duration":"19m 9s","ChapterTopicVideoID":8527,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8527.jpeg","UploadDate":"2017-02-13T03:07:27.8000000","DurationForVideoObject":"PT19M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this exercise, we have to compute the volume of"},{"Start":"00:03.120 ","End":"00:06.000","Text":"the solid bounded by the following surfaces."},{"Start":"00:06.000 ","End":"00:09.420","Text":"At first glance, we don\u0027t seem to have enough surfaces."},{"Start":"00:09.420 ","End":"00:11.550","Text":"Usually we\u0027ve had more of them."},{"Start":"00:11.550 ","End":"00:13.740","Text":"We usually project onto the x,"},{"Start":"00:13.740 ","End":"00:18.300","Text":"y plane and get some region D. Then find that the volume,"},{"Start":"00:18.300 ","End":"00:22.455","Text":"the solid is bounded between an upper and a lower surface."},{"Start":"00:22.455 ","End":"00:24.195","Text":"The thing is that here,"},{"Start":"00:24.195 ","End":"00:26.954","Text":"the middle equation is going to do double duty."},{"Start":"00:26.954 ","End":"00:29.460","Text":"You\u0027ll see what I mean in a moment."},{"Start":"00:29.460 ","End":"00:35.580","Text":"Let\u0027s first of all try and compute the projection onto the x, y plane."},{"Start":"00:35.580 ","End":"00:42.195","Text":"What we can do to get a closed region D is to take this one,"},{"Start":"00:42.195 ","End":"00:46.800","Text":"which I can write as x equals 2y squared."},{"Start":"00:46.800 ","End":"00:49.910","Text":"I can take the projection of this onto the x,"},{"Start":"00:49.910 ","End":"00:55.515","Text":"y plane, which just means letting z equals 0."},{"Start":"00:55.515 ","End":"00:59.880","Text":"The other one would be x/ 4 plus y /2,"},{"Start":"00:59.880 ","End":"01:02.865","Text":"this is 0 equals 1."},{"Start":"01:02.865 ","End":"01:07.580","Text":"We\u0027ve actually done this in a previous exercise in a different section."},{"Start":"01:07.580 ","End":"01:12.370","Text":"Why don\u0027t I just copy the diagram I did there."},{"Start":"01:12.370 ","End":"01:15.980","Text":"Here\u0027s the picture I copy pasted from the other exercise,"},{"Start":"01:15.980 ","End":"01:20.690","Text":"but I\u0027m going to explain it and do the calculations we did there."},{"Start":"01:20.690 ","End":"01:23.690","Text":"First of all we drew the parabola x equals"},{"Start":"01:23.690 ","End":"01:26.690","Text":"2y squared just by putting in a couple of points."},{"Start":"01:26.690 ","End":"01:30.290","Text":"For example, when x is a function of y."},{"Start":"01:30.290 ","End":"01:32.570","Text":"For example, if y is 0,"},{"Start":"01:32.570 ","End":"01:35.060","Text":"x is 0, if y was 1,"},{"Start":"01:35.060 ","End":"01:37.870","Text":"x was 2, and so on."},{"Start":"01:37.870 ","End":"01:42.450","Text":"The straight line we drew by doing the intercepts,"},{"Start":"01:42.450 ","End":"01:47.640","Text":"if y is 0, we get x/4 is 1 so x is 4."},{"Start":"01:47.640 ","End":"01:49.870","Text":"That gave that point."},{"Start":"01:50.810 ","End":"01:53.670","Text":"If x is 0,"},{"Start":"01:53.670 ","End":"01:55.320","Text":"then we get y/2 is 1,"},{"Start":"01:55.320 ","End":"01:57.995","Text":"so y is 2, that gave this point."},{"Start":"01:57.995 ","End":"02:03.110","Text":"Next, we found out where these 2 curves intersected here and here."},{"Start":"02:03.110 ","End":"02:08.180","Text":"This we did by letting x equals 2y squared and substituting in here."},{"Start":"02:08.180 ","End":"02:18.635","Text":"We got this equation became 2y squared over 4 plus y over 2 equals 1."},{"Start":"02:18.635 ","End":"02:24.320","Text":"Then we multiplied and got a quadratic equation."},{"Start":"02:24.320 ","End":"02:27.425","Text":"If I multiply by 2 everywhere,"},{"Start":"02:27.425 ","End":"02:31.895","Text":"then I get 4 over 4 becomes 1."},{"Start":"02:31.895 ","End":"02:39.305","Text":"We get y squared and then plus y minus 2 equals 0."},{"Start":"02:39.305 ","End":"02:50.155","Text":"This gave the solutions that y is either equal to 1,"},{"Start":"02:50.155 ","End":"02:54.110","Text":"or y is minus 2."},{"Start":"02:54.110 ","End":"02:59.385","Text":"When y is equal to 1,"},{"Start":"02:59.385 ","End":"03:03.300","Text":"then we got that x was equal to 2."},{"Start":"03:03.300 ","End":"03:12.675","Text":"We got the point lets just write that x equals 2 by substituting x equals 2y squared."},{"Start":"03:12.675 ","End":"03:18.210","Text":"Twice 1 squared is 2 and twice minus 2 squared gives 8."},{"Start":"03:18.210 ","End":"03:21.780","Text":"This gave us the point here, 1,2,"},{"Start":"03:21.780 ","End":"03:25.815","Text":"and the point minus 2,8, that explains everything."},{"Start":"03:25.815 ","End":"03:30.160","Text":"The picture is already ready."},{"Start":"03:30.440 ","End":"03:35.830","Text":"The D is the area that\u0027s shaded it\u0027s between the 2 curves."},{"Start":"03:35.830 ","End":"03:37.300","Text":"That\u0027s our projection."},{"Start":"03:37.300 ","End":"03:41.840","Text":"I like to call the solid or call it B for body."},{"Start":"03:41.960 ","End":"03:48.175","Text":"The theorem on triple integrals and volume says that the volume,"},{"Start":"03:48.175 ","End":"03:49.435","Text":"I\u0027ll just write it in words,"},{"Start":"03:49.435 ","End":"03:55.630","Text":"the volume of the solid B in this setup is"},{"Start":"03:55.630 ","End":"04:02.960","Text":"equal to the triple integral over B of just 1dv."},{"Start":"04:03.140 ","End":"04:07.690","Text":"The idea is to describe the solid in such a way that this triple"},{"Start":"04:07.690 ","End":"04:11.770","Text":"integral becomes an iterated integral where x,"},{"Start":"04:11.770 ","End":"04:17.030","Text":"y, and z not necessarily in this order, x, y, and z."},{"Start":"04:17.030 ","End":"04:18.800","Text":"Then we have some limits,"},{"Start":"04:18.800 ","End":"04:21.850","Text":"and then it becomes dx, dy, dz."},{"Start":"04:21.850 ","End":"04:29.720","Text":"Now z is always the inner 1 because we take it between 2 surfaces an upper and a lower."},{"Start":"04:29.720 ","End":"04:32.585","Text":"In our case, we have 1 surface,"},{"Start":"04:32.585 ","End":"04:36.770","Text":"z equals 0, and that\u0027s 1 surface."},{"Start":"04:36.770 ","End":"04:37.790","Text":"But like I said,"},{"Start":"04:37.790 ","End":"04:40.475","Text":"this middle equation is going to do double duty."},{"Start":"04:40.475 ","End":"04:46.990","Text":"We can also use it to extract z in terms of x and y. I first multiply by 4."},{"Start":"04:46.990 ","End":"04:52.674","Text":"Then I would get from there that"},{"Start":"04:52.674 ","End":"05:01.430","Text":"x plus 2y plus z equals 4,"},{"Start":"05:01.430 ","End":"05:07.640","Text":"from which I would get that z is equal"},{"Start":"05:07.640 ","End":"05:16.245","Text":"to 4 minus x minus 2y."},{"Start":"05:16.245 ","End":"05:18.395","Text":"The other equation, like I said,"},{"Start":"05:18.395 ","End":"05:22.290","Text":"is the z equals 0."},{"Start":"05:22.290 ","End":"05:29.249","Text":"Now we have 2 surfaces and we have to check which is the upper and which is the lower."},{"Start":"05:30.610 ","End":"05:34.010","Text":"I\u0027d like to know in my domain,"},{"Start":"05:34.010 ","End":"05:36.770","Text":"if this is bigger or equal to 0,"},{"Start":"05:36.770 ","End":"05:39.874","Text":"or is it less than or equal to 0?"},{"Start":"05:39.874 ","End":"05:42.920","Text":"Now, if you set it equal to 0,"},{"Start":"05:42.920 ","End":"05:44.120","Text":"if you set z equals 0,"},{"Start":"05:44.120 ","End":"05:47.570","Text":"it\u0027s like setting z equal 0 on the original equation,"},{"Start":"05:47.570 ","End":"05:49.249","Text":"and that\u0027s this line."},{"Start":"05:49.249 ","End":"05:53.690","Text":"On this line, 4 minus x minus 2y equals 0,"},{"Start":"05:53.690 ","End":"05:57.885","Text":"it\u0027s essentially the same as this just rearranged."},{"Start":"05:57.885 ","End":"06:02.045","Text":"On one side it\u0027s going to be positive and on one side it\u0027s negative."},{"Start":"06:02.045 ","End":"06:05.150","Text":"What we have to do is pick a point in our region D,"},{"Start":"06:05.150 ","End":"06:07.400","Text":"which is all on one side of the line."},{"Start":"06:07.400 ","End":"06:09.785","Text":"For example, I could pick the origin."},{"Start":"06:09.785 ","End":"06:12.350","Text":"If I plug the origin in x equals 0,"},{"Start":"06:12.350 ","End":"06:17.030","Text":"y equals 0, I get that 4 minus 0 minus 0."},{"Start":"06:17.030 ","End":"06:19.460","Text":"This is positive and this is 0."},{"Start":"06:19.460 ","End":"06:26.130","Text":"This is the upper region and this is the lower."},{"Start":"06:26.130 ","End":"06:29.510","Text":"This one corresponds to the G in this picture,"},{"Start":"06:29.510 ","End":"06:39.230","Text":"and this one corresponds to the upper is the F. Now I can start completing this."},{"Start":"06:39.230 ","End":"06:47.780","Text":"The upper and the lower will tell me the limits for z. I\u0027ll know that z goes from the"},{"Start":"06:47.780 ","End":"06:55.935","Text":"lower to the upper 4 minus x minus 2y of 1."},{"Start":"06:55.935 ","End":"06:58.860","Text":"Now, I\u0027ll need the outer one,"},{"Start":"06:58.860 ","End":"07:01.140","Text":"the inner one will be dz."},{"Start":"07:01.140 ","End":"07:03.610","Text":"Then I have to do these 2."},{"Start":"07:03.610 ","End":"07:06.770","Text":"But up to now we\u0027ve been doing it always as x,"},{"Start":"07:06.770 ","End":"07:09.800","Text":"y but there\u0027s no reason why it couldn\u0027t be y, x."},{"Start":"07:09.800 ","End":"07:13.985","Text":"Essentially, it depends on whether we want to slice this horizontally or vertically,"},{"Start":"07:13.985 ","End":"07:17.130","Text":"type 2 region or type 1 region."},{"Start":"07:17.130 ","End":"07:20.630","Text":"Just looking at it, it will be easier to slice it horizontally,"},{"Start":"07:20.630 ","End":"07:23.254","Text":"because otherwise we\u0027d have to divide into 2 cases,"},{"Start":"07:23.254 ","End":"07:25.430","Text":"the left of 2 and to the right of 2."},{"Start":"07:25.430 ","End":"07:31.170","Text":"Let\u0027s take it as a type 2 region where if I have"},{"Start":"07:31.170 ","End":"07:39.320","Text":"a typical say y here and I slice through the region corresponding to this y,"},{"Start":"07:39.320 ","End":"07:41.650","Text":"it hits a 2 points."},{"Start":"07:41.650 ","End":"07:45.275","Text":"These 2 points, I can compute what they are."},{"Start":"07:45.275 ","End":"07:49.670","Text":"Because what I need to do is take these 2 equations and put x in terms of y."},{"Start":"07:49.670 ","End":"07:53.260","Text":"Now this one is already x in terms of y."},{"Start":"07:53.260 ","End":"07:57.710","Text":"All I need is to put this line in the form x equals something."},{"Start":"07:57.710 ","End":"07:59.570","Text":"If you do a little bit of manipulation,"},{"Start":"07:59.570 ","End":"08:04.580","Text":"you\u0027ll get that this is x equals 4 minus 2y."},{"Start":"08:04.580 ","End":"08:10.535","Text":"Just multiply everything by 4 and bring the y to the other side."},{"Start":"08:10.535 ","End":"08:13.315","Text":"I have these points here."},{"Start":"08:13.315 ","End":"08:18.135","Text":"Now I can write that y goes,"},{"Start":"08:18.135 ","End":"08:21.810","Text":"I can see it goes in this region,"},{"Start":"08:21.810 ","End":"08:28.200","Text":"from 1 to minus 2 or the other way round really,"},{"Start":"08:28.200 ","End":"08:31.535","Text":"y goes, I have to switch the order here,"},{"Start":"08:31.535 ","End":"08:33.620","Text":"this will happen sometimes."},{"Start":"08:33.620 ","End":"08:35.915","Text":"The outer loop is y,"},{"Start":"08:35.915 ","End":"08:40.245","Text":"which is from minus 2 to 1."},{"Start":"08:40.245 ","End":"08:47.355","Text":"For each such y, x goes from the parabola,"},{"Start":"08:47.355 ","End":"08:51.130","Text":"which is 2y squared"},{"Start":"08:54.170 ","End":"09:00.600","Text":"up to 4 minus 2y."},{"Start":"09:01.330 ","End":"09:03.620","Text":"I hope this is legible."},{"Start":"09:03.620 ","End":"09:06.800","Text":"X goes from 2y squared like here,"},{"Start":"09:06.800 ","End":"09:09.770","Text":"up to 4 minus 2y like here."},{"Start":"09:09.770 ","End":"09:11.810","Text":"Now it\u0027s just technical."},{"Start":"09:11.810 ","End":"09:14.750","Text":"Now we just have to compute this integral."},{"Start":"09:14.750 ","End":"09:16.730","Text":"I almost forgot to write,"},{"Start":"09:16.730 ","End":"09:18.505","Text":"of course, dx, dy,"},{"Start":"09:18.505 ","End":"09:20.480","Text":"the outer one is the y,"},{"Start":"09:20.480 ","End":"09:23.880","Text":"and then the middle one is the x."},{"Start":"09:24.200 ","End":"09:28.260","Text":"We start from the inside That\u0027s the dz integral."},{"Start":"09:28.260 ","End":"09:30.250","Text":"We do that one first."},{"Start":"09:30.250 ","End":"09:32.150","Text":"The rest of it, I will just copy."},{"Start":"09:32.150 ","End":"09:37.460","Text":"I have the integral from y equals minus 2 to 1."},{"Start":"09:37.460 ","End":"09:45.405","Text":"Then the integral from x equals 2y squared to 4 minus 2y."},{"Start":"09:45.405 ","End":"09:47.300","Text":"This integral we could do in our heads,"},{"Start":"09:47.300 ","End":"09:51.845","Text":"the integral of 1, we said is always the upper limit minus the lower limit."},{"Start":"09:51.845 ","End":"09:56.280","Text":"It\u0027s just 4 minus x minus 2y."},{"Start":"09:56.280 ","End":"09:59.700","Text":"Now we still have dx, dy."},{"Start":"09:59.700 ","End":"10:03.120","Text":"Now we go 1 more layer inwards."},{"Start":"10:03.120 ","End":"10:09.420","Text":"Let\u0027s do now this integral, the dx integral."},{"Start":"10:09.420 ","End":"10:11.700","Text":"Lets see what we get here."},{"Start":"10:11.700 ","End":"10:14.510","Text":"I prefer to do this at the side."},{"Start":"10:14.510 ","End":"10:17.315","Text":"Let me do this one over here."},{"Start":"10:17.315 ","End":"10:21.110","Text":"The integral dx will be,"},{"Start":"10:21.110 ","End":"10:25.340","Text":"I can already I won\u0027t copy it I\u0027ll just start integrating straight away."},{"Start":"10:25.340 ","End":"10:32.160","Text":"With respect to x, I get 4x minus 1/2x squared."},{"Start":"10:32.330 ","End":"10:37.650","Text":"2y is 2xy, because remember x is the variable,"},{"Start":"10:37.650 ","End":"10:39.900","Text":"y is a constant here."},{"Start":"10:39.900 ","End":"10:46.160","Text":"We take it between the lower limit is x equals 2y squared."},{"Start":"10:46.160 ","End":"10:49.750","Text":"The upper limit, x equals 4 minus 2y."},{"Start":"10:49.750 ","End":"10:52.210","Text":"I\u0027m emphasizing writing the x equals."},{"Start":"10:52.210 ","End":"10:56.855","Text":"What this gives us, if I put the upper one first, for x,"},{"Start":"10:56.855 ","End":"11:05.805","Text":"I\u0027ve got 4 times x is 4 minus 2y minus 1/2,"},{"Start":"11:05.805 ","End":"11:12.045","Text":"x squared is 4 minus 2y squared."},{"Start":"11:12.045 ","End":"11:16.605","Text":"Then minus 2xy minus 2x"},{"Start":"11:16.605 ","End":"11:24.525","Text":"and x is 4 minus 2y."},{"Start":"11:24.525 ","End":"11:27.330","Text":"That\u0027s why all this,"},{"Start":"11:27.330 ","End":"11:31.440","Text":"I\u0027ll just stress, is the top limit,"},{"Start":"11:31.440 ","End":"11:33.015","Text":"the 4 minus 2y,"},{"Start":"11:33.015 ","End":"11:39.045","Text":"I have to subtract the other one with 2y squared so 4 times"},{"Start":"11:39.045 ","End":"11:48.340","Text":"2y squared minus 1/2 of 2y squared."},{"Start":"11:49.080 ","End":"11:59.545","Text":"Then minus 2x is 2y squared, y close brackets."},{"Start":"11:59.545 ","End":"12:01.825","Text":"This is going to be messy."},{"Start":"12:01.825 ","End":"12:07.090","Text":"Let\u0027s try and be careful so let\u0027s start opening brackets from this one"},{"Start":"12:07.090 ","End":"12:15.110","Text":"I\u0027ve got 4 times 4 is 16 minus 8y."},{"Start":"12:15.180 ","End":"12:19.255","Text":"Now let\u0027s see if I square this one,"},{"Start":"12:19.255 ","End":"12:21.370","Text":"want to do this mentally."},{"Start":"12:21.370 ","End":"12:24.940","Text":"I\u0027m going to use in general,"},{"Start":"12:24.940 ","End":"12:27.535","Text":"we have a minus b squared is"},{"Start":"12:27.535 ","End":"12:35.949","Text":"a squared minus 2ab plus b squared and if I do that here,"},{"Start":"12:35.949 ","End":"12:38.890","Text":"then the a squared is 16,"},{"Start":"12:38.890 ","End":"12:47.710","Text":"but there\u0027s always a minus 1/2 so it\u0027s 16 becomes minus 8 and then minus twice this times"},{"Start":"12:47.710 ","End":"12:53.230","Text":"this is minus 16y"},{"Start":"12:53.230 ","End":"12:59.170","Text":"but there\u0027s also a minus 1/2 so it\u0027s plus 8y."},{"Start":"12:59.170 ","End":"13:04.180","Text":"The last term, b squared is 4 y squared but again with"},{"Start":"13:04.180 ","End":"13:10.210","Text":"a minus 1/2 it\u0027s minus 2y squared so we\u0027re up to here."},{"Start":"13:10.210 ","End":"13:14.770","Text":"Now I have to take the minus 2y and multiply it by each of them minus"},{"Start":"13:14.770 ","End":"13:19.210","Text":"2y times 4 is minus 8y and minus"},{"Start":"13:19.210 ","End":"13:23.650","Text":"2y times minus 2y is plus 4y"},{"Start":"13:23.650 ","End":"13:28.975","Text":"squared and now we\u0027re just at the point where we\u0027ve done the first bit."},{"Start":"13:28.975 ","End":"13:31.705","Text":"Now let\u0027s do the second bit,"},{"Start":"13:31.705 ","End":"13:41.680","Text":"4 times 2y squared is 8y squared and then we"},{"Start":"13:41.680 ","End":"13:46.420","Text":"have 2y squared squared is 4y to the"},{"Start":"13:46.420 ","End":"13:52.585","Text":"fourth with the minus 1/2 is minus 2y to the fourth."},{"Start":"13:52.585 ","End":"14:00.160","Text":"In the last one we have minus 2 times 2 is 4y squared times y"},{"Start":"14:00.160 ","End":"14:08.005","Text":"is y cubed so this is what we get and now we want to collect like terms."},{"Start":"14:08.005 ","End":"14:12.370","Text":"Let\u0027s see maybe we\u0027ll do them in reverse order,"},{"Start":"14:12.370 ","End":"14:15.295","Text":"first the constants, constants here,"},{"Start":"14:15.295 ","End":"14:23.940","Text":"I have just 16 minus 8 is 8, that\u0027s the constants."},{"Start":"14:23.940 ","End":"14:30.935","Text":"Now let\u0027s look for y I have minus 8y plus 8y is"},{"Start":"14:30.935 ","End":"14:38.995","Text":"nothing and then I have another minus 8y so all together minus 8y,"},{"Start":"14:38.995 ","End":"14:41.380","Text":"there\u0027s no y in the second brackets."},{"Start":"14:41.380 ","End":"14:44.199","Text":"Now let\u0027s collect y squared terms,"},{"Start":"14:44.199 ","End":"14:48.055","Text":"I have minus 2y squared plus 4y squared,"},{"Start":"14:48.055 ","End":"14:56.275","Text":"that\u0027s 2y squared and here I have minus 8y squared so minus 6y squared."},{"Start":"14:56.275 ","End":"14:58.720","Text":"Next, let\u0027s look for y cubed."},{"Start":"14:58.720 ","End":"15:05.530","Text":"Nothing here, here I have y cubed it\u0027s minus and a minus 4,"},{"Start":"15:05.530 ","End":"15:13.270","Text":"so it\u0027s 4y cubed and lastly, y_4."},{"Start":"15:13.270 ","End":"15:19.970","Text":"I have from here and here, plus 2y_4."},{"Start":"15:22.050 ","End":"15:26.680","Text":"All this is this bit here so now we can"},{"Start":"15:26.680 ","End":"15:32.050","Text":"return and all these terms are even so why don\u0027t I take 2 outside"},{"Start":"15:32.050 ","End":"15:41.500","Text":"the brackets and then I\u0027ll get twice the integral from minus 2 to 1 of all this,"},{"Start":"15:41.500 ","End":"15:51.670","Text":"but divided by 2 so I\u0027ve got 4 minus 4y minus 3y squared"},{"Start":"15:51.670 ","End":"15:57.220","Text":"plus 2y cubed plus y to the"},{"Start":"15:57.220 ","End":"16:03.910","Text":"fourth and all this dy."},{"Start":"16:03.910 ","End":"16:12.280","Text":"Let\u0027s see, we\u0027ve got twice and then do the integral with 4 I get 4y minus here,"},{"Start":"16:12.280 ","End":"16:14.665","Text":"I get 2y squared."},{"Start":"16:14.665 ","End":"16:18.820","Text":"Here, I get minus y cubed,"},{"Start":"16:18.820 ","End":"16:22.090","Text":"here I raise it by 1 is 4,"},{"Start":"16:22.090 ","End":"16:29.690","Text":"2 over 4 1/2y_4, and next 1/5y_5."},{"Start":"16:31.500 ","End":"16:37.690","Text":"All this from minus 2 to 1."},{"Start":"16:37.690 ","End":"16:47.450","Text":"We\u0027re getting there so what I get is twice. Now let\u0027s see."},{"Start":"16:47.780 ","End":"16:50.900","Text":"If I put in 1,"},{"Start":"16:50.900 ","End":"16:56.385","Text":"I\u0027ve got 4 minus 2, minus 1,"},{"Start":"16:56.385 ","End":"17:04.525","Text":"plus 1/2, plus 1/5 and if I put in minus 2,"},{"Start":"17:04.525 ","End":"17:06.175","Text":"I\u0027ve got, let\u0027s see."},{"Start":"17:06.175 ","End":"17:16.150","Text":"Minus 8 minus 2 times 4 is minus 8 minus y cubed so that will become"},{"Start":"17:16.150 ","End":"17:23.430","Text":"a plus 8 and 1/2y_4 16 over"},{"Start":"17:23.430 ","End":"17:30.069","Text":"2 is plus 8 then 1/5 times 32,"},{"Start":"17:30.069 ","End":"17:33.745","Text":"or just leave it as 32 over"},{"Start":"17:33.745 ","End":"17:41.200","Text":"5 and then this equals twice."},{"Start":"17:41.200 ","End":"17:48.955","Text":"The first brackets gives me 4 minus 2 minus 1 is just 1."},{"Start":"17:48.955 ","End":"17:52.345","Text":"1/2 and 1/5 probably best to do it in decimals,"},{"Start":"17:52.345 ","End":"18:00.890","Text":"that\u0027s 0.5 and that\u0027s 0.2 put together 0.7 so that\u0027s 1 and let us do it in decimals,"},{"Start":"18:02.100 ","End":"18:07.540","Text":"1.7 or 1 and 7 10ths and then here,"},{"Start":"18:07.540 ","End":"18:10.465","Text":"minus 8, minus 8 plus 8 plus 8,"},{"Start":"18:10.465 ","End":"18:20.150","Text":"all I\u0027m left with is the 32 over 5, which is 6.4."},{"Start":"18:26.790 ","End":"18:29.439","Text":"Just noted there is tiny mistake,"},{"Start":"18:29.439 ","End":"18:34.180","Text":"this one is actually a minus because minus 2_ 5 is"},{"Start":"18:34.180 ","End":"18:40.060","Text":"minus and that means that this is a plus so let\u0027s see,"},{"Start":"18:40.060 ","End":"18:44.665","Text":"1.7 plus 6.4 will be,"},{"Start":"18:44.665 ","End":"18:52.730","Text":"see that\u0027s 8.1 twice 8.1 is going to be 16.2."},{"Start":"18:53.880 ","End":"18:59.800","Text":"This is the answer and for those who like fractions,"},{"Start":"18:59.800 ","End":"19:02.710","Text":"16 and 1/5 if you want,"},{"Start":"19:02.710 ","End":"19:05.590","Text":"it\u0027s also equal to 81 over 5."},{"Start":"19:05.590 ","End":"19:08.870","Text":"Take your pick. We are done."}],"ID":8735},{"Watched":false,"Name":"Exercise 3 part e","Duration":"7m 38s","ChapterTopicVideoID":8528,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8528.jpeg","UploadDate":"2017-02-13T03:08:29.9100000","DurationForVideoObject":"PT7M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, you have to compute the volume of the solid"},{"Start":"00:03.060 ","End":"00:06.285","Text":"bounded by this just 2 surfaces."},{"Start":"00:06.285 ","End":"00:13.725","Text":"This one is actually a cylindrical ellipse or it is an elliptical cylinder,"},{"Start":"00:13.725 ","End":"00:17.370","Text":"and this is a plane and I brought"},{"Start":"00:17.370 ","End":"00:23.330","Text":"my usual picture that I like to use when we project onto the xy plane,"},{"Start":"00:23.330 ","End":"00:26.900","Text":"but I\u0027d like to call the solid body B"},{"Start":"00:26.900 ","End":"00:31.100","Text":"rather than V. Then we have the usual formula that the volume of B,"},{"Start":"00:31.100 ","End":"00:35.415","Text":"is the triple integral over B of 1 dv."},{"Start":"00:35.415 ","End":"00:42.680","Text":"We then have to try and get this into an iterated integral as integral of x something,"},{"Start":"00:42.680 ","End":"00:44.795","Text":"integral of y something,"},{"Start":"00:44.795 ","End":"00:46.880","Text":"integral of z something,"},{"Start":"00:46.880 ","End":"00:49.860","Text":"and that\u0027s going to be our task."},{"Start":"00:50.500 ","End":"00:55.130","Text":"Now there was a similar exercise to this in the chapter on double integrals,"},{"Start":"00:55.130 ","End":"00:59.330","Text":"and I\u0027d like to recycle the sketch I made there."},{"Start":"00:59.330 ","End":"01:01.090","Text":"I\u0027m going to put it over here."},{"Start":"01:01.090 ","End":"01:04.345","Text":"First. I put the sketch. Now I\u0027ll explain it."},{"Start":"01:04.345 ","End":"01:08.060","Text":"This is the ellipse, and in general,"},{"Start":"01:08.060 ","End":"01:11.640","Text":"the equation of an ellipse in 2 dimensions,"},{"Start":"01:11.640 ","End":"01:17.345","Text":"x squared over a squared plus y squared over b squared equals 1."},{"Start":"01:17.345 ","End":"01:22.070","Text":"A is the point here and here where plus"},{"Start":"01:22.070 ","End":"01:26.825","Text":"or minus a is where it cuts the x-axis and plus or minus b for the y-axis."},{"Start":"01:26.825 ","End":"01:31.405","Text":"This is like over 1 squared and this 4 is 2 squared,"},{"Start":"01:31.405 ","End":"01:36.030","Text":"so we get the plus or minus 2 and the plus or minus 1."},{"Start":"01:36.030 ","End":"01:38.400","Text":"That explains the ellipse."},{"Start":"01:38.400 ","End":"01:40.830","Text":"That\u0027s where this solid,"},{"Start":"01:40.830 ","End":"01:44.675","Text":"if we project it onto the xy plane, this is what we get."},{"Start":"01:44.675 ","End":"01:47.705","Text":"Now, why have I only shaded half of it?"},{"Start":"01:47.705 ","End":"01:51.515","Text":"That\u0027s because of this."},{"Start":"01:51.515 ","End":"01:57.545","Text":"This z equals y is actually a plane that cuts the xy plane along the x-axis."},{"Start":"01:57.545 ","End":"01:59.240","Text":"Because when z is 0,"},{"Start":"01:59.240 ","End":"02:01.265","Text":"y is 0 and that\u0027s the x-axis,"},{"Start":"02:01.265 ","End":"02:05.735","Text":"and on 1 side it\u0027s going to be bigger or equal to 0 or less than or equal to 0."},{"Start":"02:05.735 ","End":"02:09.110","Text":"Since z equals y and z is bigger or equal to 0,"},{"Start":"02:09.110 ","End":"02:11.240","Text":"then y is bigger or equal to 0,"},{"Start":"02:11.240 ","End":"02:17.645","Text":"so it\u0027s going to be this half of the plane is where y is bigger or equal to 0,"},{"Start":"02:17.645 ","End":"02:21.020","Text":"and intersecting with the ellipse gives us this"},{"Start":"02:21.020 ","End":"02:24.935","Text":"D. That explains the picture and the shading."},{"Start":"02:24.935 ","End":"02:26.675","Text":"Now what about the rest of it?"},{"Start":"02:26.675 ","End":"02:29.820","Text":"Well, we have 2 surfaces."},{"Start":"02:29.950 ","End":"02:36.630","Text":"The z bigger or equal to 0 is like y bigger or equal to 0."},{"Start":"02:37.150 ","End":"02:41.430","Text":"That explains this line here."},{"Start":"02:43.030 ","End":"02:49.385","Text":"This part of the curve here is when I extract from here."},{"Start":"02:49.385 ","End":"02:52.820","Text":"If I isolate from here y, first of all,"},{"Start":"02:52.820 ","End":"03:00.050","Text":"I would get y squared over 4 equals 1 minus x squared,"},{"Start":"03:00.050 ","End":"03:07.190","Text":"and then multiply by 4 and get y squared equals 4 times 1 minus x squared."},{"Start":"03:07.190 ","End":"03:09.185","Text":"Then when you take the square root,"},{"Start":"03:09.185 ","End":"03:14.570","Text":"you would get that y normally plus or minus square root of 4."},{"Start":"03:14.570 ","End":"03:19.325","Text":"But only get the plus square root."},{"Start":"03:19.325 ","End":"03:23.045","Text":"So it\u0027ll be twice square root of 1 minus x squared,"},{"Start":"03:23.045 ","End":"03:24.905","Text":"and that explains this."},{"Start":"03:24.905 ","End":"03:27.740","Text":"We\u0027re going to take it as a type 1 region."},{"Start":"03:27.740 ","End":"03:31.520","Text":"We take slices that x goes from minus 1 to 1."},{"Start":"03:31.520 ","End":"03:32.905","Text":"I can already write that,"},{"Start":"03:32.905 ","End":"03:38.940","Text":"x goes from minus 1 to 1,"},{"Start":"03:38.940 ","End":"03:42.515","Text":"and then y goes from this line here,"},{"Start":"03:42.515 ","End":"03:45.485","Text":"which is y equals 0 up to here,"},{"Start":"03:45.485 ","End":"03:50.400","Text":"which is twice the square root of 1 minus x squared."},{"Start":"03:50.400 ","End":"03:54.735","Text":"As for z, it has to go between 2 surfaces."},{"Start":"03:54.735 ","End":"03:56.775","Text":"Now we have the 2 surfaces."},{"Start":"03:56.775 ","End":"04:07.830","Text":"1 of the surfaces is z equals 0 and the other surface is z equals y."},{"Start":"04:08.830 ","End":"04:12.980","Text":"Now the upper 1 is this 1."},{"Start":"04:12.980 ","End":"04:16.325","Text":"Because of y bigger or equal to 0,"},{"Start":"04:16.325 ","End":"04:18.740","Text":"because z is y and z is bigger or equal to 0."},{"Start":"04:18.740 ","End":"04:20.930","Text":"This is the upper and the lower.,"},{"Start":"04:20.930 ","End":"04:22.760","Text":"and when we have this case,"},{"Start":"04:22.760 ","End":"04:25.130","Text":"we put the upper 1 here."},{"Start":"04:25.130 ","End":"04:30.359","Text":"Z goes from 0 to y of 1,"},{"Start":"04:30.359 ","End":"04:32.690","Text":"and now we have to put them in the right order."},{"Start":"04:32.690 ","End":"04:34.895","Text":"First dz to close this,"},{"Start":"04:34.895 ","End":"04:39.320","Text":"then dy, and then dx."},{"Start":"04:39.320 ","End":"04:44.215","Text":"From here on, we don\u0027t need any sketches, It\u0027s just computational."},{"Start":"04:44.215 ","End":"04:47.220","Text":"We start with the inner 1."},{"Start":"04:47.220 ","End":"04:54.080","Text":"That\u0027s easy because the integral of 1 is always the upper minus the lower."},{"Start":"04:54.080 ","End":"04:56.420","Text":"I can just copy the rest of it."},{"Start":"04:56.420 ","End":"05:00.729","Text":"X goes from minus 1 to 1,"},{"Start":"05:00.729 ","End":"05:08.795","Text":"y goes from 0 to twice root 1 minus x squared."},{"Start":"05:08.795 ","End":"05:11.615","Text":"This thing becomes y minus 0,"},{"Start":"05:11.615 ","End":"05:13.700","Text":"which is just y."},{"Start":"05:13.700 ","End":"05:17.330","Text":"Then dy dx. Once again,"},{"Start":"05:17.330 ","End":"05:18.980","Text":"we do the inner 1."},{"Start":"05:18.980 ","End":"05:23.315","Text":"This time it\u0027s the dy integral."},{"Start":"05:23.315 ","End":"05:27.190","Text":"Let me just do this as a side exercise over here."},{"Start":"05:27.190 ","End":"05:34.090","Text":"What we get, the integral of y dy is 1/2y squared,"},{"Start":"05:34.090 ","End":"05:40.430","Text":"and we need to take this from y equals 0 up"},{"Start":"05:40.430 ","End":"05:47.809","Text":"to y equals twice square root of 1 minus x squared."},{"Start":"05:47.809 ","End":"05:52.650","Text":"This will equal, if I put the upper limit,"},{"Start":"05:54.170 ","End":"05:56.250","Text":"we\u0027ll actually have it here,"},{"Start":"05:56.250 ","End":"05:59.480","Text":"y squared is 4 times 1 minus x squared,"},{"Start":"05:59.480 ","End":"06:05.850","Text":"but the half makes it just twice 1 minus x squared."},{"Start":"06:05.850 ","End":"06:10.515","Text":"When I put in 0, I got the 0, I can write it."},{"Start":"06:10.515 ","End":"06:14.900","Text":"Now I got the answer to this part here."},{"Start":"06:14.900 ","End":"06:18.680","Text":"I\u0027m going to return with the twice 1 minus x squared,"},{"Start":"06:18.680 ","End":"06:23.820","Text":"and now we have the integral from minus 1 to 1 of twice,"},{"Start":"06:23.820 ","End":"06:26.595","Text":"I can put the 2 in front,"},{"Start":"06:26.595 ","End":"06:36.730","Text":"and I\u0027ve got 1 minus x squared dx from minus 1 to 1."},{"Start":"06:37.520 ","End":"06:40.205","Text":"Putting the equals here,"},{"Start":"06:40.205 ","End":"06:44.015","Text":"continuing, this is twice."},{"Start":"06:44.015 ","End":"06:48.740","Text":"Now the integral of this is x minus"},{"Start":"06:48.740 ","End":"06:57.930","Text":"1/3x cubed from minus 1 to 1."},{"Start":"06:57.930 ","End":"06:59.465","Text":"Let\u0027s see what I get."},{"Start":"06:59.465 ","End":"07:01.355","Text":"Twice I can take outside."},{"Start":"07:01.355 ","End":"07:03.365","Text":"Now, I plug in 1,"},{"Start":"07:03.365 ","End":"07:07.430","Text":"I get 1 minus 1/3."},{"Start":"07:07.430 ","End":"07:09.695","Text":"If I plug in minus 1,"},{"Start":"07:09.695 ","End":"07:13.980","Text":"I get minus 1 plus 1/3."},{"Start":"07:13.980 ","End":"07:24.150","Text":"Altogether, I\u0027ve got 2/3 minus minus 2/3 is 4/3."},{"Start":"07:24.150 ","End":"07:29.840","Text":"4/3 times 2 comes out to be 8/3."},{"Start":"07:29.840 ","End":"07:31.370","Text":"If you like it as a mixed fraction,"},{"Start":"07:31.370 ","End":"07:33.485","Text":"you could write it as 2 and 2/3,"},{"Start":"07:33.485 ","End":"07:37.010","Text":"but I prefer just to leave it like this."},{"Start":"07:37.010 ","End":"07:39.270","Text":"Anyway, we\u0027re d1."}],"ID":8736},{"Watched":false,"Name":"Exercise 3 part f","Duration":"8m 26s","ChapterTopicVideoID":8529,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8529.jpeg","UploadDate":"2017-02-13T03:09:40.9670000","DurationForVideoObject":"PT8M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Here\u0027s another 1 of these problems where we have to"},{"Start":"00:02.730 ","End":"00:06.450","Text":"compute the volume of a solid bounded by surfaces."},{"Start":"00:06.450 ","End":"00:10.050","Text":"I put my usual picture"},{"Start":"00:10.050 ","End":"00:14.850","Text":"where the solid is between an upper and a lower surface and it\u0027s projected onto"},{"Start":"00:14.850 ","End":"00:23.130","Text":"the x-y plane in a region D. I prefer to use the letter B for body or solid body."},{"Start":"00:23.130 ","End":"00:26.040","Text":"We know that the volume of B is the standard formula,"},{"Start":"00:26.040 ","End":"00:32.650","Text":"the triple integral of 1 over this region B, solid B."},{"Start":"00:33.710 ","End":"00:39.080","Text":"We had a very similar question in the chapter on double integrals,"},{"Start":"00:39.080 ","End":"00:43.885","Text":"and I\u0027d like to reuse the picture I did there."},{"Start":"00:43.885 ","End":"00:47.840","Text":"Here\u0027s the picture I brought in and now let me explain it."},{"Start":"00:47.840 ","End":"00:54.005","Text":"To find the projection of this body onto the x-y plane,"},{"Start":"00:54.005 ","End":"00:58.295","Text":"we needed some closed lines."},{"Start":"00:58.295 ","End":"01:01.460","Text":"If I just took x equals 0 and y equals 0,"},{"Start":"01:01.460 ","End":"01:03.455","Text":"that\u0027s not enough, it doesn\u0027t close."},{"Start":"01:03.455 ","End":"01:04.970","Text":"I need a third line."},{"Start":"01:04.970 ","End":"01:07.655","Text":"I want to show you where this third line comes from."},{"Start":"01:07.655 ","End":"01:10.760","Text":"We basically have 2 surfaces."},{"Start":"01:10.760 ","End":"01:14.734","Text":"1 of them is z equals x plus y,"},{"Start":"01:14.734 ","End":"01:18.335","Text":"and 1 of them is z equals 6."},{"Start":"01:18.335 ","End":"01:21.110","Text":"I don\u0027t know yet which is upper and which is lower,"},{"Start":"01:21.110 ","End":"01:25.410","Text":"but part of the boundary will be the intersection of these 2."},{"Start":"01:25.410 ","End":"01:33.860","Text":"The intersection of these 2 will be when x plus y equals 6 and that is this line here."},{"Start":"01:33.860 ","End":"01:35.945","Text":"I can do it by intercepts."},{"Start":"01:35.945 ","End":"01:39.015","Text":"When x is 0, y is 6,"},{"Start":"01:39.015 ","End":"01:40.980","Text":"and when y is 0, x is 6."},{"Start":"01:40.980 ","End":"01:46.155","Text":"I get these 2.06 and 60 and draw a line through them."},{"Start":"01:46.155 ","End":"01:49.350","Text":"That gives me the third 1 and then I shade the region"},{"Start":"01:49.350 ","End":"01:55.490","Text":"D. There\u0027s a couple of things I still have to decide."},{"Start":"01:55.490 ","End":"01:59.760","Text":"1 of them is which is upper and which is lower."},{"Start":"02:00.700 ","End":"02:04.240","Text":"This can be decided as follows."},{"Start":"02:04.240 ","End":"02:07.600","Text":"If this is the line x plus y equals 6,"},{"Start":"02:07.600 ","End":"02:10.360","Text":"then on 1 side it\u0027s going to be all bigger than 6,"},{"Start":"02:10.360 ","End":"02:12.145","Text":"and 1 side smaller than 6."},{"Start":"02:12.145 ","End":"02:16.000","Text":"What I do is pick some point in my domain."},{"Start":"02:16.000 ","End":"02:20.740","Text":"Let\u0027s say I could choose the point 0,0 which is in the domain,"},{"Start":"02:20.740 ","End":"02:23.285","Text":"and substitute it here and here."},{"Start":"02:23.285 ","End":"02:25.110","Text":"Here it\u0027s always 6."},{"Start":"02:25.110 ","End":"02:28.680","Text":"Here the origin x plus y is 0."},{"Start":"02:28.680 ","End":"02:31.500","Text":"This is less than this."},{"Start":"02:31.500 ","End":"02:35.770","Text":"This 1 is the upper surface,"},{"Start":"02:35.770 ","End":"02:40.750","Text":"and this 1 is the lower in the sketch,"},{"Start":"02:40.750 ","End":"02:50.050","Text":"the lower is g and the upper is f. I can already start writing this integral,"},{"Start":"02:50.050 ","End":"02:52.615","Text":"which I want to write as an iterated integral,"},{"Start":"02:52.615 ","End":"02:55.360","Text":"x, y, and z."},{"Start":"02:55.360 ","End":"02:57.460","Text":"Possibly I might have to change the order,"},{"Start":"02:57.460 ","End":"03:07.700","Text":"but I already know that z goes from the lower surface to the upper surface."},{"Start":"03:08.280 ","End":"03:11.589","Text":"The lower is x plus y,"},{"Start":"03:11.589 ","End":"03:19.665","Text":"and the upper is 6 and then it\u0027s going to be 1 here and then dz. What about x and y?"},{"Start":"03:19.665 ","End":"03:21.340","Text":"As you can see from the picture,"},{"Start":"03:21.340 ","End":"03:27.650","Text":"I\u0027ve decided to take it as a type 1 region with vertical slices."},{"Start":"03:28.070 ","End":"03:31.935","Text":"Here I have y equals 0 is the x-axis."},{"Start":"03:31.935 ","End":"03:38.555","Text":"Here, this equation came from here by isolating y as a function of x."},{"Start":"03:38.555 ","End":"03:41.300","Text":"As I go through the domain,"},{"Start":"03:41.300 ","End":"03:43.765","Text":"x goes from 0 to 6,"},{"Start":"03:43.765 ","End":"03:48.535","Text":"which I can write here and y goes from"},{"Start":"03:48.535 ","End":"03:56.420","Text":"0 up to 6 minus x. I can write this 0 to 6 minus x."},{"Start":"03:56.420 ","End":"04:01.970","Text":"We do the inside integral first, the dz."},{"Start":"04:01.970 ","End":"04:10.920","Text":"Oh, I forgot to write here the dy to close this y and the dx for the x."},{"Start":"04:10.920 ","End":"04:17.825","Text":"This integral is easy because the integral of 1 is always the upper minus the lower."},{"Start":"04:17.825 ","End":"04:24.664","Text":"This part is just 6 minus x minus y is the answer to this integral,"},{"Start":"04:24.664 ","End":"04:26.210","Text":"I\u0027ll put it in brackets."},{"Start":"04:26.210 ","End":"04:32.525","Text":"I still need this dydx, same limits."},{"Start":"04:32.525 ","End":"04:36.005","Text":"Y goes from 0 to 6 minus x,"},{"Start":"04:36.005 ","End":"04:39.740","Text":"and x goes from 0 to 6."},{"Start":"04:39.740 ","End":"04:41.855","Text":"The inner 1 is this,"},{"Start":"04:41.855 ","End":"04:46.400","Text":"the dy integral here and here."},{"Start":"04:46.400 ","End":"04:49.700","Text":"Let me do this integral at the side here."},{"Start":"04:49.700 ","End":"04:53.240","Text":"What I get, if I do the integral of this dy,"},{"Start":"04:53.240 ","End":"04:58.145","Text":"it becomes 6y minus xy,"},{"Start":"04:58.145 ","End":"05:00.620","Text":"y is the variable,"},{"Start":"05:00.620 ","End":"05:05.375","Text":"so x is a constant and then minus a 0.5y squared."},{"Start":"05:05.375 ","End":"05:15.150","Text":"All this taken from y equals 0 up to y equals 6 minus x."},{"Start":"05:18.610 ","End":"05:27.710","Text":"We first do the 6 minus x and so we get 6 times 6 minus x,"},{"Start":"05:27.710 ","End":"05:33.830","Text":"minus x times 6 minus x,"},{"Start":"05:34.570 ","End":"05:42.135","Text":"and then minus 0.5 6 minus x squared."},{"Start":"05:42.135 ","End":"05:44.915","Text":"When I plug in y equals 0,"},{"Start":"05:44.915 ","End":"05:47.625","Text":"I just get 0."},{"Start":"05:47.625 ","End":"05:50.595","Text":"We\u0027ll just add a minus 0 here."},{"Start":"05:50.595 ","End":"05:52.805","Text":"Let\u0027s open up the brackets."},{"Start":"05:52.805 ","End":"05:57.965","Text":"I\u0027ve got 36 minus 6x"},{"Start":"05:57.965 ","End":"06:04.180","Text":"minus another 6x plus x squared."},{"Start":"06:04.180 ","End":"06:07.880","Text":"Let\u0027s see, 6 minus x squared."},{"Start":"06:07.880 ","End":"06:11.990","Text":"I\u0027ll do using the special binomial expansion of a minus b"},{"Start":"06:11.990 ","End":"06:16.475","Text":"squared is a squared minus 2ab plus b squared."},{"Start":"06:16.475 ","End":"06:18.200","Text":"In our case, let\u0027s see,"},{"Start":"06:18.200 ","End":"06:20.470","Text":"6 squared is 36,"},{"Start":"06:20.470 ","End":"06:21.780","Text":"but it\u0027s a minus 0.5,"},{"Start":"06:21.780 ","End":"06:23.790","Text":"so it\u0027s minus 18."},{"Start":"06:23.790 ","End":"06:29.775","Text":"Then minus 2 times 6 times x minus 12x times minus 0.5,"},{"Start":"06:29.775 ","End":"06:37.905","Text":"so it\u0027s plus 6x and then I get minus 0.5x squared."},{"Start":"06:37.905 ","End":"06:41.875","Text":"If I collect like terms together,"},{"Start":"06:41.875 ","End":"06:47.180","Text":"let\u0027s see x squared minus 0.5x squared is 0.5x squared."},{"Start":"06:47.180 ","End":"06:56.340","Text":"For x\u0027s, I get minus 6 minus 6 plus 6 altogether minus 6x and constants,"},{"Start":"06:56.340 ","End":"07:07.260","Text":"let\u0027s see 36 minus 18 is plus 18 and I can put this here."},{"Start":"07:07.260 ","End":"07:13.650","Text":"Going back here, we get the integral from 0 to"},{"Start":"07:13.650 ","End":"07:23.080","Text":"6 of 0.5 x squared minus 6x plus 18dx,"},{"Start":"07:23.210 ","End":"07:28.394","Text":"which is equal to 0.5x squared."},{"Start":"07:28.394 ","End":"07:36.325","Text":"The integral of that 1/6 x cubed,"},{"Start":"07:36.325 ","End":"07:41.000","Text":"x becomes x squared divided by 2 and minus 3x"},{"Start":"07:41.000 ","End":"07:48.895","Text":"squared plus 18x all this from 0 to 6."},{"Start":"07:48.895 ","End":"07:51.450","Text":"When I plug in 0, I get nothing."},{"Start":"07:51.450 ","End":"07:53.820","Text":"I just have to plug in the 6,"},{"Start":"07:53.820 ","End":"07:57.660","Text":"1/6 times 6 cubed it\u0027s just like 6 squared,"},{"Start":"07:57.660 ","End":"08:06.460","Text":"so this is 36, 3 times 36 is 108."},{"Start":"08:06.710 ","End":"08:15.795","Text":"18 times 6 turns out also to be 108."},{"Start":"08:15.795 ","End":"08:18.195","Text":"The 108 cancels."},{"Start":"08:18.195 ","End":"08:21.485","Text":"The answer is just 36."},{"Start":"08:21.485 ","End":"08:25.890","Text":"I\u0027ll just highlight that and declare that we are done."}],"ID":8737},{"Watched":false,"Name":"Exercise 4 part 1","Duration":"7m 12s","ChapterTopicVideoID":8530,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8530.jpeg","UploadDate":"2017-02-13T03:10:46.7830000","DurationForVideoObject":"PT7M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we have to compute the mass and the center of"},{"Start":"00:04.530 ","End":"00:08.940","Text":"mass of a cylinder whose height is h. I"},{"Start":"00:08.940 ","End":"00:13.080","Text":"brought in a picture of a cylinder from somewhere so let\u0027s just indicate that"},{"Start":"00:13.080 ","End":"00:17.835","Text":"the height is h. It\u0027s not an infinite cylinder,"},{"Start":"00:17.835 ","End":"00:21.149","Text":"z goes from 0 to h,"},{"Start":"00:21.149 ","End":"00:25.380","Text":"the equation of a cylinder is here that\u0027s centered on the z-axis,"},{"Start":"00:25.380 ","End":"00:26.760","Text":"okay, so back here."},{"Start":"00:26.760 ","End":"00:33.765","Text":"The base has a radius r and we\u0027re given the density function, or more specifically,"},{"Start":"00:33.765 ","End":"00:38.100","Text":"we\u0027re given the density function delta and it\u0027s"},{"Start":"00:38.100 ","End":"00:43.475","Text":"proportional to the height of a point from the base."},{"Start":"00:43.475 ","End":"00:46.865","Text":"Proportional means some constant times,"},{"Start":"00:46.865 ","End":"00:49.765","Text":"positive constant time z."},{"Start":"00:49.765 ","End":"00:53.000","Text":"What we need, and we\u0027re going to deal with the mass first,"},{"Start":"00:53.000 ","End":"00:55.910","Text":"later we\u0027ll talk about center of mass, is the equation."},{"Start":"00:55.910 ","End":"00:58.430","Text":"You don\u0027t have to know physics, you just have to know the equation."},{"Start":"00:58.430 ","End":"01:00.395","Text":"The equation of the mass,"},{"Start":"01:00.395 ","End":"01:06.375","Text":"let\u0027s call that M, is the triple integral over the body,"},{"Start":"01:06.375 ","End":"01:09.260","Text":"here it\u0027s a cylinder but let\u0027s just say in general,"},{"Start":"01:09.260 ","End":"01:11.420","Text":"the body is b,"},{"Start":"01:11.420 ","End":"01:16.700","Text":"the solid, over b of the density function."},{"Start":"01:16.700 ","End":"01:18.230","Text":"That\u0027s the function of x, y,"},{"Start":"01:18.230 ","End":"01:20.209","Text":"and z dv."},{"Start":"01:20.209 ","End":"01:21.969","Text":"That\u0027s the formula."},{"Start":"01:21.969 ","End":"01:29.360","Text":"Now we want to write this b in such a way that we have an iterated integral,"},{"Start":"01:29.360 ","End":"01:33.665","Text":"like the integral of x from something to something,"},{"Start":"01:33.665 ","End":"01:38.780","Text":"integral of y, integral of z, and so on."},{"Start":"01:38.780 ","End":"01:42.010","Text":"Let\u0027s analyze this."},{"Start":"01:42.010 ","End":"01:46.325","Text":"What would help is to get a sketch of the base."},{"Start":"01:46.325 ","End":"01:48.590","Text":"Let\u0027s say that this is the base."},{"Start":"01:48.590 ","End":"01:51.830","Text":"I hope this shading won\u0027t confuse,"},{"Start":"01:51.830 ","End":"01:56.650","Text":"that this is a domain and this is in the x, y plane."},{"Start":"01:56.650 ","End":"01:58.990","Text":"Here I\u0027ve brought in a sketch."},{"Start":"01:58.990 ","End":"02:01.705","Text":"This is going to be our D, that\u0027s the base."},{"Start":"02:01.705 ","End":"02:10.750","Text":"This is r and this is r and so now we can start writing this integral."},{"Start":"02:10.750 ","End":"02:17.410","Text":"The easiest thing is the z. z goes from 0 to h, always,"},{"Start":"02:17.410 ","End":"02:24.160","Text":"everywhere along D. Whenever x and y is in this disk,"},{"Start":"02:24.160 ","End":"02:34.485","Text":"the z goes from 0 to h so I can write 0 to h and also copy this delta of x, y, and z."},{"Start":"02:34.485 ","End":"02:37.820","Text":"In fact, why don\u0027t I just already put it as kz."},{"Start":"02:37.820 ","End":"02:43.770","Text":"So kz, and then I have dz."},{"Start":"02:43.770 ","End":"02:46.390","Text":"Now let\u0027s see about the x and the y,"},{"Start":"02:46.390 ","End":"02:51.769","Text":"basically I have to describe this region in the x, y plane."},{"Start":"02:51.769 ","End":"02:54.170","Text":"I can do it as a type 1 or type 2."},{"Start":"02:54.170 ","End":"03:04.205","Text":"Let\u0027s take vertical slices so x will go from minus r to r and if I take a typical x,"},{"Start":"03:04.205 ","End":"03:07.310","Text":"let\u0027s say here, and I want a vertical slice,"},{"Start":"03:07.310 ","End":"03:09.755","Text":"I\u0027ll just try drawing that,"},{"Start":"03:09.755 ","End":"03:16.055","Text":"what I need to know is that where this thing enters and where it leaves."},{"Start":"03:16.055 ","End":"03:22.400","Text":"Actually, the circle that\u0027s the border of the region,"},{"Start":"03:22.400 ","End":"03:24.050","Text":"is made up of 2 functions,"},{"Start":"03:24.050 ","End":"03:27.320","Text":"the upper semicircle and the lower semicircle."},{"Start":"03:27.320 ","End":"03:28.760","Text":"If I take this equation,"},{"Start":"03:28.760 ","End":"03:30.890","Text":"x squared plus y squared equals r squared,"},{"Start":"03:30.890 ","End":"03:39.180","Text":"I can write it as y squared equals r squared minus x"},{"Start":"03:39.180 ","End":"03:47.975","Text":"squared and then y equals plus or minus the square root of r squared minus x squared."},{"Start":"03:47.975 ","End":"03:50.900","Text":"Where of course, the plus is for"},{"Start":"03:50.900 ","End":"03:56.720","Text":"the upper semicircle so here I have square root of r squared minus x"},{"Start":"03:56.720 ","End":"04:01.385","Text":"squared and the minus is for the lowest semicircle so here it\u0027s minus"},{"Start":"04:01.385 ","End":"04:07.460","Text":"the square root of r squared minus x squared and this is exactly what I write here."},{"Start":"04:07.460 ","End":"04:10.410","Text":"It\u0027s a bit cramped."},{"Start":"04:14.920 ","End":"04:19.190","Text":"Wait a minute, I\u0027ve got a bit more space there,"},{"Start":"04:19.190 ","End":"04:23.885","Text":"so minus the square root of r squared minus"},{"Start":"04:23.885 ","End":"04:29.645","Text":"x squared 2 plus the square root of r squared minus x squared,"},{"Start":"04:29.645 ","End":"04:35.645","Text":"and that\u0027s the mass and now I don\u0027t need the sketches anymore,"},{"Start":"04:35.645 ","End":"04:38.390","Text":"I can just do the computation."},{"Start":"04:38.390 ","End":"04:40.760","Text":"Oh, and I forgot to write, of course,"},{"Start":"04:40.760 ","End":"04:46.160","Text":"I need the dy to close the y and then the dx."},{"Start":"04:46.160 ","End":"04:49.205","Text":"We do the innermost 1 first."},{"Start":"04:49.205 ","End":"04:52.175","Text":"That\u0027s the dz integral."},{"Start":"04:52.175 ","End":"04:57.495","Text":"We\u0027ll start there and let me do this as a side exercise."},{"Start":"04:57.495 ","End":"05:05.660","Text":"What we have is the integral from 0 to h of kz, dz."},{"Start":"05:05.660 ","End":"05:08.855","Text":"This is just 1.5 kz"},{"Start":"05:08.855 ","End":"05:17.610","Text":"squared from 0 to h and when z is 0,"},{"Start":"05:17.610 ","End":"05:19.950","Text":"it gives me nothing, when z is h,"},{"Start":"05:19.950 ","End":"05:25.465","Text":"I get 1.5 kh squared."},{"Start":"05:25.465 ","End":"05:29.435","Text":"Since this thing evaluates to a constant,"},{"Start":"05:29.435 ","End":"05:31.055","Text":"instead of putting it in here,"},{"Start":"05:31.055 ","End":"05:38.605","Text":"I can bring the constant in front of the integral so I get 1.5 kh"},{"Start":"05:38.605 ","End":"05:43.265","Text":"squared times this double integral"},{"Start":"05:43.265 ","End":"05:49.760","Text":"of 1 dy dx."},{"Start":"05:49.760 ","End":"05:55.880","Text":"Now, I\u0027m deliberately not copying this because really this double integral"},{"Start":"05:55.880 ","End":"06:02.390","Text":"is just the integral over the domain D. I could just go back to saying D,"},{"Start":"06:02.390 ","End":"06:04.130","Text":"I have a reason for doing this,"},{"Start":"06:04.130 ","End":"06:08.810","Text":"because the integral of 1 over a region is just the area of"},{"Start":"06:08.810 ","End":"06:15.630","Text":"that region so this is equal to 1.5 kh squared and as I said,"},{"Start":"06:15.630 ","End":"06:24.260","Text":"because the 1, it\u0027s just the area of D. Write that symbolically."},{"Start":"06:24.260 ","End":"06:30.920","Text":"Now, I know the area of D because I know from basic geometry that the area of D,"},{"Start":"06:30.920 ","End":"06:38.295","Text":"because the formula pi r squared so I just get 1.5 kh"},{"Start":"06:38.295 ","End":"06:42.645","Text":"squared and everyone knows that\u0027s a circle of radius r is"},{"Start":"06:42.645 ","End":"06:48.080","Text":"pi r squared and so if I multiply it out,"},{"Start":"06:48.080 ","End":"06:53.030","Text":"I get that the mass is equal to"},{"Start":"06:53.030 ","End":"07:04.610","Text":"1.5 kh squared r squared pi."},{"Start":"07:04.610 ","End":"07:09.230","Text":"Anyway, this answers the first part about the mass."},{"Start":"07:09.230 ","End":"07:12.690","Text":"Now let\u0027s move on to the center of mass."}],"ID":8738},{"Watched":false,"Name":"Exercise 4 part 2","Duration":"17m 2s","ChapterTopicVideoID":8531,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8531.jpeg","UploadDate":"2017-03-29T03:11:47.9600000","DurationForVideoObject":"PT17M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"Next, you want to move on to the center of mass."},{"Start":"00:03.030 ","End":"00:07.230","Text":"I\u0027m going to erase some of the stuff I don\u0027t need. Let\u0027s see."},{"Start":"00:07.230 ","End":"00:08.520","Text":"I want to keep the result,"},{"Start":"00:08.520 ","End":"00:11.295","Text":"but not all the computations."},{"Start":"00:11.295 ","End":"00:15.120","Text":"Now I need to give you the formula for the center of mass."},{"Start":"00:15.120 ","End":"00:18.540","Text":"Well, let\u0027s give the center of mass some coordinates."},{"Start":"00:18.540 ","End":"00:21.690","Text":"I don\u0027t just want to use any x, y, and z."},{"Start":"00:21.690 ","End":"00:25.740","Text":"Let\u0027s call them x-bar, y-bar, and z-bar."},{"Start":"00:25.740 ","End":"00:29.040","Text":"The special x, y, z for the center of mass."},{"Start":"00:29.040 ","End":"00:31.879","Text":"Each coordinate has a separate formula."},{"Start":"00:31.879 ","End":"00:38.130","Text":"The x-coordinate has the formula that this is equal to 1/m,"},{"Start":"00:38.130 ","End":"00:43.130","Text":"where m is the mass that we found here times the triple"},{"Start":"00:43.130 ","End":"00:48.815","Text":"integral over B. I\u0027m giving it in general."},{"Start":"00:48.815 ","End":"00:50.510","Text":"It\u0027s very similar to this,"},{"Start":"00:50.510 ","End":"00:52.550","Text":"but we put an x in front of here,"},{"Start":"00:52.550 ","End":"00:57.275","Text":"so it\u0027s x times Delta of x, y,"},{"Start":"00:57.275 ","End":"01:08.020","Text":"z, dV, and y-bar and z-bar have similar formulas."},{"Start":"01:08.020 ","End":"01:16.145","Text":"The only difference for y-bar and z-bar is that here there\u0027s a y and here there is a z."},{"Start":"01:16.145 ","End":"01:21.684","Text":"Each of these is actually similar to the original integral."},{"Start":"01:21.684 ","End":"01:26.215","Text":"As you see, there\u0027s a 1/m in front and either an x or y or z."},{"Start":"01:26.215 ","End":"01:27.850","Text":"Let\u0027s take the first one,"},{"Start":"01:27.850 ","End":"01:32.050","Text":"x-bar and see if we can figure that one."},{"Start":"01:32.050 ","End":"01:36.195","Text":"I\u0027ll begin with x-bar equals."},{"Start":"01:36.195 ","End":"01:39.180","Text":"I already know about the shape B,"},{"Start":"01:39.180 ","End":"01:41.825","Text":"you have to write it as an iterated integral."},{"Start":"01:41.825 ","End":"01:44.240","Text":"This part is just going to be the same."},{"Start":"01:44.240 ","End":"01:53.600","Text":"It\u0027s going to be that x goes from minus r to r. Then y is going to go from minus root r"},{"Start":"01:53.600 ","End":"01:57.980","Text":"squared minus x squared to plus root r squared minus x"},{"Start":"01:57.980 ","End":"02:03.415","Text":"squared and z is going to go from 0-h."},{"Start":"02:03.415 ","End":"02:07.505","Text":"The k, I\u0027m going to put in front, it\u0027s a constant."},{"Start":"02:07.505 ","End":"02:12.740","Text":"Also, the 1/m is going to have k/m in front."},{"Start":"02:12.740 ","End":"02:19.830","Text":"Here, all I\u0027m left with is the x and the z from here."},{"Start":"02:19.830 ","End":"02:27.920","Text":"It\u0027s xz, and that\u0027s dz dy dx."},{"Start":"02:27.920 ","End":"02:33.584","Text":"I just added an extra x here and took the k out front,"},{"Start":"02:33.584 ","End":"02:37.900","Text":"and I used this formula and with the 1/m."},{"Start":"02:37.900 ","End":"02:41.685","Text":"Let\u0027s start doing it from inside out."},{"Start":"02:41.685 ","End":"02:47.590","Text":"First, we\u0027ll do the dz integral."},{"Start":"02:48.430 ","End":"02:53.520","Text":"Let me do this integral here at the side."},{"Start":"02:54.220 ","End":"03:01.790","Text":"Because it\u0027s dz, I get 1/2xz squared,"},{"Start":"03:01.790 ","End":"03:11.825","Text":"which is the integral of this taken from 0-h. That\u0027s z going from 0-h."},{"Start":"03:11.825 ","End":"03:19.735","Text":"If I put in h, I get 1/2xh squared."},{"Start":"03:19.735 ","End":"03:23.860","Text":"If I put in 0, I\u0027ll just get minus 0."},{"Start":"03:25.820 ","End":"03:29.600","Text":"I\u0027ll just copy the answer over here,"},{"Start":"03:29.600 ","End":"03:31.490","Text":"just so it\u0027s close at hand."},{"Start":"03:31.490 ","End":"03:36.845","Text":"I got 1/2xh squared here."},{"Start":"03:36.845 ","End":"03:41.735","Text":"What I want to do is also take the constants in front."},{"Start":"03:41.735 ","End":"03:47.030","Text":"What I get now is this k/m together with the 1/2h squared,"},{"Start":"03:47.030 ","End":"03:52.245","Text":"I get kh squared."},{"Start":"03:52.245 ","End":"03:56.830","Text":"The 2 goes in the denominator with the m. Then I\u0027ve"},{"Start":"03:56.830 ","End":"04:02.305","Text":"got the integral from minus minus r to r"},{"Start":"04:02.305 ","End":"04:09.000","Text":"and then the integral from"},{"Start":"04:09.000 ","End":"04:16.350","Text":"minus root r squared minus x squared to root r squared minus x squared,"},{"Start":"04:16.350 ","End":"04:18.045","Text":"this is y, this is x."},{"Start":"04:18.045 ","End":"04:23.160","Text":"What else? Not only could I take the 1/2h squared upfront,"},{"Start":"04:23.160 ","End":"04:25.620","Text":"but the x, it\u0027s not a constant,"},{"Start":"04:25.620 ","End":"04:27.525","Text":"I can\u0027t bring it all the way upfront."},{"Start":"04:27.525 ","End":"04:37.335","Text":"But if I\u0027m going to be doing an integral dy then I can bring the x in front of here,"},{"Start":"04:37.335 ","End":"04:40.050","Text":"that will maybe save me a little bit."},{"Start":"04:40.050 ","End":"04:47.330","Text":"Then I\u0027ll get just dy dx."},{"Start":"04:47.330 ","End":"04:48.935","Text":"There\u0027s nothing left."},{"Start":"04:48.935 ","End":"04:54.120","Text":"I could even write this is 1 just so there\u0027s something there."},{"Start":"04:55.520 ","End":"04:57.920","Text":"I hope you followed that."},{"Start":"04:57.920 ","End":"05:00.170","Text":"The x, which would have been here,"},{"Start":"05:00.170 ","End":"05:02.780","Text":"I brought in front because the next integral is going to be"},{"Start":"05:02.780 ","End":"05:06.740","Text":"dy and x is a constant as far as y goes."},{"Start":"05:06.740 ","End":"05:10.980","Text":"The next integral will be the dy integral,"},{"Start":"05:12.590 ","End":"05:19.430","Text":"but the integral of 1 is always the upper limit minus the lower limit."},{"Start":"05:19.430 ","End":"05:23.900","Text":"In general, if I have something minus its negative,"},{"Start":"05:23.900 ","End":"05:26.390","Text":"I just get twice this."},{"Start":"05:26.390 ","End":"05:34.400","Text":"This minus this is twice the square root of r squared minus x squared."},{"Start":"05:34.400 ","End":"05:37.745","Text":"In general, a minus minus a is 2a."},{"Start":"05:37.745 ","End":"05:40.205","Text":"I still get a dx."},{"Start":"05:40.205 ","End":"05:46.655","Text":"I still have the integral from"},{"Start":"05:46.655 ","End":"05:56.175","Text":"minus r to r. I need to put also an x here."},{"Start":"05:56.175 ","End":"06:03.630","Text":"I\u0027ll write the x here for the moment and then after,"},{"Start":"06:03.630 ","End":"06:10.125","Text":"copy the constant kh squared over 2m."},{"Start":"06:10.125 ","End":"06:14.670","Text":"But what I want to do is just bring this"},{"Start":"06:14.670 ","End":"06:20.000","Text":"2 upfront and then say that it will cancel with this 2,"},{"Start":"06:20.000 ","End":"06:25.970","Text":"so I get kh squared over m times the integral from"},{"Start":"06:25.970 ","End":"06:33.900","Text":"minus r to r of x root r squared minus x squared dx."},{"Start":"06:33.900 ","End":"06:40.100","Text":"I\u0027m now going to use a trick based on odd and even functions."},{"Start":"06:40.100 ","End":"06:42.530","Text":"I\u0027ll remind you what an odd function is."},{"Start":"06:42.530 ","End":"06:50.960","Text":"In general, if I have f of minus x equals minus f of x,"},{"Start":"06:50.960 ","End":"06:52.490","Text":"then it\u0027s an odd function."},{"Start":"06:52.490 ","End":"06:55.775","Text":"I\u0027m claiming that this function here,"},{"Start":"06:55.775 ","End":"06:57.890","Text":"if this is my f of x, it\u0027s odd."},{"Start":"06:57.890 ","End":"07:01.325","Text":"Because if I replace x by minus x,"},{"Start":"07:01.325 ","End":"07:03.860","Text":"this part doesn\u0027t change when I square it,"},{"Start":"07:03.860 ","End":"07:05.855","Text":"but this part becomes negative,"},{"Start":"07:05.855 ","End":"07:07.820","Text":"so this is an odd function."},{"Start":"07:07.820 ","End":"07:12.325","Text":"The formula is that if I have an integral over a symmetrical interval,"},{"Start":"07:12.325 ","End":"07:20.405","Text":"say from minus a to a of some odd function of f of x dx,"},{"Start":"07:20.405 ","End":"07:24.445","Text":"where f is odd,"},{"Start":"07:24.445 ","End":"07:29.680","Text":"then this is equal to 0."},{"Start":"07:30.290 ","End":"07:33.020","Text":"Actually after coming this far,"},{"Start":"07:33.020 ","End":"07:35.150","Text":"we can just say that because of this theorem,"},{"Start":"07:35.150 ","End":"07:37.325","Text":"this integral is 0,"},{"Start":"07:37.325 ","End":"07:41.790","Text":"and so this whole thing is equal to 0."},{"Start":"07:42.510 ","End":"07:46.465","Text":"The computation for y bar is going to be very similar."},{"Start":"07:46.465 ","End":"07:47.890","Text":"I could do it alongside,"},{"Start":"07:47.890 ","End":"07:50.245","Text":"so I\u0027m going to erase this."},{"Start":"07:50.245 ","End":"08:00.520","Text":"Here it goes, y bar is equal to k over m. Is pretty much the same integral"},{"Start":"08:00.520 ","End":"08:10.780","Text":"from minus r to r. Integral from minus root r squared minus x squared to root r"},{"Start":"08:10.780 ","End":"08:18.955","Text":"squared minus x squared integral from 0 to h. The difference is that"},{"Start":"08:18.955 ","End":"08:21.365","Text":"here it will be at"},{"Start":"08:21.365 ","End":"08:30.700","Text":"yz dz dy dx."},{"Start":"08:30.700 ","End":"08:35.450","Text":"That was before we start off with the dz integral."},{"Start":"08:36.240 ","End":"08:40.420","Text":"Just like x was a constant as far as z goes so is y."},{"Start":"08:40.420 ","End":"08:41.680","Text":"We\u0027re going to get the same thing."},{"Start":"08:41.680 ","End":"08:47.350","Text":"Instead of 1/2 xh squared we\u0027re going to get 1/2 yh squared,"},{"Start":"08:47.350 ","End":"08:49.450","Text":"and I\u0027ll just save that step."},{"Start":"08:49.450 ","End":"08:55.360","Text":"Only this time, I can\u0027t bring the x in front because the next step is"},{"Start":"08:55.360 ","End":"09:02.410","Text":"dy so what I get is that this equals and still bring the constants out in front."},{"Start":"09:02.410 ","End":"09:08.680","Text":"I\u0027ve still got kh squared over 2m."},{"Start":"09:08.680 ","End":"09:14.755","Text":"This time I have the integral from minus r to r, this is x,"},{"Start":"09:14.755 ","End":"09:22.400","Text":"and then the integral from minus the square root to the square root."},{"Start":"09:22.530 ","End":"09:29.424","Text":"The y is going to stay here this time it\u0027s going to be y dy, and then dx."},{"Start":"09:29.424 ","End":"09:34.940","Text":"Let me write this in r squared minus x squared."},{"Start":"09:36.480 ","End":"09:42.830","Text":"Now the inner integral is this 1 dy."},{"Start":"09:43.020 ","End":"09:46.270","Text":"Again they use the same trick I did before."},{"Start":"09:46.270 ","End":"09:50.470","Text":"We said before that because this is odd and because"},{"Start":"09:50.470 ","End":"09:55.590","Text":"this interval is symmetric from minus something to plus something,"},{"Start":"09:55.590 ","End":"09:57.315","Text":"that this integral was 0."},{"Start":"09:57.315 ","End":"09:58.620","Text":"The same thing here."},{"Start":"09:58.620 ","End":"10:02.030","Text":"The function y is an odd function."},{"Start":"10:02.030 ","End":"10:04.225","Text":"If I replace y by minus y,"},{"Start":"10:04.225 ","End":"10:06.430","Text":"then it just becomes negative."},{"Start":"10:06.430 ","End":"10:08.920","Text":"This is symmetrical from minus to plus,"},{"Start":"10:08.920 ","End":"10:13.725","Text":"so this whole thing is going to be 0 by the same trick,"},{"Start":"10:13.725 ","End":"10:19.360","Text":"and so this is also equal to 0."},{"Start":"10:20.070 ","End":"10:24.820","Text":"I\u0027m going to need this result for the z part."},{"Start":"10:24.820 ","End":"10:30.100","Text":"Let me just copy it down here before I scroll away."},{"Start":"10:30.100 ","End":"10:36.280","Text":"Let me just write it at the side that we found that m was equal to"},{"Start":"10:36.280 ","End":"10:43.195","Text":"1.5 kh squared r squared Pi."},{"Start":"10:43.195 ","End":"10:47.410","Text":"Now I can scroll and we\u0027ll do the part with"},{"Start":"10:47.410 ","End":"10:55.160","Text":"z. I\u0027ll do it over here."},{"Start":"10:55.740 ","End":"10:58.690","Text":"I\u0027ll choose a different color."},{"Start":"10:58.690 ","End":"11:07.390","Text":"Just write what was there before that z bar was equal to 1 over m,"},{"Start":"11:07.390 ","End":"11:11.815","Text":"and the integral, the integral, the integral."},{"Start":"11:11.815 ","End":"11:17.979","Text":"Here I recall we had from minus r to r. Here we had from minus"},{"Start":"11:17.979 ","End":"11:21.160","Text":"the square root of r squared minus x"},{"Start":"11:21.160 ","End":"11:25.285","Text":"squared to the square root of r squared minus x squared."},{"Start":"11:25.285 ","End":"11:33.925","Text":"Here we had from 0 to h. The difference is that this time we have a z."},{"Start":"11:33.925 ","End":"11:37.690","Text":"Once we had an x, then we had a y, now we have a z,"},{"Start":"11:37.690 ","End":"11:39.550","Text":"and the second bit is the same"},{"Start":"11:39.550 ","End":"11:50.560","Text":"z dz dy dx."},{"Start":"11:50.560 ","End":"11:53.630","Text":"I forgot we took the k out front."},{"Start":"11:54.090 ","End":"11:58.525","Text":"I can write this as z squared."},{"Start":"11:58.525 ","End":"12:03.920","Text":"Let\u0027s do this first integral at the side."},{"Start":"12:04.200 ","End":"12:10.690","Text":"I have the integral from 0 to h of z squared dz."},{"Start":"12:10.690 ","End":"12:19.570","Text":"This is equal to 1/3 z cubed taken from 0 to h,"},{"Start":"12:19.570 ","End":"12:25.060","Text":"and that just comes out to be 1/3 h cubed."},{"Start":"12:25.060 ","End":"12:29.695","Text":"Now if instead of this I put 1/3 h cubed,"},{"Start":"12:29.695 ","End":"12:33.100","Text":"then this is all a constant."},{"Start":"12:33.100 ","End":"12:37.280","Text":"I can take it out front,"},{"Start":"12:37.860 ","End":"12:46.250","Text":"and so I get kh cubed 3m,"},{"Start":"12:46.250 ","End":"12:52.620","Text":"and then the integral from minus r to r. The integral from"},{"Start":"12:52.620 ","End":"13:00.070","Text":"minus the square root of r squared minus x squared root r squared minus x squared."},{"Start":"13:00.070 ","End":"13:07.015","Text":"Nothing left here but the 1 dy dx."},{"Start":"13:07.015 ","End":"13:10.360","Text":"Now we\u0027re going to do a little trick,"},{"Start":"13:10.360 ","End":"13:12.310","Text":"we\u0027re not going to do a lot of work here."},{"Start":"13:12.310 ","End":"13:18.655","Text":"I want to just bring in a picture from the beginning of the clip."},{"Start":"13:18.655 ","End":"13:20.320","Text":"This is what I\u0027m talking about."},{"Start":"13:20.320 ","End":"13:22.330","Text":"This is what we had before."},{"Start":"13:22.330 ","End":"13:32.305","Text":"The reason we had these limits minus r to r for x and y from minus square root,"},{"Start":"13:32.305 ","End":"13:42.340","Text":"square root was exactly how we made the disk of radius r into an iterative integral."},{"Start":"13:42.340 ","End":"13:45.460","Text":"We\u0027ve said x goes from minus r to r,"},{"Start":"13:45.460 ","End":"13:50.350","Text":"and then y goes from minus square root to square root."},{"Start":"13:50.350 ","End":"13:54.710","Text":"In other words, this is exactly equal to"},{"Start":"13:56.520 ","End":"14:07.510","Text":"kh cubed over 3m times the double integral over d. It looks I\u0027m going backwards,"},{"Start":"14:07.510 ","End":"14:09.085","Text":"but I have a reason for this,"},{"Start":"14:09.085 ","End":"14:14.620","Text":"of 1 dy dx."},{"Start":"14:14.620 ","End":"14:18.850","Text":"Now here\u0027s the thing. When we have the integral of 1 as a"},{"Start":"14:18.850 ","End":"14:23.574","Text":"2-dimensional integral over a region in the plane,"},{"Start":"14:23.574 ","End":"14:28.735","Text":"then this is just equal to the area of the region."},{"Start":"14:28.735 ","End":"14:35.035","Text":"What I can say is this is equal to kh cubed"},{"Start":"14:35.035 ","End":"14:44.695","Text":"over 3m times the area of d. But the area of d,"},{"Start":"14:44.695 ","End":"14:47.875","Text":"this is a circle of radius r,"},{"Start":"14:47.875 ","End":"14:50.950","Text":"is just Pi r squared."},{"Start":"14:50.950 ","End":"14:58.430","Text":"What we get, this is equal to"},{"Start":"15:00.090 ","End":"15:05.005","Text":"kh cubed times"},{"Start":"15:05.005 ","End":"15:12.830","Text":"Pi r squared over 3m."},{"Start":"15:14.670 ","End":"15:19.465","Text":"The Pi r squared is just this bit here."},{"Start":"15:19.465 ","End":"15:22.960","Text":"Now I want to use finally this."},{"Start":"15:22.960 ","End":"15:27.100","Text":"I have m here and I\u0027m going to substitute that here."},{"Start":"15:27.100 ","End":"15:35.200","Text":"What we get is kh cubed Pi r squared over 3."},{"Start":"15:35.200 ","End":"15:44.970","Text":"Now here I\u0027ll put what m is times 1.5 kh squared r squared Pi."},{"Start":"15:44.970 ","End":"15:46.940","Text":"A lot of stuff cancels."},{"Start":"15:46.940 ","End":"15:49.895","Text":"Notice k cancels with k,"},{"Start":"15:49.895 ","End":"15:53.190","Text":"Pi cancels with Pi,"},{"Start":"15:53.950 ","End":"15:59.020","Text":"r squared cancels with r squared,"},{"Start":"15:59.020 ","End":"16:07.265","Text":"and h cubed over h squared is just h. What I end up getting,"},{"Start":"16:07.265 ","End":"16:09.155","Text":"I can put this 2 on the top,"},{"Start":"16:09.155 ","End":"16:14.735","Text":"I get 2h over 3,"},{"Start":"16:14.735 ","End":"16:20.120","Text":"or perhaps could write it as 2/3 of"},{"Start":"16:20.120 ","End":"16:27.530","Text":"h. This is the answer for the z bar."},{"Start":"16:27.530 ","End":"16:33.260","Text":"Finally, I want to collect it all together and say that"},{"Start":"16:33.260 ","End":"16:40.900","Text":"the center of mass is at the point,"},{"Start":"16:40.900 ","End":"16:44.080","Text":"remember, x bar was 0,"},{"Start":"16:44.080 ","End":"16:46.750","Text":"y bar was 0,"},{"Start":"16:46.750 ","End":"16:50.585","Text":"and z bar is just this,"},{"Start":"16:50.585 ","End":"16:58.350","Text":"2/3 h. This is our final answer for the center of mass."},{"Start":"16:58.350 ","End":"16:59.970","Text":"We already did the mass,"},{"Start":"16:59.970 ","End":"17:02.950","Text":"and so we are done."}],"ID":8739},{"Watched":false,"Name":"Exercise 5","Duration":"11m 21s","ChapterTopicVideoID":8532,"CourseChapterTopicPlaylistID":21192,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8532.jpeg","UploadDate":"2017-02-13T03:14:27.7770000","DurationForVideoObject":"PT11M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Here we have an exercise from physics,"},{"Start":"00:02.700 ","End":"00:06.525","Text":"but don\u0027t worry, I\u0027ll give all the formulas that we need."},{"Start":"00:06.525 ","End":"00:13.680","Text":"We have to compute the moment of inertia about the z-axis of the homogeneous box,"},{"Start":"00:13.680 ","End":"00:16.410","Text":"which is described as follows."},{"Start":"00:16.410 ","End":"00:19.230","Text":"I won\u0027t read it aloud."},{"Start":"00:19.230 ","End":"00:23.850","Text":"We have to give our answer in terms of the mass of the board."},{"Start":"00:23.850 ","End":"00:26.010","Text":"There\u0027s actually 2 computation\u0027s."},{"Start":"00:26.010 ","End":"00:29.835","Text":"First, the mass and then the moment of inertia."},{"Start":"00:29.835 ","End":"00:35.920","Text":"Homogeneous just means that the density function is a constant."},{"Start":"00:35.960 ","End":"00:42.075","Text":"Let\u0027s see. Maybe I\u0027ll do a rough sketch."},{"Start":"00:42.075 ","End":"00:47.120","Text":"Let\u0027s see, the box here are some axes and these are the positive directions."},{"Start":"00:47.120 ","End":"00:51.635","Text":"X goes from 0 to some a,"},{"Start":"00:51.635 ","End":"00:52.925","Text":"which might be here."},{"Start":"00:52.925 ","End":"00:55.190","Text":"Y goes from 0 to,"},{"Start":"00:55.190 ","End":"00:57.274","Text":"let\u0027s say this is b,"},{"Start":"00:57.274 ","End":"01:04.320","Text":"and z goes from 0 to c. If we\u0027re just looking at the x,"},{"Start":"01:04.320 ","End":"01:07.115","Text":"y plane, that would be easier to sketch."},{"Start":"01:07.115 ","End":"01:11.515","Text":"We\u0027d get a rectangle though it looks like a parallelogram,"},{"Start":"01:11.515 ","End":"01:13.070","Text":"and I would shade this,"},{"Start":"01:13.070 ","End":"01:14.255","Text":"that would be the base,"},{"Start":"01:14.255 ","End":"01:16.820","Text":"but we have a third dimension."},{"Start":"01:16.820 ","End":"01:21.260","Text":"Let\u0027s see if I can get that somehow in there."},{"Start":"01:21.260 ","End":"01:26.290","Text":"Something like this, something like this."},{"Start":"01:27.370 ","End":"01:29.555","Text":"Think you get the idea."},{"Start":"01:29.555 ","End":"01:37.635","Text":"This is a box, and it\u0027s just a box whose sides are A, B,"},{"Start":"01:37.635 ","End":"01:41.120","Text":"and C. The volume of the box, for example,"},{"Start":"01:41.120 ","End":"01:44.720","Text":"would be A times B times C. Not the greatest sketch,"},{"Start":"01:44.720 ","End":"01:47.250","Text":"but enough to give you the idea."},{"Start":"01:48.020 ","End":"01:51.830","Text":"Back here, we want to do the mass."},{"Start":"01:51.830 ","End":"01:57.320","Text":"The mass in general is given by the formula for"},{"Start":"01:57.320 ","End":"02:03.320","Text":"any 3-dimensional region is the integral over the region,"},{"Start":"02:03.320 ","End":"02:08.510","Text":"in our case a B for box or B for body in general,"},{"Start":"02:08.510 ","End":"02:11.050","Text":"of the density function,"},{"Start":"02:11.050 ","End":"02:12.830","Text":"and it\u0027s usually called Delta,"},{"Start":"02:12.830 ","End":"02:15.905","Text":"sometimes row Delta of x, y,"},{"Start":"02:15.905 ","End":"02:20.050","Text":"and z and dv."},{"Start":"02:20.050 ","End":"02:22.890","Text":"It\u0027s a 3-dimensional integral."},{"Start":"02:22.890 ","End":"02:24.665","Text":"Now, in our case,"},{"Start":"02:24.665 ","End":"02:27.290","Text":"we have that the box is homogeneous,"},{"Start":"02:27.290 ","End":"02:31.190","Text":"which means that this Delta function is a constant."},{"Start":"02:31.190 ","End":"02:34.400","Text":"If this is a constant, let\u0027s say k,"},{"Start":"02:34.400 ","End":"02:39.200","Text":"then I can take k outside of the integral and"},{"Start":"02:39.200 ","End":"02:44.240","Text":"say that this is k times the triple integral over,"},{"Start":"02:44.240 ","End":"02:50.340","Text":"in our case, it\u0027s the box B of 1 dv,"},{"Start":"02:50.340 ","End":"02:54.380","Text":"and the integral of 1 over a"},{"Start":"02:54.380 ","End":"03:00.260","Text":"solid has the meaning that it\u0027s the volume I\u0027ll write it in words,"},{"Start":"03:00.260 ","End":"03:04.630","Text":"it\u0027s the volume of that solid B."},{"Start":"03:04.630 ","End":"03:08.180","Text":"Now, in our case, the volume of a box,"},{"Start":"03:08.180 ","End":"03:11.690","Text":"we all know it\u0027s just length times width times height."},{"Start":"03:11.690 ","End":"03:17.100","Text":"It\u0027s a, b, c. In our case,"},{"Start":"03:17.100 ","End":"03:20.730","Text":"it\u0027s just going to equal k,"},{"Start":"03:20.730 ","End":"03:28.160","Text":"a, b, c. Now,"},{"Start":"03:28.160 ","End":"03:31.565","Text":"how about the moment of inertia?"},{"Start":"03:31.565 ","End":"03:34.190","Text":"Well, I think I\u0027ll do it over here."},{"Start":"03:34.190 ","End":"03:36.520","Text":"I want to get rid of this picture."},{"Start":"03:36.520 ","End":"03:42.325","Text":"Now, the formula for moment of inertia about the z-axis,"},{"Start":"03:42.325 ","End":"03:47.810","Text":"we call it I for inertia and Z for the z-axis is equal to,"},{"Start":"03:47.810 ","End":"03:49.460","Text":"it\u0027s a similar triple integral,"},{"Start":"03:49.460 ","End":"03:51.470","Text":"just a bit more involved."},{"Start":"03:51.470 ","End":"03:53.105","Text":"It\u0027s a triple integral."},{"Start":"03:53.105 ","End":"03:57.595","Text":"This formula works in general of a body B not just our box"},{"Start":"03:57.595 ","End":"04:07.700","Text":"of x squared plus y squared times the rest of it Delta of x,"},{"Start":"04:07.700 ","End":"04:11.895","Text":"y, and z dv."},{"Start":"04:11.895 ","End":"04:15.080","Text":"There\u0027s a pattern here, if it was the x-axis,"},{"Start":"04:15.080 ","End":"04:17.930","Text":"and here we\u0027d have y squared plus z squared."},{"Start":"04:17.930 ","End":"04:19.670","Text":"If it was the y-axis,"},{"Start":"04:19.670 ","End":"04:26.370","Text":"you\u0027d have x squared plus z squared wherever the letter is here,"},{"Start":"04:26.370 ","End":"04:28.910","Text":"that\u0027s one that\u0027s missing with the squares."},{"Start":"04:28.910 ","End":"04:32.900","Text":"It\u0027s actually 3 formulas for this is the one we want anyway."},{"Start":"04:32.900 ","End":"04:36.205","Text":"Now, as before, this delta,"},{"Start":"04:36.205 ","End":"04:38.580","Text":"it\u0027s the same deltas before,"},{"Start":"04:38.580 ","End":"04:41.805","Text":"and it\u0027s homogeneous. Deltas are constant."},{"Start":"04:41.805 ","End":"04:45.740","Text":"This Delta function is a constant k. As such,"},{"Start":"04:45.740 ","End":"04:48.124","Text":"we can take it in front of the integral,"},{"Start":"04:48.124 ","End":"04:55.415","Text":"and so we get k times the triple integral of"},{"Start":"04:55.415 ","End":"05:05.090","Text":"x squared plus y squared dv."},{"Start":"05:05.090 ","End":"05:08.330","Text":"However, I don\u0027t just want it in general like this."},{"Start":"05:08.330 ","End":"05:11.690","Text":"I want it as an iterated integral."},{"Start":"05:11.690 ","End":"05:13.700","Text":"Doesn\u0027t matter really in which order x,"},{"Start":"05:13.700 ","End":"05:15.305","Text":"y, or z for a box."},{"Start":"05:15.305 ","End":"05:20.965","Text":"But let\u0027s say we do it as x here,"},{"Start":"05:20.965 ","End":"05:25.475","Text":"y here, and z here,"},{"Start":"05:25.475 ","End":"05:31.940","Text":"then x will go from 0 to a."},{"Start":"05:31.940 ","End":"05:33.785","Text":"These are limits of x,"},{"Start":"05:33.785 ","End":"05:38.150","Text":"y will go from 0 to b,"},{"Start":"05:38.150 ","End":"05:42.600","Text":"and z will go from 0 to c,"},{"Start":"05:42.600 ","End":"05:44.760","Text":"the k goes here,"},{"Start":"05:44.760 ","End":"05:51.270","Text":"and I have here x squared plus y squared,"},{"Start":"05:51.270 ","End":"05:53.900","Text":"and then I do the d\u0027s in the opposite order."},{"Start":"05:53.900 ","End":"05:56.615","Text":"This dz goes with this."},{"Start":"05:56.615 ","End":"05:58.685","Text":"Then we have dy,"},{"Start":"05:58.685 ","End":"06:01.860","Text":"and then we have dx."},{"Start":"06:03.190 ","End":"06:08.360","Text":"Now, let me start with the innermost integral,"},{"Start":"06:08.360 ","End":"06:10.970","Text":"that would be this one,"},{"Start":"06:10.970 ","End":"06:16.395","Text":"dz, and let me do this at the side lets say here."},{"Start":"06:16.395 ","End":"06:18.529","Text":"What I have, actually,"},{"Start":"06:18.529 ","End":"06:23.120","Text":"because x squared plus y squared is a constant as far as z goes,"},{"Start":"06:23.120 ","End":"06:29.570","Text":"this is equal to x squared plus y squared times the"},{"Start":"06:29.570 ","End":"06:37.880","Text":"integral from 0 to c. There\u0027s nothing left here, just 1 dz."},{"Start":"06:37.880 ","End":"06:44.350","Text":"The integral of 1 is just the upper limit minus the lower limit."},{"Start":"06:44.350 ","End":"06:46.780","Text":"I mean, if I take z and substitute c,"},{"Start":"06:46.780 ","End":"06:48.640","Text":"it\u0027s c, substitute 0 is 0."},{"Start":"06:48.640 ","End":"06:57.840","Text":"Subtract it\u0027s just c. This is actually equal to c times x squared plus y squared."},{"Start":"06:57.840 ","End":"07:03.450","Text":"Now I\u0027m going to put that back where this is shaded."},{"Start":"07:03.450 ","End":"07:07.830","Text":"In fact, I can take the c out in front again."},{"Start":"07:07.830 ","End":"07:12.764","Text":"Now, I get the c goes with the k,"},{"Start":"07:12.764 ","End":"07:16.375","Text":"and then I have the integral from 0 to a,"},{"Start":"07:16.375 ","End":"07:24.180","Text":"integral from 0 to b of x squared plus y squared,"},{"Start":"07:24.180 ","End":"07:29.980","Text":"and it\u0027s dy, dx."},{"Start":"07:29.980 ","End":"07:32.930","Text":"Once again, we do the inner one first."},{"Start":"07:32.930 ","End":"07:37.015","Text":"That would be this one, the dy integral."},{"Start":"07:37.015 ","End":"07:39.829","Text":"Again, I like to do these at the side,"},{"Start":"07:39.829 ","End":"07:42.080","Text":"so I\u0027ll do this somewhere over here."},{"Start":"07:42.080 ","End":"07:44.615","Text":"Let\u0027s see if I take the integral dy,"},{"Start":"07:44.615 ","End":"07:46.640","Text":"x is a constant,"},{"Start":"07:46.640 ","End":"07:55.460","Text":"so I get x squared y plus 1/3y cubed,"},{"Start":"07:55.460 ","End":"08:00.890","Text":"and all this from 0 to b."},{"Start":"08:00.890 ","End":"08:04.920","Text":"These are the limits, of course, for y."},{"Start":"08:05.870 ","End":"08:09.250","Text":"If I plug in the lower one,"},{"Start":"08:09.250 ","End":"08:11.485","Text":"y equals 0, I just get 0."},{"Start":"08:11.485 ","End":"08:13.300","Text":"That doesn\u0027t give me anything."},{"Start":"08:13.300 ","End":"08:16.655","Text":"I just need to plug in y equals b,"},{"Start":"08:16.655 ","End":"08:19.545","Text":"and so this becomes,"},{"Start":"08:19.545 ","End":"08:28.260","Text":"when y is b I get bx squared plus 1/3 b cubed."},{"Start":"08:28.260 ","End":"08:31.260","Text":"Now I go back here,"},{"Start":"08:31.260 ","End":"08:33.750","Text":"and I need some more space."},{"Start":"08:33.750 ","End":"08:42.850","Text":"What we get is ck times just the integral from 0 to a. We got to write there."},{"Start":"08:42.850 ","End":"08:44.425","Text":"It doesn\u0027t really matter actually."},{"Start":"08:44.425 ","End":"08:46.210","Text":"That was the limit for y in for x."},{"Start":"08:46.210 ","End":"08:48.310","Text":"Here we have x from 0 to y."},{"Start":"08:48.310 ","End":"08:50.800","Text":"Of course it\u0027s x because it\u0027s dx."},{"Start":"08:50.800 ","End":"08:54.350","Text":"We have bx squared,"},{"Start":"08:54.350 ","End":"08:55.610","Text":"I\u0027m copying from here,"},{"Start":"08:55.610 ","End":"09:00.755","Text":"plus 1/3b cubed dx."},{"Start":"09:00.755 ","End":"09:03.350","Text":"This is straightforward enough."},{"Start":"09:03.350 ","End":"09:08.140","Text":"This is equal to ck,"},{"Start":"09:08.140 ","End":"09:11.040","Text":"and then I\u0027ve got, with respect to x,"},{"Start":"09:11.040 ","End":"09:16.885","Text":"this is 1/3, b is a constant, it stays."},{"Start":"09:16.885 ","End":"09:19.860","Text":"Here it\u0027s also a constant,"},{"Start":"09:19.860 ","End":"09:24.360","Text":"so it just gets 1/3b cubed x,"},{"Start":"09:24.360 ","End":"09:32.610","Text":"all this, when x goes from 0 up to a."},{"Start":"09:32.610 ","End":"09:37.095","Text":"Now, when x is 0,"},{"Start":"09:37.095 ","End":"09:38.295","Text":"I just get 0."},{"Start":"09:38.295 ","End":"09:41.520","Text":"Once again, I just need to plug in a."},{"Start":"09:41.520 ","End":"09:50.370","Text":"We get ck times 1/3, now when x is a,"},{"Start":"09:50.370 ","End":"09:54.760","Text":"this gives ba cubed,"},{"Start":"09:56.120 ","End":"10:05.310","Text":"and here I get 1/3b cubed a or a,"},{"Start":"10:05.310 ","End":"10:07.355","Text":"well, let\u0027s leave it as,"},{"Start":"10:07.355 ","End":"10:10.920","Text":"yeah, leave the order b cubed a."},{"Start":"10:11.450 ","End":"10:13.850","Text":"Now, we\u0027re not quite done yet."},{"Start":"10:13.850 ","End":"10:15.515","Text":"We want to simplify,"},{"Start":"10:15.515 ","End":"10:21.030","Text":"and we also want to use the fact from here."},{"Start":"10:21.370 ","End":"10:24.590","Text":"Yeah, we had that the mass was equal to k,"},{"Start":"10:24.590 ","End":"10:26.555","Text":"a, b, c, and we\u0027re going to use that."},{"Start":"10:26.555 ","End":"10:30.530","Text":"Continuing to simplify, what I\u0027m going to"},{"Start":"10:30.530 ","End":"10:36.470","Text":"do is take the common stuff out, which is 1/3ab."},{"Start":"10:36.470 ","End":"10:39.725","Text":"I\u0027ve got 1/3 ab,"},{"Start":"10:39.725 ","End":"10:41.780","Text":"which I\u0027ll remove from here,"},{"Start":"10:41.780 ","End":"10:43.715","Text":"I still have the ck."},{"Start":"10:43.715 ","End":"10:46.820","Text":"After I take out 1/3ab,"},{"Start":"10:46.820 ","End":"10:49.670","Text":"I\u0027m left here with a squared,"},{"Start":"10:49.670 ","End":"10:53.120","Text":"and I\u0027m left here with b squared."},{"Start":"10:53.120 ","End":"10:57.410","Text":"At this point, I look at this and see that k, a, b,"},{"Start":"10:57.410 ","End":"11:00.390","Text":"c, or a, b,"},{"Start":"11:00.390 ","End":"11:02.415","Text":"c k is the mass."},{"Start":"11:02.415 ","End":"11:10.170","Text":"This is exactly 1/3a squared plus b squared,"},{"Start":"11:10.170 ","End":"11:13.050","Text":"and this here is m,"},{"Start":"11:13.050 ","End":"11:15.420","Text":"which I can put at the end,"},{"Start":"11:15.420 ","End":"11:17.730","Text":"and I\u0027ll just highlight it,"},{"Start":"11:17.730 ","End":"11:21.180","Text":"and this is our answer. We are done."}],"ID":8740}],"Thumbnail":null,"ID":21192},{"Name":"Triple Integrals, Cylindrical and Spherical","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"13m 51s","ChapterTopicVideoID":9246,"CourseChapterTopicPlaylistID":21193,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9246.jpeg","UploadDate":"2017-03-30T09:46:14.2300000","DurationForVideoObject":"PT13M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.445","Text":"In this exercise, we have to compute this triple integral."},{"Start":"00:05.445 ","End":"00:13.500","Text":"Let\u0027s take a look at the region on which this is defined."},{"Start":"00:13.500 ","End":"00:20.410","Text":"The outer loop is x and x goes from 0-1."},{"Start":"00:20.750 ","End":"00:26.610","Text":"Inside that, we have y going from minus root of"},{"Start":"00:26.610 ","End":"00:34.320","Text":"1 minus x squared to plus square root of 1 minus x squared."},{"Start":"00:34.320 ","End":"00:40.850","Text":"Less importantly, z goes from minus x"},{"Start":"00:40.850 ","End":"00:47.750","Text":"squared plus y squared up to x squared plus y squared."},{"Start":"00:47.750 ","End":"00:52.145","Text":"What I\u0027d like to do is just focus on the xy for the moment."},{"Start":"00:52.145 ","End":"00:58.455","Text":"Let\u0027s see what this gives us in the xy-plane."},{"Start":"00:58.455 ","End":"01:04.325","Text":"I brought in a sketch and I\u0027ll just say a couple of words of how I got to this sketch."},{"Start":"01:04.325 ","End":"01:09.995","Text":"Well, in general, if I just took the unit circle,"},{"Start":"01:09.995 ","End":"01:16.220","Text":"that would be x squared plus y squared equals 1."},{"Start":"01:16.220 ","End":"01:23.630","Text":"If you extract y, you get that y is plus or minus the square root of 1 minus x squared."},{"Start":"01:23.630 ","End":"01:25.610","Text":"You would think that this might be"},{"Start":"01:25.610 ","End":"01:34.970","Text":"the full circle and y would go between plus the square root and minus the square root,"},{"Start":"01:34.970 ","End":"01:39.290","Text":"except that we have that x is restricted to be between 0 and 1,"},{"Start":"01:39.290 ","End":"01:43.060","Text":"so we only get the right 1/2 of it."},{"Start":"01:43.060 ","End":"01:49.380","Text":"This is in fact the region in the xy plane defined by these 2."},{"Start":"01:49.380 ","End":"01:52.230","Text":"As x goes from 0-1,"},{"Start":"01:52.230 ","End":"01:54.195","Text":"let\u0027s say this is a typical x,"},{"Start":"01:54.195 ","End":"02:00.195","Text":"then y goes from"},{"Start":"02:00.195 ","End":"02:04.250","Text":"minus square root of 1 minus x squared up"},{"Start":"02:04.250 ","End":"02:09.380","Text":"to plus square root of 1 minus x squared and everything in between."},{"Start":"02:09.380 ","End":"02:11.450","Text":"It sweeps the semicircle,"},{"Start":"02:11.450 ","End":"02:13.730","Text":"as x sweeps from 0-1,"},{"Start":"02:13.730 ","End":"02:18.380","Text":"y goes along these vertical slices and that\u0027s how we get it."},{"Start":"02:18.380 ","End":"02:20.330","Text":"If you\u0027re working in 2D,"},{"Start":"02:20.330 ","End":"02:22.340","Text":"I would just say at this point,"},{"Start":"02:22.340 ","End":"02:26.645","Text":"let\u0027s move to polar coordinates because we have a circular symmetry,"},{"Start":"02:26.645 ","End":"02:30.350","Text":"but we\u0027re in 3D, so the equivalent of polar would"},{"Start":"02:30.350 ","End":"02:34.520","Text":"be cylindrical since cylindrical doesn\u0027t really touch z,"},{"Start":"02:34.520 ","End":"02:36.990","Text":"it just leaves it alone."},{"Start":"02:37.040 ","End":"02:41.765","Text":"I\u0027d just like to remark that on way of doing this question,"},{"Start":"02:41.765 ","End":"02:44.150","Text":"an alternative way, and you might try it,"},{"Start":"02:44.150 ","End":"02:47.065","Text":"is to do the dz integral first,"},{"Start":"02:47.065 ","End":"02:49.140","Text":"with all this is a constant,"},{"Start":"02:49.140 ","End":"02:50.715","Text":"and that comes in front."},{"Start":"02:50.715 ","End":"02:55.650","Text":"It\u0027s easy to do the integral dz and then you\u0027ll just get dydx."},{"Start":"02:55.650 ","End":"03:02.505","Text":"Then you could do a polar transformation in just 2 variables,"},{"Start":"03:02.505 ","End":"03:06.300","Text":"but I\u0027ll leave this as is."},{"Start":"03:06.300 ","End":"03:12.365","Text":"Not only does the region beg to be done in polar,"},{"Start":"03:12.365 ","End":"03:14.990","Text":"but I also have x squared plus y squared,"},{"Start":"03:14.990 ","End":"03:19.245","Text":"which is a good expression for polar or cylindrical,"},{"Start":"03:19.245 ","End":"03:20.960","Text":"they\u0027re both, as I said, very similar."},{"Start":"03:20.960 ","End":"03:24.830","Text":"Let me write the equations of the cylindrical transformation."},{"Start":"03:24.830 ","End":"03:27.990","Text":"It starts out like polar,"},{"Start":"03:28.330 ","End":"03:33.665","Text":"x equals r cosine Theta,"},{"Start":"03:33.665 ","End":"03:38.525","Text":"y equals r sine Theta."},{"Start":"03:38.525 ","End":"03:41.435","Text":"The differences between polar and cylindrical,"},{"Start":"03:41.435 ","End":"03:42.770","Text":"we get this extra one,"},{"Start":"03:42.770 ","End":"03:47.430","Text":"z equals z, which means z just stays as it is."},{"Start":"03:49.670 ","End":"03:55.695","Text":"Well, this thing here is actually dV."},{"Start":"03:55.695 ","End":"03:58.280","Text":"The 3-dimensional equivalent in 2D,"},{"Start":"03:58.280 ","End":"04:00.890","Text":"we would just have dydx, which is dA."},{"Start":"04:00.890 ","End":"04:04.760","Text":"The difference is that the dV, it\u0027s just like the dA."},{"Start":"04:04.760 ","End":"04:09.865","Text":"It\u0027s r dr d Theta,"},{"Start":"04:09.865 ","End":"04:14.045","Text":"except that we also have an extra dz in here,"},{"Start":"04:14.045 ","End":"04:16.620","Text":"so let me just rewrite that."},{"Start":"04:16.870 ","End":"04:24.040","Text":"It\u0027s r dz and then dr d Theta."},{"Start":"04:24.040 ","End":"04:27.050","Text":"Just like with the polar,"},{"Start":"04:27.050 ","End":"04:33.770","Text":"there was the extra useful formula that x squared plus y squared equals r squared,"},{"Start":"04:33.770 ","End":"04:41.180","Text":"so all these might be of use to us when we do the transformation."},{"Start":"04:41.180 ","End":"04:44.255","Text":"I don\u0027t need this sketch anymore."},{"Start":"04:44.255 ","End":"04:48.965","Text":"I would like to start the conversion. Let\u0027s see."},{"Start":"04:48.965 ","End":"04:54.160","Text":"The first 2 are going to be Theta something,"},{"Start":"04:54.160 ","End":"04:57.420","Text":"r something, we\u0027ll see in a moment."},{"Start":"04:57.420 ","End":"05:01.430","Text":"The last one, the z is going to stay unchanged."},{"Start":"05:01.430 ","End":"05:08.430","Text":"I\u0027m still going to get z from minus x squared."},{"Start":"05:08.430 ","End":"05:11.350","Text":"Well, not quit."},{"Start":"05:12.290 ","End":"05:15.000","Text":"The limits for z don\u0027t change,"},{"Start":"05:15.000 ","End":"05:18.960","Text":"but the expression changes,"},{"Start":"05:18.960 ","End":"05:21.330","Text":"x squared plus y squared is r squared,"},{"Start":"05:21.330 ","End":"05:26.220","Text":"so it goes from minus r squared to r squared."},{"Start":"05:26.220 ","End":"05:31.270","Text":"As for these 2, I just have to rewrite this 2-dimensional region,"},{"Start":"05:31.270 ","End":"05:33.850","Text":"instead of xy is r Theta."},{"Start":"05:33.850 ","End":"05:43.780","Text":"Well, we see that here the negative y-axis Theta is minus 90 degrees or minus Pi over 2."},{"Start":"05:43.780 ","End":"05:50.050","Text":"Then we go all the way round up to here,"},{"Start":"05:50.050 ","End":"05:55.435","Text":"where Theta is 90 degrees or Pi over 2."},{"Start":"05:55.435 ","End":"05:57.340","Text":"There are other ways of doing it,"},{"Start":"05:57.340 ","End":"05:59.335","Text":"but this is the way I like doing it."},{"Start":"05:59.335 ","End":"06:02.605","Text":"Remember, counterclockwise is the positive direction."},{"Start":"06:02.605 ","End":"06:06.710","Text":"For each particular Theta in this range,"},{"Start":"06:06.710 ","End":"06:11.070","Text":"r always goes from 0-1."},{"Start":"06:11.070 ","End":"06:14.040","Text":"It\u0027s a circle of radius 1."},{"Start":"06:14.040 ","End":"06:22.830","Text":"I can write this as Theta goes from minus Pi over 2-Pi over 2,"},{"Start":"06:22.830 ","End":"06:26.925","Text":"and r goes from 0-1,"},{"Start":"06:26.925 ","End":"06:29.375","Text":"and now we need what\u0027s here."},{"Start":"06:29.375 ","End":"06:36.630","Text":"We have 21, x is r cosine Theta,"},{"Start":"06:37.760 ","End":"06:40.755","Text":"y is r sine Theta,"},{"Start":"06:40.755 ","End":"06:47.655","Text":"but it\u0027s squared, and then we have the dv,"},{"Start":"06:47.655 ","End":"07:04.590","Text":"which is r times dzdrd Theta."},{"Start":"07:04.590 ","End":"07:07.840","Text":"Now, I\u0027d like to simplify this."},{"Start":"07:08.060 ","End":"07:16.170","Text":"If I just look at this part here, if I collect stuff,"},{"Start":"07:16.170 ","End":"07:20.490","Text":"it\u0027s 21, and then I have r,"},{"Start":"07:20.490 ","End":"07:24.780","Text":"r squared, and r, so that\u0027s r^4."},{"Start":"07:24.780 ","End":"07:36.070","Text":"Then I have cosine Theta sine squared Theta."},{"Start":"07:37.190 ","End":"07:43.430","Text":"I just meant to simplify this bit without the dzdrd Theta."},{"Start":"07:43.430 ","End":"07:49.400","Text":"Now, what I like to do is not only do I like to bring the constant out front,"},{"Start":"07:49.400 ","End":"07:53.660","Text":"but I noticed that this first integral is dz,"},{"Start":"07:53.660 ","End":"07:58.385","Text":"so anything with r and Theta can be brought in front of the integral."},{"Start":"07:58.385 ","End":"08:01.860","Text":"In fact, this is also the dr,"},{"Start":"08:01.860 ","End":"08:04.500","Text":"so Theta can be brought in front."},{"Start":"08:04.500 ","End":"08:06.585","Text":"Let me show you how I would do it."},{"Start":"08:06.585 ","End":"08:08.450","Text":"Not everyone does it this way,"},{"Start":"08:08.450 ","End":"08:14.519","Text":"but I find it simplifies things to take everything upfront as much as possible."},{"Start":"08:14.519 ","End":"08:20.920","Text":"I have Theta goes from minus Pi over 2-Pi over 2."},{"Start":"08:20.920 ","End":"08:25.675","Text":"Now, the cosine Theta sine squared Theta can go here"},{"Start":"08:25.675 ","End":"08:33.235","Text":"because it\u0027s a constant as far as z and r go."},{"Start":"08:33.235 ","End":"08:36.820","Text":"Then the next, I can take r from"},{"Start":"08:36.820 ","End":"08:46.695","Text":"0-1 and I pull the r^4 in front of the dz integral,"},{"Start":"08:46.695 ","End":"08:56.715","Text":"r^4, and then z goes from minus r squared to r squared of just dz,"},{"Start":"08:56.715 ","End":"08:59.010","Text":"and then I close the dr,"},{"Start":"08:59.010 ","End":"09:01.590","Text":"and then I close the d Theta."},{"Start":"09:01.590 ","End":"09:03.200","Text":"That\u0027s 1 way of doing it,"},{"Start":"09:03.200 ","End":"09:09.280","Text":"is bringing stuff as much as possible outwards to the left."},{"Start":"09:11.510 ","End":"09:14.100","Text":"We start with the inside,"},{"Start":"09:14.100 ","End":"09:17.040","Text":"which is the dz integral."},{"Start":"09:17.040 ","End":"09:19.170","Text":"Maybe I\u0027ll just write a 1 here."},{"Start":"09:19.170 ","End":"09:20.880","Text":"I don\u0027t like to leave a blank there."},{"Start":"09:20.880 ","End":"09:23.850","Text":"Now, the integral of 1 from the upper limit"},{"Start":"09:23.850 ","End":"09:27.155","Text":"to the lower limit is just the upper minus the lower."},{"Start":"09:27.155 ","End":"09:30.365","Text":"If I do r squared minus minus r squared,"},{"Start":"09:30.365 ","End":"09:35.805","Text":"this bit here is just 2r squared."},{"Start":"09:35.805 ","End":"09:40.350","Text":"What I can do now is put the 2 in front,"},{"Start":"09:40.350 ","End":"09:43.905","Text":"and then that will give me,"},{"Start":"09:43.905 ","End":"09:48.840","Text":"2 times 21 is 42."},{"Start":"09:48.840 ","End":"09:52.915","Text":"The integral Theta goes from minus Pi over"},{"Start":"09:52.915 ","End":"10:00.030","Text":"2-Pi over 2 cosine Theta sine squared Theta."},{"Start":"10:00.030 ","End":"10:02.385","Text":"Then I have the integral,"},{"Start":"10:02.385 ","End":"10:06.375","Text":"the r squared with the r^4 is r^6,"},{"Start":"10:06.375 ","End":"10:08.385","Text":"the 2 we took in front already,"},{"Start":"10:08.385 ","End":"10:13.200","Text":"from 0-1 of r^6 dr,"},{"Start":"10:13.200 ","End":"10:15.750","Text":"and then the d Theta."},{"Start":"10:15.750 ","End":"10:21.930","Text":"Now the inner integral is the dr. That\u0027s here."},{"Start":"10:21.930 ","End":"10:25.115","Text":"Again, I can do this at the side."},{"Start":"10:25.115 ","End":"10:32.110","Text":"What we get is the integral of r^6 is 1/7 r^7."},{"Start":"10:32.110 ","End":"10:36.150","Text":"If I take this between 0 and 1,"},{"Start":"10:36.150 ","End":"10:38.325","Text":"when I plug in 1, I get 1/7,"},{"Start":"10:38.325 ","End":"10:42.180","Text":"when I plug in 0, I just get 0 as nothing."},{"Start":"10:42.180 ","End":"10:47.060","Text":"Altogether, this is equal to 1/7."},{"Start":"10:47.060 ","End":"10:49.440","Text":"If this is a 1/7,"},{"Start":"10:49.930 ","End":"10:54.770","Text":"then that will combine with the constant in front,"},{"Start":"10:54.770 ","End":"10:58.880","Text":"and so I will just get that this equals,"},{"Start":"10:58.880 ","End":"11:05.240","Text":"42 over 7 is 6 times the integral from"},{"Start":"11:05.240 ","End":"11:14.209","Text":"minus Pi over 2-Pi over 2."},{"Start":"11:14.209 ","End":"11:16.280","Text":"Let me just change the order here."},{"Start":"11:16.280 ","End":"11:24.295","Text":"I\u0027d rather write it as sine squared Theta cosine Theta d Theta."},{"Start":"11:24.295 ","End":"11:27.790","Text":"Now, we have just an integral in Theta."},{"Start":"11:27.790 ","End":"11:31.925","Text":"This integral we can do with a substitution."},{"Start":"11:31.925 ","End":"11:35.690","Text":"What I\u0027d like to substitute is sine Theta,"},{"Start":"11:35.690 ","End":"11:42.590","Text":"so let\u0027s let t equals sine Theta and"},{"Start":"11:42.590 ","End":"11:51.095","Text":"then dt will just be cosine Theta d Theta."},{"Start":"11:51.095 ","End":"11:56.340","Text":"Also, I don\u0027t necessarily want to come back from t to Theta,"},{"Start":"11:56.340 ","End":"11:58.610","Text":"so if I substitute the limits too,"},{"Start":"11:58.610 ","End":"12:01.145","Text":"and let\u0027s remind ourselves this is Theta,"},{"Start":"12:01.145 ","End":"12:06.930","Text":"when Theta equals Pi over 2,"},{"Start":"12:06.930 ","End":"12:12.465","Text":"the upper one, then t is sine of Pi over 2."},{"Start":"12:12.465 ","End":"12:15.015","Text":"Sine of Pi over 2 is 1."},{"Start":"12:15.015 ","End":"12:19.155","Text":"When Theta is minus Pi over 2,"},{"Start":"12:19.155 ","End":"12:23.555","Text":"then t is sine of minus Pi over 2,"},{"Start":"12:23.555 ","End":"12:26.045","Text":"and that\u0027s negative 1."},{"Start":"12:26.045 ","End":"12:29.855","Text":"Sine of minus 90 degrees is minus 1."},{"Start":"12:29.855 ","End":"12:39.640","Text":"Altogether, we get now 6 times the integral from minus 1-1,"},{"Start":"12:39.640 ","End":"12:48.600","Text":"sine squared Theta is just t squared and cosine Theta d Theta is just dt."},{"Start":"12:50.020 ","End":"12:54.365","Text":"Just to save space, let me continue over here."},{"Start":"12:54.365 ","End":"12:58.120","Text":"I\u0027ve got 6 times,"},{"Start":"12:58.120 ","End":"13:05.220","Text":"the integral of t squared is 1/3 of"},{"Start":"13:05.220 ","End":"13:14.295","Text":"t cubed between minus 1 and 1."},{"Start":"13:14.295 ","End":"13:17.490","Text":"Let me take the 1/3 out,"},{"Start":"13:17.490 ","End":"13:24.930","Text":"so it\u0027s twice t cubed from minus 1-1."},{"Start":"13:24.930 ","End":"13:29.290","Text":"If I plug in 1, I get 1."},{"Start":"13:29.290 ","End":"13:32.435","Text":"If I plug in minus 1,"},{"Start":"13:32.435 ","End":"13:35.840","Text":"I\u0027ve got minus 1."},{"Start":"13:35.840 ","End":"13:41.670","Text":"1 minus minus 1 is 2,"},{"Start":"13:41.670 ","End":"13:46.380","Text":"2 times 2 is 4."},{"Start":"13:46.380 ","End":"13:51.340","Text":"This is our final answer. We\u0027re done."}],"ID":9572},{"Watched":false,"Name":"Exercise 2","Duration":"7m 41s","ChapterTopicVideoID":9245,"CourseChapterTopicPlaylistID":21193,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9245.jpeg","UploadDate":"2017-03-30T09:43:54.4630000","DurationForVideoObject":"PT7M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"In this exercise, we have to compute the following triple integral."},{"Start":"00:05.085 ","End":"00:09.810","Text":"Remember that this can also be written as dv,"},{"Start":"00:09.810 ","End":"00:12.195","Text":"which is more symmetrical,"},{"Start":"00:12.195 ","End":"00:18.010","Text":"doesn\u0027t depend on the order; dz, dy, dx."},{"Start":"00:18.520 ","End":"00:23.339","Text":"What I\u0027d like to do is write down the limits."},{"Start":"00:23.339 ","End":"00:28.230","Text":"This is the x goes from the outside in,"},{"Start":"00:28.230 ","End":"00:33.135","Text":"so x goes from minus 1 to 1,"},{"Start":"00:33.135 ","End":"00:35.765","Text":"and then this one belongs to y."},{"Start":"00:35.765 ","End":"00:41.060","Text":"So y goes from minus root 1 minus"},{"Start":"00:41.060 ","End":"00:47.345","Text":"x squared up to root 1 minus x squared."},{"Start":"00:47.345 ","End":"00:51.095","Text":"I\u0027m less concerned with z. I\u0027ll write it,"},{"Start":"00:51.095 ","End":"01:00.240","Text":"that z goes from square root of x squared plus y squared to 1."},{"Start":"01:00.240 ","End":"01:05.603","Text":"But I\u0027m really concern now with x and y."},{"Start":"01:05.603 ","End":"01:11.880","Text":"I\u0027m going to bring in a sketch of this in the xy-plane."},{"Start":"01:11.880 ","End":"01:18.125","Text":"Here it is. You probably identified it already from the equations as the unit circle."},{"Start":"01:18.125 ","End":"01:21.705","Text":"X goes from minus 1 to 1."},{"Start":"01:21.705 ","End":"01:29.395","Text":"The unit circle, which is normally written as x squared plus y squared equals 1,"},{"Start":"01:29.395 ","End":"01:34.134","Text":"if you isolate y from this by bringing the x squared to the other side,"},{"Start":"01:34.134 ","End":"01:37.585","Text":"then we get plus or minus the square root."},{"Start":"01:37.585 ","End":"01:43.820","Text":"So that the upper semicircle is y equals square root of 1 minus x squared,"},{"Start":"01:43.820 ","End":"01:48.850","Text":"and the lower semi circle is minus the square root of 1 minus x squared."},{"Start":"01:48.850 ","End":"01:52.810","Text":"For each particular, say we have"},{"Start":"01:52.810 ","End":"02:01.065","Text":"a typical x in this interval from minus 1 to 1,"},{"Start":"02:01.065 ","End":"02:04.380","Text":"and we take a vertical slice,"},{"Start":"02:04.380 ","End":"02:08.650","Text":"then it cuts at these 2 points,"},{"Start":"02:08.650 ","End":"02:14.090","Text":"and everything less than or equal to the upper and the lower gives us this bit."},{"Start":"02:14.090 ","End":"02:17.390","Text":"So what we sweep across from minus 1 to 1 with"},{"Start":"02:17.390 ","End":"02:20.705","Text":"x and y goes to plus or minus the square root,"},{"Start":"02:20.705 ","End":"02:23.825","Text":"we get the whole unit disk."},{"Start":"02:23.825 ","End":"02:28.760","Text":"Now this tells us we\u0027re much better off in polar coordinates,"},{"Start":"02:28.760 ","End":"02:32.435","Text":"only we\u0027re not in 2 dimensions, we\u0027re in 3-dimensions."},{"Start":"02:32.435 ","End":"02:33.920","Text":"So instead of polar,"},{"Start":"02:33.920 ","End":"02:37.745","Text":"we\u0027ll go to cylindrical because cylindrical is like polar,"},{"Start":"02:37.745 ","End":"02:43.290","Text":"just as also a z which is left as is."},{"Start":"02:43.290 ","End":"02:49.970","Text":"I copy-pasted all the formulas we need from the my previous exercise."},{"Start":"02:49.970 ","End":"02:53.620","Text":"It\u0027s just like polar,"},{"Start":"02:53.620 ","End":"02:59.990","Text":"except that we also have a z which essentially remains unchanged."},{"Start":"02:59.990 ","End":"03:04.295","Text":"Now, if we transform this integral,"},{"Start":"03:04.295 ","End":"03:14.210","Text":"what we get is instead of the x and the y going like this,"},{"Start":"03:14.210 ","End":"03:17.130","Text":"we\u0027ll get an r and a Theta."},{"Start":"03:20.200 ","End":"03:22.700","Text":"Well, what do they go from?"},{"Start":"03:22.700 ","End":"03:26.045","Text":"A typical Theta."},{"Start":"03:26.045 ","End":"03:32.630","Text":"Well, it starts here where Theta equals 0 and then go all the way"},{"Start":"03:32.630 ","End":"03:40.470","Text":"around and end up at Theta equals 360 degrees or 2 Pi."},{"Start":"03:40.470 ","End":"03:43.065","Text":"That\u0027s the full circle."},{"Start":"03:43.065 ","End":"03:48.100","Text":"As for r, it goes from 0 to 1 always."},{"Start":"03:49.040 ","End":"03:51.259","Text":"I got these backwards."},{"Start":"03:51.259 ","End":"03:55.055","Text":"We usually take the Theta on the outside,"},{"Start":"03:55.055 ","End":"03:57.890","Text":"that\u0027s from 0 to 2 Pi."},{"Start":"03:57.890 ","End":"04:01.740","Text":"R is from 0 to 1."},{"Start":"04:01.740 ","End":"04:03.360","Text":"So that transforms x,"},{"Start":"04:03.360 ","End":"04:05.115","Text":"y into r Theta,"},{"Start":"04:05.115 ","End":"04:09.154","Text":"and the z just stays as is."},{"Start":"04:09.154 ","End":"04:11.255","Text":"Z goes from what was it,"},{"Start":"04:11.255 ","End":"04:18.830","Text":"root x squared plus y squared to 1 but not exactly,"},{"Start":"04:18.830 ","End":"04:23.165","Text":"because we want to write x and y in terms of r and Theta."},{"Start":"04:23.165 ","End":"04:25.415","Text":"So if we use this formula,"},{"Start":"04:25.415 ","End":"04:28.160","Text":"x squared plus y squared,"},{"Start":"04:28.160 ","End":"04:38.080","Text":"square root is just R. So I\u0027ll erase this and write r in its place."},{"Start":"04:38.080 ","End":"04:45.390","Text":"Then we have the dv also to transform, so that\u0027s r,"},{"Start":"04:45.390 ","End":"04:55.285","Text":"and then we have dz dr d Theta."},{"Start":"04:55.285 ","End":"05:02.575","Text":"I can slightly simplify this because the first integral is dz and r doesn\u0027t depend on z,"},{"Start":"05:02.575 ","End":"05:09.095","Text":"I could pull it in front of the integral sign here."},{"Start":"05:09.095 ","End":"05:14.150","Text":"Let\u0027s pretend this is not r, it\u0027s just 1."},{"Start":"05:14.150 ","End":"05:16.805","Text":"Then the r goes here."},{"Start":"05:16.805 ","End":"05:19.400","Text":"Let\u0027s start with the inner one."},{"Start":"05:19.400 ","End":"05:23.045","Text":"We always start from the inside and work our way outwards."},{"Start":"05:23.045 ","End":"05:26.990","Text":"The first integral is just the integral of 1dz,"},{"Start":"05:29.030 ","End":"05:34.640","Text":"and the integral of 1 is just the upper limit minus the lower limit."},{"Start":"05:34.640 ","End":"05:38.000","Text":"This whole thing, if I did the integral,"},{"Start":"05:38.000 ","End":"05:46.355","Text":"this comes out to be 1 minus r. So if I substitute that here,"},{"Start":"05:46.355 ","End":"05:51.730","Text":"what I get is the integral from 0 to 2 Pi,"},{"Start":"05:51.730 ","End":"05:55.770","Text":"the integral from 0 to 1."},{"Start":"05:55.770 ","End":"06:01.755","Text":"I have this r with this 1 minus r. Let me just also do that."},{"Start":"06:01.755 ","End":"06:06.610","Text":"R times 1 minus r is just r minus r squared."},{"Start":"06:06.610 ","End":"06:12.420","Text":"Here I have r minus r squared, dr,"},{"Start":"06:12.420 ","End":"06:14.354","Text":"and then d Theta,"},{"Start":"06:14.354 ","End":"06:20.930","Text":"and now the inner integral is this one, the dr."},{"Start":"06:20.930 ","End":"06:23.340","Text":"I\u0027ll highlight it."},{"Start":"06:23.600 ","End":"06:27.220","Text":"Let me also do this one at the side."},{"Start":"06:27.220 ","End":"06:36.180","Text":"This integral is 1/2 r squared minus 1/3 r cubed."},{"Start":"06:36.180 ","End":"06:39.695","Text":"I want this from 0 to 1."},{"Start":"06:39.695 ","End":"06:41.620","Text":"The 0 gives me 0,"},{"Start":"06:41.620 ","End":"06:44.059","Text":"so I just have to plug in 1,"},{"Start":"06:44.059 ","End":"06:50.470","Text":"and what I get is 1/2 minus 1/3,"},{"Start":"06:50.470 ","End":"06:54.465","Text":"which is equal to 1/6."},{"Start":"06:54.465 ","End":"06:59.200","Text":"All this inner integral is just 1/6."},{"Start":"06:59.930 ","End":"07:05.255","Text":"I can pull this 1/6 in front of the integral,"},{"Start":"07:05.255 ","End":"07:13.050","Text":"what we\u0027re left with is 0 to 2 Pi of just d Theta or rather 1 d Theta."},{"Start":"07:14.360 ","End":"07:18.950","Text":"Just like before, when we have the integral of 1,"},{"Start":"07:18.950 ","End":"07:24.805","Text":"it\u0027s just the upper minus the lower 2 Pi minus 0 is 2 Pi,"},{"Start":"07:24.805 ","End":"07:27.865","Text":"and if I simplify this,"},{"Start":"07:27.865 ","End":"07:30.365","Text":"2/6 is a 1/3,"},{"Start":"07:30.365 ","End":"07:32.585","Text":"so it\u0027s 1/3 Pi,"},{"Start":"07:32.585 ","End":"07:36.180","Text":"or Pi/3, whichever you prefer."},{"Start":"07:36.650 ","End":"07:40.870","Text":"This is our final answer."}],"ID":9573},{"Watched":false,"Name":"Exercise 3","Duration":"9m 27s","ChapterTopicVideoID":9244,"CourseChapterTopicPlaylistID":21193,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9244.jpeg","UploadDate":"2017-03-30T09:42:39.9700000","DurationForVideoObject":"PT9M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we\u0027re given a triple integral and we\u0027re told"},{"Start":"00:04.650 ","End":"00:08.760","Text":"that we want to switch this to convert it to cylindrical coordinates,"},{"Start":"00:08.760 ","End":"00:11.939","Text":"but not to actually compute the result."},{"Start":"00:11.939 ","End":"00:18.689","Text":"I just want to emphasize here that the limits for the different variables,"},{"Start":"00:18.689 ","End":"00:22.470","Text":"so this one belongs to x because it\u0027s the outer one,"},{"Start":"00:22.470 ","End":"00:25.200","Text":"this one belongs to y the middle one,"},{"Start":"00:25.200 ","End":"00:27.630","Text":"and the inner one is for z."},{"Start":"00:27.630 ","End":"00:34.050","Text":"Let me in fact write down what this region is for this integral."},{"Start":"00:34.050 ","End":"00:36.120","Text":"X goes from 0-2,"},{"Start":"00:36.120 ","End":"00:40.905","Text":"so I write that x is some which between 0 and 2."},{"Start":"00:40.905 ","End":"00:46.230","Text":"Y goes from 0 to this expression,"},{"Start":"00:46.230 ","End":"00:50.930","Text":"square root of 2x minus x squared and z less important for me,"},{"Start":"00:50.930 ","End":"00:52.070","Text":"but I\u0027ll write it anyway,"},{"Start":"00:52.070 ","End":"01:00.940","Text":"goes from minus the square root of this thing up to the same thing with a plus sign,"},{"Start":"01:00.940 ","End":"01:05.945","Text":"the square root of 4 minus x squared minus y squared."},{"Start":"01:05.945 ","End":"01:09.410","Text":"For starters I want to see in the x,"},{"Start":"01:09.410 ","End":"01:16.235","Text":"y plane how I can express this x and y limits."},{"Start":"01:16.235 ","End":"01:24.260","Text":"Well, I brought in a picture and I\u0027ll explain why this region that I highlighted,"},{"Start":"01:24.260 ","End":"01:28.590","Text":"the upper 1/2 of a disk like this,"},{"Start":"01:28.590 ","End":"01:30.605","Text":"why this corresponds to this?"},{"Start":"01:30.605 ","End":"01:34.340","Text":"Well, the 0 less than or equal to x less than or equal to 2,"},{"Start":"01:34.340 ","End":"01:37.230","Text":"we can see x goes from 0-2."},{"Start":"01:37.270 ","End":"01:44.360","Text":"What I still have to show you is that this here really is"},{"Start":"01:44.360 ","End":"01:52.895","Text":"y equals the square root of 2x minus x squared."},{"Start":"01:52.895 ","End":"01:56.330","Text":"This is obviously y equals 0,"},{"Start":"01:56.330 ","End":"02:01.160","Text":"the x axis and y goes between the 2."},{"Start":"02:01.160 ","End":"02:03.365","Text":"Let me explain why this is so."},{"Start":"02:03.365 ","End":"02:07.995","Text":"If I look at this and I square,"},{"Start":"02:07.995 ","End":"02:09.585","Text":"well, I can start with this."},{"Start":"02:09.585 ","End":"02:12.355","Text":"Let me do the computations over here."},{"Start":"02:12.355 ","End":"02:14.885","Text":"If I square both sides,"},{"Start":"02:14.885 ","End":"02:21.925","Text":"I would get y squared equals 2x minus x squared."},{"Start":"02:21.925 ","End":"02:25.550","Text":"Then if I bring this stuff over here to the left,"},{"Start":"02:25.550 ","End":"02:27.110","Text":"I write the x\u0027s first,"},{"Start":"02:27.110 ","End":"02:36.175","Text":"it\u0027s going to be plus x squared minus 2x plus y squared equals 0."},{"Start":"02:36.175 ","End":"02:39.809","Text":"Now we do what we call completing the square,"},{"Start":"02:39.809 ","End":"02:42.385","Text":"I write x squared minus 2x,"},{"Start":"02:42.385 ","End":"02:45.500","Text":"I want it to be x minus something squared."},{"Start":"02:45.500 ","End":"02:53.385","Text":"I take 1/2 this co-efficient x minus 1 and square it and if we square this,"},{"Start":"02:53.385 ","End":"02:57.735","Text":"we get x squared minus 2x plus 1."},{"Start":"02:57.735 ","End":"03:01.515","Text":"I\u0027ve added an extra 1 on this side,"},{"Start":"03:01.515 ","End":"03:04.920","Text":"so I also add an extra 1 on the right-hand side and now"},{"Start":"03:04.920 ","End":"03:08.415","Text":"I\u0027m all right if I put here plus 1 and here plus 1,"},{"Start":"03:08.415 ","End":"03:10.170","Text":"then this works out,"},{"Start":"03:10.170 ","End":"03:14.510","Text":"and we see that this is the equation of a circle,"},{"Start":"03:14.510 ","End":"03:18.000","Text":"1 is also 1 squared."},{"Start":"03:22.160 ","End":"03:26.250","Text":"It\u0027s a circle with center 1,"},{"Start":"03:26.250 ","End":"03:34.050","Text":"0 and radius of 1."},{"Start":"03:34.050 ","End":"03:37.700","Text":"That would normally be the full circle."},{"Start":"03:37.700 ","End":"03:39.345","Text":"But when we squared some things,"},{"Start":"03:39.345 ","End":"03:47.245","Text":"we got extra solutions because we see that y is the square root so it\u0027s non-negative."},{"Start":"03:47.245 ","End":"03:53.015","Text":"The other 1/2 of the circle would be y equals minus the square root."},{"Start":"03:53.015 ","End":"03:55.010","Text":"Although this is a full circle,"},{"Start":"03:55.010 ","End":"03:58.520","Text":"we only have 1/2 the circle."},{"Start":"03:58.520 ","End":"04:02.960","Text":"I\u0027ll just add just to tie in the concepts that we took this as"},{"Start":"04:02.960 ","End":"04:08.060","Text":"a type 1 region meaning if we had a particular x here,"},{"Start":"04:08.060 ","End":"04:16.115","Text":"for this particular x, we took y going from this point here to this point here."},{"Start":"04:16.115 ","End":"04:20.040","Text":"This was on the lower curve, 0,"},{"Start":"04:20.040 ","End":"04:24.750","Text":"this was on the upper curve 2x minus x squared square root and that gave"},{"Start":"04:24.750 ","End":"04:30.815","Text":"us these limits for y as x sweeps across from 0-2."},{"Start":"04:30.815 ","End":"04:34.050","Text":"Now we\u0027re ready to convert,"},{"Start":"04:34.050 ","End":"04:35.460","Text":"I was going to say polar,"},{"Start":"04:35.460 ","End":"04:37.110","Text":"2-dimensions it would be polar,"},{"Start":"04:37.110 ","End":"04:39.820","Text":"but in 3-dimensions it\u0027s cylindrical."},{"Start":"04:39.820 ","End":"04:42.155","Text":"Let me bring in the formulas."},{"Start":"04:42.155 ","End":"04:45.080","Text":"Here are all the formulas I copy pasted from"},{"Start":"04:45.080 ","End":"04:49.490","Text":"a previous exercise and I also want to add that in general,"},{"Start":"04:49.490 ","End":"04:55.060","Text":"this dxdydz in any particular order is dv,"},{"Start":"04:55.060 ","End":"04:57.290","Text":"I\u0027ll need that for substitution."},{"Start":"04:57.290 ","End":"05:00.575","Text":"I also want to describe this in polar coordinates."},{"Start":"05:00.575 ","End":"05:04.195","Text":"It is circular or semicircular but,"},{"Start":"05:04.195 ","End":"05:08.990","Text":"it\u0027s not centered at the origin so we have to work a little bit."},{"Start":"05:08.990 ","End":"05:13.130","Text":"Now let\u0027s see where does Theta go from and to."},{"Start":"05:13.130 ","End":"05:22.860","Text":"Well, Theta, if I take a particular Theta we can see that it starts off at the x-axis"},{"Start":"05:22.860 ","End":"05:31.580","Text":"where Theta equals 0 and as this point goes along the angle gets steeper and steeper"},{"Start":"05:31.580 ","End":"05:40.830","Text":"until ultimately it reaches Theta equals 90 degrees or Pi over 2,"},{"Start":"05:40.830 ","End":"05:42.955","Text":"so that\u0027s our limits for Theta."},{"Start":"05:42.955 ","End":"05:47.510","Text":"I can start writing the integral as the integral,"},{"Start":"05:47.510 ","End":"05:52.445","Text":"the Theta I have from 0 to Pi over 2."},{"Start":"05:52.445 ","End":"05:54.730","Text":"Now let\u0027s see what about r?"},{"Start":"05:54.730 ","End":"06:02.700","Text":"Now this r goes from 0 but to this point here."},{"Start":"06:02.700 ","End":"06:04.800","Text":"What this point here?"},{"Start":"06:04.800 ","End":"06:08.520","Text":"I\u0027ll give it a name let\u0027s call it say r_1,"},{"Start":"06:08.520 ","End":"06:12.405","Text":"which varies as I go along so it\u0027s a function of"},{"Start":"06:12.405 ","End":"06:18.920","Text":"Theta and what I\u0027m going to do is use these formulas."},{"Start":"06:18.920 ","End":"06:24.440","Text":"Meanwhile, I\u0027ll just write that we know that r goes from 0,"},{"Start":"06:24.440 ","End":"06:26.630","Text":"but I\u0027ll leave a question mark here,"},{"Start":"06:26.630 ","End":"06:32.285","Text":"which is this r_1 and we\u0027re going to compute this with the help of,"},{"Start":"06:32.285 ","End":"06:35.645","Text":"let\u0027s see, this particular equation."},{"Start":"06:35.645 ","End":"06:45.439","Text":"If I convert this using the cylindrical or polar conversion formulas,"},{"Start":"06:45.439 ","End":"06:51.185","Text":"we see that x squared plus y squared is r squared."},{"Start":"06:51.185 ","End":"06:58.260","Text":"This gives me that r squared minus 2x,"},{"Start":"06:58.260 ","End":"07:07.905","Text":"which is minus 2r cosine Theta equals 0."},{"Start":"07:07.905 ","End":"07:12.270","Text":"Now, I just didn\u0027t want to use this letter r twice,"},{"Start":"07:12.270 ","End":"07:13.545","Text":"so I called it r_1."},{"Start":"07:13.545 ","End":"07:15.975","Text":"I\u0027ll call it r_1 here,"},{"Start":"07:15.975 ","End":"07:23.630","Text":"and I can divide both sides by r_1."},{"Start":"07:23.630 ","End":"07:27.380","Text":"Well, let me first of all bring this to the other side and say that r_1"},{"Start":"07:27.380 ","End":"07:32.850","Text":"squared is 2r_1cosine Theta."},{"Start":"07:32.890 ","End":"07:43.280","Text":"Now I\u0027m going to divide both sides by r_1 and get that r_1 equals 2 cosine Theta."},{"Start":"07:43.280 ","End":"07:45.350","Text":"You might say, well,"},{"Start":"07:45.350 ","End":"07:48.890","Text":"how do you know that r_1 isn\u0027t 0?"},{"Start":"07:48.890 ","End":"07:54.380","Text":"r_1 is actually 0 only when we finally get to this point here where"},{"Start":"07:54.380 ","End":"08:00.965","Text":"Theta\u0027s 90 degrees or Pi over 2 and it works because cosine of 90 degrees is 0."},{"Start":"08:00.965 ","End":"08:04.580","Text":"It works for that too so in case you were"},{"Start":"08:04.580 ","End":"08:09.425","Text":"wondering why I\u0027m allowed to divide by r_1, it\u0027s fine."},{"Start":"08:09.425 ","End":"08:13.370","Text":"Now I have this expression, 2 cosine Theta,"},{"Start":"08:13.370 ","End":"08:20.940","Text":"so I can replace this question mark by 2 cosine Theta."},{"Start":"08:20.940 ","End":"08:24.670","Text":"Now I have the third integral,"},{"Start":"08:25.580 ","End":"08:33.470","Text":"this is the limits for z and since x squared plus y squared is r squared,"},{"Start":"08:33.470 ","End":"08:35.725","Text":"what I have is the square root,"},{"Start":"08:35.725 ","End":"08:40.020","Text":"here it\u0027s 4 minus r squared and"},{"Start":"08:40.020 ","End":"08:46.130","Text":"here it\u0027s just the same thing with a plus 4 minus r squared."},{"Start":"08:46.130 ","End":"08:49.205","Text":"Finally, I have to convert dv,"},{"Start":"08:49.205 ","End":"08:51.860","Text":"and here I have the formula here so this"},{"Start":"08:51.860 ","End":"08:55.940","Text":"is r and now I have to close them in the right order."},{"Start":"08:55.940 ","End":"08:57.320","Text":"This was the integral,"},{"Start":"08:57.320 ","End":"09:00.740","Text":"I should have written z equals."},{"Start":"09:00.740 ","End":"09:05.120","Text":"Here we have the dz which closes this one,"},{"Start":"09:05.120 ","End":"09:08.175","Text":"then for this one we need the dr,"},{"Start":"09:08.175 ","End":"09:12.675","Text":"and for the outer one we need d Theta."},{"Start":"09:12.675 ","End":"09:18.885","Text":"This is the same integral in cylindrical coordinates."},{"Start":"09:18.885 ","End":"09:27.550","Text":"I\u0027ll just highlight it and this is our final answer. We\u0027re done."}],"ID":9574},{"Watched":false,"Name":"Exercise 4","Duration":"22m 16s","ChapterTopicVideoID":9243,"CourseChapterTopicPlaylistID":21193,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9243.jpeg","UploadDate":"2017-03-30T09:41:02.8470000","DurationForVideoObject":"PT22M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this exercise, we\u0027re given a description of a solid body that it\u0027s"},{"Start":"00:04.200 ","End":"00:11.040","Text":"pounded the sides by the cylinder x squared plus y squared equals 9."},{"Start":"00:11.040 ","End":"00:18.075","Text":"Below it\u0027s the xy-plane and above it\u0027s the hemisphere given by this formula."},{"Start":"00:18.075 ","End":"00:21.240","Text":"We have to compute the volume of the body."},{"Start":"00:21.240 ","End":"00:22.650","Text":"That\u0027s the first part."},{"Start":"00:22.650 ","End":"00:27.050","Text":"Then later we\u0027ll compute also its centroid."},{"Start":"00:27.050 ","End":"00:30.425","Text":"Sometimes call center of mass."},{"Start":"00:30.425 ","End":"00:38.390","Text":"I\u0027d like to sketch the cylinder or at least the projection onto the xy-plane."},{"Start":"00:38.870 ","End":"00:41.530","Text":"I say that in the xy-plane,"},{"Start":"00:41.530 ","End":"00:45.830","Text":"this is just the equation of a circle because 9 is 3 squared."},{"Start":"00:45.830 ","End":"00:48.770","Text":"It\u0027s a circle centered at the origin,"},{"Start":"00:48.770 ","End":"00:51.605","Text":"and with radius 3,"},{"Start":"00:51.605 ","End":"01:00.110","Text":"and I\u0027m called it d. What we have now is the solid which is below is just this,"},{"Start":"01:00.110 ","End":"01:04.950","Text":"this is just the bottom surface of this body and above,"},{"Start":"01:05.120 ","End":"01:07.815","Text":"well, in the third dimension,"},{"Start":"01:07.815 ","End":"01:09.660","Text":"it\u0027s in the z-direction,"},{"Start":"01:09.660 ","End":"01:11.330","Text":"it\u0027s equal to this."},{"Start":"01:11.330 ","End":"01:14.740","Text":"The way I\u0027m going to describe the body,"},{"Start":"01:14.740 ","End":"01:18.560","Text":"well, before that, let me just write the formula,"},{"Start":"01:19.220 ","End":"01:24.825","Text":"what we need for the volume is the integral,"},{"Start":"01:24.825 ","End":"01:31.050","Text":"and let\u0027s call the body B that the"},{"Start":"01:31.050 ","End":"01:38.860","Text":"integral over B of 1 dv,"},{"Start":"01:40.470 ","End":"01:48.055","Text":"that this expression is the volume of the body of,"},{"Start":"01:48.055 ","End":"01:49.885","Text":"in this case, B."},{"Start":"01:49.885 ","End":"01:53.255","Text":"Now, this triple integral,"},{"Start":"01:53.255 ","End":"01:56.300","Text":"I want to write it as an iterated integral."},{"Start":"01:56.300 ","End":"02:00.490","Text":"I\u0027m going to do it in a slightly peculiar way."},{"Start":"02:00.490 ","End":"02:07.235","Text":"What I\u0027m going to do is say that let\u0027s take the first two together."},{"Start":"02:07.235 ","End":"02:10.490","Text":"I don\u0027t want to do it as x and y. I could,"},{"Start":"02:10.490 ","End":"02:14.180","Text":"I could say x goes from minus 3-3 and then"},{"Start":"02:14.180 ","End":"02:19.275","Text":"compute plus or minus the square root of something and say, y."},{"Start":"02:19.275 ","End":"02:25.980","Text":"But I\u0027m going to be converting to cylindrical coordinates or like polar in 2D."},{"Start":"02:25.980 ","End":"02:31.550","Text":"I\u0027ll do these 2 combined and say this is the double integral over D,"},{"Start":"02:31.550 ","End":"02:34.974","Text":"and then I\u0027ll take the third dimension z,"},{"Start":"02:34.974 ","End":"02:41.610","Text":"going from below the xy-plane is where z equals 0,"},{"Start":"02:41.610 ","End":"02:46.325","Text":"so I\u0027ll go from 0 up to the square root"},{"Start":"02:46.325 ","End":"02:52.915","Text":"of 25 minus x squared minus y squared,"},{"Start":"02:52.915 ","End":"02:58.760","Text":"and that\u0027s going to be times 1 dv."},{"Start":"02:58.760 ","End":"03:02.130","Text":"I\u0027m going to convert to,"},{"Start":"03:02.180 ","End":"03:04.605","Text":"keeps wanting to say polar,"},{"Start":"03:04.605 ","End":"03:06.530","Text":"now it\u0027s cylindrical because we have,"},{"Start":"03:06.530 ","End":"03:07.820","Text":"but it\u0027s almost the same."},{"Start":"03:07.820 ","End":"03:09.185","Text":"The z doesn\u0027t change."},{"Start":"03:09.185 ","End":"03:13.740","Text":"Let me write the equations of cylindrical conversion."},{"Start":"03:13.910 ","End":"03:17.120","Text":"Here they are even more than we need."},{"Start":"03:17.120 ","End":"03:20.450","Text":"This 1 doesn\u0027t strictly have to be in there,"},{"Start":"03:20.450 ","End":"03:24.290","Text":"but it\u0027s so useful that I included and z equals z seem silly,"},{"Start":"03:24.290 ","End":"03:27.770","Text":"but just to emphasize that z doesn\u0027t change."},{"Start":"03:27.770 ","End":"03:33.140","Text":"What I want to do now is express this region in cylindrical,"},{"Start":"03:33.140 ","End":"03:39.185","Text":"which is really polar on the x and y changing to r and Theta and the z stays unchanged."},{"Start":"03:39.185 ","End":"03:44.610","Text":"What I have to do is describe this d in r and Theta."},{"Start":"03:45.530 ","End":"03:49.150","Text":"Since it\u0027s a circle centered at the origin,"},{"Start":"03:49.150 ","End":"03:53.170","Text":"we know that Theta here goes,"},{"Start":"03:53.170 ","End":"03:58.415","Text":"I can say Theta goes from 0-2Pi."},{"Start":"03:58.415 ","End":"03:59.680","Text":"We\u0027ve done this before."},{"Start":"03:59.680 ","End":"04:02.110","Text":"We start off here."},{"Start":"04:02.130 ","End":"04:05.020","Text":"Theta equals 0."},{"Start":"04:05.020 ","End":"04:07.090","Text":"We have a typical Theta here,"},{"Start":"04:07.090 ","End":"04:14.750","Text":"and it goes from 0 all the way around until it ends up the same place only."},{"Start":"04:14.750 ","End":"04:18.300","Text":"This is also Theta equals 2-Pi."},{"Start":"04:18.300 ","End":"04:20.670","Text":"Theta goes from 0-2Pi."},{"Start":"04:20.670 ","End":"04:21.930","Text":"For any given Theta,"},{"Start":"04:21.930 ","End":"04:26.355","Text":"r goes from 0 to the radius which is."},{"Start":"04:26.355 ","End":"04:30.905","Text":"Already I have a start that this is equal to"},{"Start":"04:30.905 ","End":"04:38.920","Text":"the integral as Theta goes from 0-2Pi."},{"Start":"04:38.920 ","End":"04:43.530","Text":"Then r goes from 0-3."},{"Start":"04:43.530 ","End":"04:51.390","Text":"Now I\u0027ve expressed the d in polar or cylindrical terms, I just need the z,"},{"Start":"04:51.620 ","End":"04:57.830","Text":"and z doesn\u0027t change when we go to cylindrical z equals z,"},{"Start":"04:57.830 ","End":"05:01.010","Text":"and it\u0027s just from below,"},{"Start":"05:01.010 ","End":"05:07.055","Text":"z equals 0 and above the square root"},{"Start":"05:07.055 ","End":"05:15.250","Text":"of 25 minus x squared minus y squared."},{"Start":"05:15.250 ","End":"05:17.805","Text":"I\u0027ll mention this though it\u0027s not important,"},{"Start":"05:17.805 ","End":"05:25.865","Text":"that this is the equation of an upper hemisphere of radius 5."},{"Start":"05:25.865 ","End":"05:31.935","Text":"It hovers somewhere, a part of it above this disk of radius 3,"},{"Start":"05:31.935 ","End":"05:36.725","Text":"that doesn\u0027t matter if you\u0027re just mentioning it\u0027s a hemisphere."},{"Start":"05:36.725 ","End":"05:42.580","Text":"Then I need to convert."},{"Start":"05:42.580 ","End":"05:46.550","Text":"What I did is not quite right because I have to"},{"Start":"05:46.550 ","End":"05:50.840","Text":"convert to our Theta z and I\u0027m using this 1,"},{"Start":"05:50.840 ","End":"05:52.970","Text":"that x squared plus y squared is r squared."},{"Start":"05:52.970 ","End":"05:57.215","Text":"I\u0027m going to erase the x squared plus y squared."},{"Start":"05:57.215 ","End":"06:00.710","Text":"I mean, this is minus brackets x squared plus y squared."},{"Start":"06:00.710 ","End":"06:04.910","Text":"Instead of that, I\u0027ll write minus r squared."},{"Start":"06:04.910 ","End":"06:07.070","Text":"Now that we\u0027re better off,"},{"Start":"06:07.070 ","End":"06:10.370","Text":"now the dv, I use the formula here."},{"Start":"06:10.370 ","End":"06:14.520","Text":"It\u0027s r, and then in this order,"},{"Start":"06:14.520 ","End":"06:19.430","Text":"dz because I closed the z and then I need to close the r with a dr,"},{"Start":"06:19.430 ","End":"06:22.100","Text":"I need to close the Theta with a d Theta."},{"Start":"06:22.100 ","End":"06:26.540","Text":"This now is the integral which gives us our volume of B."},{"Start":"06:26.540 ","End":"06:31.020","Text":"Let\u0027s call it V for volume."},{"Start":"06:31.540 ","End":"06:36.280","Text":"This volume will be V. After we\u0027ve done this,"},{"Start":"06:36.280 ","End":"06:40.150","Text":"we\u0027ll have that and then we\u0027ll worry later about the centroid."},{"Start":"06:40.150 ","End":"06:48.335","Text":"Now, because the first integral is in the 1 dz and r is a constant as far as x goes."},{"Start":"06:48.335 ","End":"06:56.270","Text":"I would prefer to take the r out of here and pull it in front here and just leave 1 here."},{"Start":"06:56.670 ","End":"07:01.000","Text":"The first integral then will be this inner 1,"},{"Start":"07:01.000 ","End":"07:04.605","Text":"the integral of 1 dz,"},{"Start":"07:04.605 ","End":"07:11.080","Text":"and the integral of 1 is always just the upper minus the lower."},{"Start":"07:11.090 ","End":"07:16.515","Text":"What I get is, these in a minute."},{"Start":"07:16.515 ","End":"07:19.360","Text":"Instead of this, I still have the r,"},{"Start":"07:19.360 ","End":"07:21.920","Text":"but instead of this, I can write this minus 0,"},{"Start":"07:21.920 ","End":"07:30.120","Text":"which is just the square root of 25 minus r squared dr d Theta,"},{"Start":"07:30.120 ","End":"07:31.565","Text":"and let me just write here,"},{"Start":"07:31.565 ","End":"07:33.820","Text":"r goes from 0-3,"},{"Start":"07:33.820 ","End":"07:37.605","Text":"Theta goes from 0-2Pi,"},{"Start":"07:37.605 ","End":"07:41.115","Text":"and now we\u0027ve got it down to a double integral,"},{"Start":"07:41.115 ","End":"07:46.030","Text":"and the next step we want to compute is this dr,"},{"Start":"07:46.030 ","End":"07:50.130","Text":"and I\u0027ll do this integral here at this side."},{"Start":"07:50.130 ","End":"07:52.910","Text":"We can do this 1 by substitution."},{"Start":"07:52.910 ","End":"07:54.440","Text":"There\u0027s actually 2 possibilities."},{"Start":"07:54.440 ","End":"08:00.925","Text":"They\u0027ll both work. Either substitute 25 minus r squared or the square root of it."},{"Start":"08:00.925 ","End":"08:04.090","Text":"I think it works slightly better with the square root."},{"Start":"08:04.090 ","End":"08:11.690","Text":"Let me take t equals the square root of 25 minus r squared,"},{"Start":"08:11.690 ","End":"08:14.780","Text":"and I could actually square both sides,"},{"Start":"08:14.780 ","End":"08:16.370","Text":"get rid of the square root."},{"Start":"08:16.370 ","End":"08:21.510","Text":"I\u0027ve got t squared is 25 minus r squared."},{"Start":"08:21.510 ","End":"08:26.710","Text":"Now, if I differentiate both sides here I have 2t, dt,"},{"Start":"08:26.710 ","End":"08:30.195","Text":"and here I have minus 2r,"},{"Start":"08:30.195 ","End":"08:35.470","Text":"dr, and I\u0027ll just make some more space."},{"Start":"08:35.630 ","End":"08:38.895","Text":"Then I could certainly cancel the 2s."},{"Start":"08:38.895 ","End":"08:42.390","Text":"Now, look here I have rdr."},{"Start":"08:42.390 ","End":"08:46.160","Text":"So rdr, I can replace,"},{"Start":"08:46.160 ","End":"08:50.700","Text":"if I put the minus over as minus tdt."},{"Start":"08:51.080 ","End":"08:55.595","Text":"Also, the 25 minus r squared can be replaced by"},{"Start":"08:55.595 ","End":"09:01.355","Text":"t. But there\u0027s something else I have to substitute and that\u0027s the upper and lower limits."},{"Start":"09:01.355 ","End":"09:03.050","Text":"I\u0027ll do that in a moment."},{"Start":"09:03.050 ","End":"09:04.525","Text":"I\u0027ll replace this in this."},{"Start":"09:04.525 ","End":"09:07.370","Text":"First of all, just see what we have so far with other"},{"Start":"09:07.370 ","End":"09:10.090","Text":"said the 25 minus r squared square root is"},{"Start":"09:10.090 ","End":"09:16.540","Text":"just t. Then the rdr gives me minus tdt."},{"Start":"09:17.540 ","End":"09:20.900","Text":"Now, I\u0027ll substitute here and here."},{"Start":"09:20.900 ","End":"09:24.145","Text":"When r is 0,"},{"Start":"09:24.145 ","End":"09:26.495","Text":"you see when r is 0,"},{"Start":"09:26.495 ","End":"09:33.800","Text":"I get that t equals square root of 25 minus 0 is 5,"},{"Start":"09:33.800 ","End":"09:36.155","Text":"so the 5 is down here."},{"Start":"09:36.155 ","End":"09:38.365","Text":"When r is full,"},{"Start":"09:38.365 ","End":"09:40.590","Text":"I get 25 minus 4 squared."},{"Start":"09:40.590 ","End":"09:42.705","Text":"25 minus 16 is 9."},{"Start":"09:42.705 ","End":"09:45.790","Text":"That t is 3,"},{"Start":"09:46.700 ","End":"09:48.870","Text":"I did that the wrong way round,"},{"Start":"09:48.870 ","End":"09:52.650","Text":"what I had up there was a 3"},{"Start":"09:52.650 ","End":"09:56.760","Text":"and then this here comes out to be 4 somehow I got that backwards."},{"Start":"09:56.760 ","End":"10:01.835","Text":"This is a 4. What I can do now,"},{"Start":"10:01.835 ","End":"10:07.175","Text":"I also don\u0027t like it when there\u0027s a minus here and this is less than this,"},{"Start":"10:07.175 ","End":"10:13.474","Text":"that the standard trick is to reverse the upper and the lower."},{"Start":"10:13.474 ","End":"10:16.540","Text":"I can write it as from 4-5,"},{"Start":"10:16.540 ","End":"10:18.910","Text":"and that makes it negative,"},{"Start":"10:18.910 ","End":"10:21.085","Text":"and then I can swallow this negative,"},{"Start":"10:21.085 ","End":"10:24.710","Text":"so I\u0027ve just got t squared dt."},{"Start":"10:24.710 ","End":"10:27.590","Text":"For subtracting, subtracting the other order,"},{"Start":"10:27.590 ","End":"10:29.315","Text":"and it takes care of a minus."},{"Start":"10:29.315 ","End":"10:38.755","Text":"What we get now is that this is equal to say 1/3 of t cubed,"},{"Start":"10:38.755 ","End":"10:44.310","Text":"and this has got to be taken from 4-5."},{"Start":"10:44.310 ","End":"10:53.370","Text":"Let\u0027s see, we\u0027ve got 1/3 of 5 cubed is 125,"},{"Start":"10:53.370 ","End":"10:58.300","Text":"4 cubed is 64."},{"Start":"10:58.350 ","End":"11:01.555","Text":"This minus this is 61,"},{"Start":"11:01.555 ","End":"11:05.200","Text":"it just get 61 over 3."},{"Start":"11:05.200 ","End":"11:10.525","Text":"I\u0027ll make a note that here all this came out to be 61 over 3,"},{"Start":"11:10.525 ","End":"11:12.220","Text":"so there\u0027s a constant,"},{"Start":"11:12.220 ","End":"11:14.650","Text":"I can take it in front of the integral."},{"Start":"11:14.650 ","End":"11:19.480","Text":"In the next step, I\u0027ve just have 61 over 3 times the"},{"Start":"11:19.480 ","End":"11:24.430","Text":"integral from 0-2 Pi of just d Theta,"},{"Start":"11:24.430 ","End":"11:26.500","Text":"write that as 1 d Theta."},{"Start":"11:26.500 ","End":"11:29.155","Text":"We\u0027ve already seen this enough times the integral of 1,"},{"Start":"11:29.155 ","End":"11:31.660","Text":"you just subtract upper minus lower,"},{"Start":"11:31.660 ","End":"11:33.895","Text":"that gives me 2 Pi."},{"Start":"11:33.895 ","End":"11:36.714","Text":"This bit here is 2 Pi,"},{"Start":"11:36.714 ","End":"11:42.730","Text":"so altogether I get 61 over 3 times 2 Pi,"},{"Start":"11:42.730 ","End":"11:51.880","Text":"122 over 3 Pi."},{"Start":"11:51.880 ","End":"11:55.555","Text":"This is what we call the volume V,"},{"Start":"11:55.555 ","End":"11:57.430","Text":"and I\u0027ll highlight it,"},{"Start":"11:57.430 ","End":"12:03.305","Text":"and that just ends the first part because now we also have to find the centroid."},{"Start":"12:03.305 ","End":"12:07.625","Text":"Now I\u0027m going to erase some stuff that we don\u0027t need."},{"Start":"12:07.625 ","End":"12:11.605","Text":"Let\u0027s go back and see what the original question asked."},{"Start":"12:11.605 ","End":"12:17.050","Text":"We have the volume already and I kept the result and now we want the centroid."},{"Start":"12:17.050 ","End":"12:19.880","Text":"I\u0027II give the formula for the centroid."},{"Start":"12:19.880 ","End":"12:26.565","Text":"The centroid is a point in 3D space and it has an x and the y and a z,"},{"Start":"12:26.565 ","End":"12:28.850","Text":"I\u0027m going to call it x bar,"},{"Start":"12:28.850 ","End":"12:31.975","Text":"y bar, z bar."},{"Start":"12:31.975 ","End":"12:33.280","Text":"Just to be specific,"},{"Start":"12:33.280 ","End":"12:35.530","Text":"that it\u0027s specific x, y, and z."},{"Start":"12:35.530 ","End":"12:38.860","Text":"There\u0027s a separate formula for each of them."},{"Start":"12:38.860 ","End":"12:42.340","Text":"In general, for a body such as this,"},{"Start":"12:42.340 ","End":"12:43.720","Text":"instead of this integral,"},{"Start":"12:43.720 ","End":"12:45.565","Text":"we have something similar,"},{"Start":"12:45.565 ","End":"12:55.375","Text":"is that x bar is equal to 1 over the volume of the body,"},{"Start":"12:55.375 ","End":"12:57.880","Text":"that\u0027s V I\u0027d say,"},{"Start":"12:57.880 ","End":"13:03.850","Text":"triple integral over the same body."},{"Start":"13:03.850 ","End":"13:06.145","Text":"Instead of 1 dv,"},{"Start":"13:06.145 ","End":"13:11.680","Text":"I have x dv, and similarly,"},{"Start":"13:11.680 ","End":"13:21.160","Text":"y bar is 1 over v triple integral over the body of y dv,"},{"Start":"13:21.160 ","End":"13:24.400","Text":"and z bar, as you can guess,"},{"Start":"13:24.400 ","End":"13:31.690","Text":"is 1 over v triple integral over the body of z dv."},{"Start":"13:31.690 ","End":"13:35.665","Text":"Now, there\u0027s something that is often"},{"Start":"13:35.665 ","End":"13:40.060","Text":"done in mathematics and physics and that is to use the concept of symmetry."},{"Start":"13:40.060 ","End":"13:44.455","Text":"I mean, this whole thing has a circular symmetry about the z-axis,"},{"Start":"13:44.455 ","End":"13:47.920","Text":"the cylinder is symmetric about the z-axis,"},{"Start":"13:47.920 ","End":"13:50.215","Text":"and so is this."},{"Start":"13:50.215 ","End":"13:52.270","Text":"If I put x instead of minus x,"},{"Start":"13:52.270 ","End":"13:56.245","Text":"I\u0027ll get the same thing and y instead of minus y everywhere."},{"Start":"13:56.245 ","End":"13:58.390","Text":"As far as an x and y goes,"},{"Start":"13:58.390 ","End":"13:59.950","Text":"we have complete symmetry,"},{"Start":"13:59.950 ","End":"14:06.100","Text":"so there\u0027s no reason why the xy of the centroid should be anything other than 0."},{"Start":"14:06.100 ","End":"14:08.515","Text":"It has to be symmetrical."},{"Start":"14:08.515 ","End":"14:14.455","Text":"We often just say that this is equal to 0 and this is equal to 0,"},{"Start":"14:14.455 ","End":"14:17.410","Text":"and we just write here"},{"Start":"14:17.410 ","End":"14:25.015","Text":"for reasons of symmetry,"},{"Start":"14:25.015 ","End":"14:28.070","Text":"and usually this is good enough."},{"Start":"14:29.640 ","End":"14:33.970","Text":"If you want to, you could actually do the integral,"},{"Start":"14:33.970 ","End":"14:36.685","Text":"but we will be doing 1 of the integrals."},{"Start":"14:36.685 ","End":"14:38.560","Text":"Let\u0027s see if we can do this 1."},{"Start":"14:38.560 ","End":"14:43.150","Text":"Now, I want to make use of the work I\u0027ve done previously."},{"Start":"14:43.150 ","End":"14:52.135","Text":"I already figured that this body B is really over a disc,"},{"Start":"14:52.135 ","End":"14:57.655","Text":"which was a disc of center 3 in the xy plane centered at the origin."},{"Start":"14:57.655 ","End":"15:01.585","Text":"Then I had this z, and that\u0027s how I got this integral."},{"Start":"15:01.585 ","End":"15:04.765","Text":"The difference is that this z here,"},{"Start":"15:04.765 ","End":"15:08.200","Text":"all I have to do is take this same integral,"},{"Start":"15:08.200 ","End":"15:12.445","Text":"put a 1 over v in front and stick an extra z in here,"},{"Start":"15:12.445 ","End":"15:15.715","Text":"and I should get the integral for this."},{"Start":"15:15.715 ","End":"15:22.630","Text":"I get that z bar equals 1 over v,"},{"Start":"15:22.630 ","End":"15:25.960","Text":"that\u0027s this volume here I\u0027ll plug it in at the end."},{"Start":"15:25.960 ","End":"15:31.585","Text":"Integral same thing from 0-2 Pi, that\u0027s the Theta,"},{"Start":"15:31.585 ","End":"15:39.010","Text":"the integral from 0-3 for r. Then I have this r that I took out here."},{"Start":"15:39.010 ","End":"15:47.785","Text":"Then the integral from 0 to the square root of 25 minus r squared,"},{"Start":"15:47.785 ","End":"15:50.230","Text":"and the only difference instead of a 1,"},{"Start":"15:50.230 ","End":"15:54.110","Text":"I now have the z here,"},{"Start":"15:54.870 ","End":"15:58.975","Text":"dz, dr, d Theta."},{"Start":"15:58.975 ","End":"16:02.020","Text":"I\u0027ll remove the highlighting here,"},{"Start":"16:02.020 ","End":"16:06.700","Text":"and I\u0027ll put some over here because this is where we\u0027re working now,"},{"Start":"16:06.700 ","End":"16:11.065","Text":"and our first integral is the dz integral."},{"Start":"16:11.065 ","End":"16:16.510","Text":"I\u0027d like to do this in an integral somewhere at the side, say here,"},{"Start":"16:16.510 ","End":"16:22.120","Text":"the integral of z is 1/2 z squared,"},{"Start":"16:22.120 ","End":"16:27.820","Text":"and I need the limits from 0 to"},{"Start":"16:27.820 ","End":"16:35.425","Text":"square root of 25 minus r squared."},{"Start":"16:35.425 ","End":"16:39.925","Text":"Let\u0027s see what this equals, if z is 0,"},{"Start":"16:39.925 ","End":"16:43.990","Text":"that\u0027s just nothing, if z is the square root of this,"},{"Start":"16:43.990 ","End":"16:47.080","Text":"then z squared is just this itself."},{"Start":"16:47.080 ","End":"16:55.675","Text":"I end up getting 1/2 of 25 minus r squared without the square root."},{"Start":"16:55.675 ","End":"17:00.110","Text":"If I put that back in here,"},{"Start":"17:00.660 ","End":"17:05.305","Text":"then we will get 1 over"},{"Start":"17:05.305 ","End":"17:13.705","Text":"v times the integral from 0-2 Pi,"},{"Start":"17:13.705 ","End":"17:18.190","Text":"the integral from 0-3,"},{"Start":"17:18.190 ","End":"17:23.510","Text":"that\u0027s for r. I\u0027ll put this r together with this,"},{"Start":"17:23.820 ","End":"17:27.310","Text":"I can take the 2 in front also,"},{"Start":"17:27.310 ","End":"17:30.670","Text":"I\u0027II tell you what the 1/2 I\u0027ll pull in front."},{"Start":"17:30.670 ","End":"17:33.385","Text":"That\u0027s make that then 2v,"},{"Start":"17:33.385 ","End":"17:35.185","Text":"that takes care of the 2,"},{"Start":"17:35.185 ","End":"17:38.500","Text":"and then the r with the 25 minus r squared,"},{"Start":"17:38.500 ","End":"17:41.515","Text":"so I\u0027ll get 25 r"},{"Start":"17:41.515 ","End":"17:51.640","Text":"minus r cubed dr and d Theta, sorry."},{"Start":"17:51.640 ","End":"17:54.519","Text":"We do the integral first,"},{"Start":"17:54.519 ","End":"17:58.540","Text":"so that this 1 here,"},{"Start":"17:58.540 ","End":"18:03.595","Text":"dr, and I like to do these things at the side."},{"Start":"18:03.595 ","End":"18:06.085","Text":"Let\u0027s see, 25 r,"},{"Start":"18:06.085 ","End":"18:11.210","Text":"so we get 25 r squared over 2,"},{"Start":"18:14.490 ","End":"18:20.440","Text":"25 over 2 r squared minus"},{"Start":"18:20.440 ","End":"18:28.555","Text":"1/4 r to the 4th,"},{"Start":"18:28.555 ","End":"18:34.930","Text":"and this has to be evaluated from 0-3."},{"Start":"18:34.930 ","End":"18:38.365","Text":"Of course, when I put in r equals 0, I get nothing."},{"Start":"18:38.365 ","End":"18:41.470","Text":"What happens when I put r equals 3,"},{"Start":"18:41.470 ","End":"18:47.720","Text":"then I get here 9 times 25?"},{"Start":"18:48.030 ","End":"18:54.220","Text":"Let\u0027s say, that would be 9 times 25"},{"Start":"18:54.220 ","End":"19:00.980","Text":"is 225 over 2."},{"Start":"19:01.620 ","End":"19:11.560","Text":"I put here r equals 3,"},{"Start":"19:11.560 ","End":"19:15.580","Text":"then 3 to the 4th is 81,"},{"Start":"19:15.580 ","End":"19:18.830","Text":"so 81 over 4."},{"Start":"19:20.670 ","End":"19:22.870","Text":"Maybe I should compute this,"},{"Start":"19:22.870 ","End":"19:24.715","Text":"lets put everything over 4,"},{"Start":"19:24.715 ","End":"19:30.160","Text":"this would be 450 minus 81,"},{"Start":"19:30.160 ","End":"19:38.460","Text":"that makes it 369 is 450 minus 81 over 4."},{"Start":"19:38.460 ","End":"19:43.860","Text":"Now I go back and put this here and since it\u0027s also a constant,"},{"Start":"19:43.860 ","End":"19:46.395","Text":"I can bring it in front,"},{"Start":"19:46.395 ","End":"19:52.590","Text":"so what we get is 369 over"},{"Start":"19:52.590 ","End":"19:58.590","Text":"4 with the 2v makes it 8v."},{"Start":"19:58.590 ","End":"20:04.330","Text":"Then the integral from 0-2 Pi of just d Theta,"},{"Start":"20:04.330 ","End":"20:08.480","Text":"or I can write it as 1 d Theta."},{"Start":"20:10.530 ","End":"20:17.410","Text":"This integral here, the integral of 1 is just this minus this,"},{"Start":"20:17.410 ","End":"20:19.775","Text":"which is 2 Pi."},{"Start":"20:19.775 ","End":"20:23.535","Text":"Now I\u0027m going to collect it all together and see what we get."},{"Start":"20:23.535 ","End":"20:29.770","Text":"This I can put as this,"},{"Start":"20:29.770 ","End":"20:36.580","Text":"but upside down because it\u0027s over v. I\u0027m going to get for 1 over v,"},{"Start":"20:36.580 ","End":"20:44.560","Text":"3 over 122 Pi,"},{"Start":"20:44.560 ","End":"20:50.065","Text":"then I have 369 over 8,"},{"Start":"20:50.065 ","End":"20:53.950","Text":"then I have 2 Pi."},{"Start":"20:53.950 ","End":"20:58.240","Text":"I can cancel a little bit,"},{"Start":"20:58.240 ","End":"21:01.945","Text":"Pi can go with Pi,"},{"Start":"21:01.945 ","End":"21:08.290","Text":"and 2 goes into 8, 4 times."},{"Start":"21:08.290 ","End":"21:10.780","Text":"This leaves us with"},{"Start":"21:10.780 ","End":"21:13.915","Text":"3 times 369"},{"Start":"21:13.915 ","End":"21:23.890","Text":"is 1,107,"},{"Start":"21:23.890 ","End":"21:31.090","Text":"and 4 times 122 is 488."},{"Start":"21:31.090 ","End":"21:36.445","Text":"But this is just the answer for the z part."},{"Start":"21:36.445 ","End":"21:44.545","Text":"We have to remember that the x bar and y bar are both 0 for reasons of symmetry."},{"Start":"21:44.545 ","End":"21:49.780","Text":"Altogether, I can write the answer and I can write it in a nice read that"},{"Start":"21:49.780 ","End":"21:52.900","Text":"the centroid is at 0"},{"Start":"21:52.900 ","End":"22:01.310","Text":",0 ,1107 over 488,"},{"Start":"22:01.440 ","End":"22:04.660","Text":"and this is our answer."},{"Start":"22:04.660 ","End":"22:11.150","Text":"This is the centroid of the body."},{"Start":"22:11.460 ","End":"22:16.080","Text":"That\u0027s the volume and we are done."}],"ID":9575},{"Watched":false,"Name":"Exercise 5","Duration":"23m 43s","ChapterTopicVideoID":9242,"CourseChapterTopicPlaylistID":21193,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9242.jpeg","UploadDate":"2017-06-27T02:44:07.1900000","DurationForVideoObject":"PT23M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to compute 2 things,"},{"Start":"00:03.270 ","End":"00:06.840","Text":"both the volume and the centroid of the body,"},{"Start":"00:06.840 ","End":"00:11.025","Text":"which is bounded above by the sphere,"},{"Start":"00:11.025 ","End":"00:12.900","Text":"given by this equation."},{"Start":"00:12.900 ","End":"00:22.475","Text":"It\u0027s a sphere of radius 4 and we also have to bound it below by this equation of a cone."},{"Start":"00:22.475 ","End":"00:26.705","Text":"I\u0027ll try and bring some kind of a sketch."},{"Start":"00:26.705 ","End":"00:29.750","Text":"I\u0027ll start with a 2D sketch from the side."},{"Start":"00:29.750 ","End":"00:31.805","Text":"Let\u0027s say this is the x-y plane."},{"Start":"00:31.805 ","End":"00:37.415","Text":"For example, if I took it as the x-axis and let y equals 0,"},{"Start":"00:37.415 ","End":"00:40.640","Text":"I get x squared plus z squared equals 16,"},{"Start":"00:40.640 ","End":"00:42.050","Text":"which is 4 squared,"},{"Start":"00:42.050 ","End":"00:46.025","Text":"circle of radius 4 and the cone,"},{"Start":"00:46.025 ","End":"00:50.180","Text":"it\u0027s really only a half cone because the full cone would normally"},{"Start":"00:50.180 ","End":"00:54.260","Text":"be z squared equals x squared plus y squared."},{"Start":"00:54.260 ","End":"00:59.765","Text":"But it\u0027s only a half cone because it\u0027s only bigger or equal to 0."},{"Start":"00:59.765 ","End":"01:02.450","Text":"I\u0027ll try and give it a 3D look."},{"Start":"01:02.450 ","End":"01:07.775","Text":"If I just sort of put a bit of a circular spin on this,"},{"Start":"01:07.775 ","End":"01:10.670","Text":"I\u0027ll just highlight the outsides of it,"},{"Start":"01:10.670 ","End":"01:16.685","Text":"as it\u0027s going to turn out best in spherical coordinates."},{"Start":"01:16.685 ","End":"01:20.690","Text":"I found another picture on the internet."},{"Start":"01:20.690 ","End":"01:24.740","Text":"Now I mentioned spherical coordinates and let me write down"},{"Start":"01:24.740 ","End":"01:30.605","Text":"the conversion formulas between Cartesian and spherical."},{"Start":"01:30.605 ","End":"01:41.300","Text":"X equals r sine Phi cosine Theta."},{"Start":"01:41.300 ","End":"01:49.925","Text":"Y equals r sine Phi sine Theta"},{"Start":"01:49.925 ","End":"01:56.415","Text":"and z equals r cosine Phi."},{"Start":"01:56.415 ","End":"01:59.075","Text":"These are the 3 basic ones."},{"Start":"01:59.075 ","End":"02:02.619","Text":"Phi would be the angle."},{"Start":"02:02.619 ","End":"02:09.389","Text":"For example here, this is what Phi is for this point."},{"Start":"02:09.389 ","End":"02:14.060","Text":"Phi is the angle between"},{"Start":"02:14.060 ","End":"02:19.985","Text":"the line connecting the origin to the point and the positive z-axis."},{"Start":"02:19.985 ","End":"02:22.670","Text":"What I want to show you is that"},{"Start":"02:22.670 ","End":"02:26.720","Text":"the boundary of the cone is characterized by a constant Phi,"},{"Start":"02:26.720 ","End":"02:30.234","Text":"because we have this angle constantly rotating"},{"Start":"02:30.234 ","End":"02:34.520","Text":"around and in fact Phi will turn out to be 45 degrees."},{"Start":"02:34.520 ","End":"02:36.470","Text":"So let me show you that."},{"Start":"02:36.470 ","End":"02:38.540","Text":"But I\u0027ll also write down meanwhile,"},{"Start":"02:38.540 ","End":"02:46.024","Text":"another formula that when we have dv, in an integral,"},{"Start":"02:46.024 ","End":"02:52.040","Text":"this is r squared sine Phi and"},{"Start":"02:52.040 ","End":"02:58.395","Text":"then dr d Phi d Theta."},{"Start":"02:58.395 ","End":"03:00.870","Text":"Another formula we also use often,"},{"Start":"03:00.870 ","End":"03:02.250","Text":"very useful to have,"},{"Start":"03:02.250 ","End":"03:10.025","Text":"is that r squared equals x squared plus y squared plus z squared."},{"Start":"03:10.025 ","End":"03:11.990","Text":"Armed with all of these,"},{"Start":"03:11.990 ","End":"03:14.930","Text":"let\u0027s see what, what the cone is."},{"Start":"03:14.930 ","End":"03:17.480","Text":"The cone, which is here,"},{"Start":"03:17.480 ","End":"03:23.240","Text":"z equals square root of x squared plus y squared."},{"Start":"03:23.240 ","End":"03:26.870","Text":"If I replace each of these by what it\u0027s equal to,"},{"Start":"03:26.870 ","End":"03:34.250","Text":"we get that r cosine Phi equals"},{"Start":"03:34.250 ","End":"03:42.645","Text":"the square root i sine Phi cosine Theta,"},{"Start":"03:42.645 ","End":"03:46.640","Text":"this squared plus r"},{"Start":"03:46.640 ","End":"03:56.170","Text":"sine Phi sine Theta also squared."},{"Start":"03:56.630 ","End":"03:59.480","Text":"Then this is equal to,"},{"Start":"03:59.480 ","End":"04:04.220","Text":"because r sine Theta belongs to both,"},{"Start":"04:04.220 ","End":"04:09.160","Text":"I can say that I can take this squared r,"},{"Start":"04:09.160 ","End":"04:16.430","Text":"squared sine squared Phi outside of both of them and what I\u0027m left"},{"Start":"04:16.430 ","End":"04:24.205","Text":"with is cosine squared Theta plus sine squared Theta."},{"Start":"04:24.205 ","End":"04:28.210","Text":"But everyone knows that cosine squared plus sine squared is 1."},{"Start":"04:28.210 ","End":"04:35.925","Text":"So this is just the square root of r squared sine squared"},{"Start":"04:35.925 ","End":"04:45.690","Text":"Phi and the square root of this is just r sine of Phi."},{"Start":"04:45.690 ","End":"04:52.320","Text":"Now r is positive and sine is also positive from 0-180 degrees,"},{"Start":"04:52.320 ","End":"04:54.365","Text":"which is where Phi can be,"},{"Start":"04:54.365 ","End":"04:57.665","Text":"because otherwise I would have had to put the absolute value."},{"Start":"04:57.665 ","End":"05:04.805","Text":"Bringing this down, we have our cosine Phi equals r sine Phi."},{"Start":"05:04.805 ","End":"05:07.295","Text":"This will cancel with this."},{"Start":"05:07.295 ","End":"05:12.020","Text":"If I divide this I\u0027ll get tangent of Phi,"},{"Start":"05:12.020 ","End":"05:15.575","Text":"which is sine over cosine, is equal to 1."},{"Start":"05:15.575 ","End":"05:18.350","Text":"Phi is 45 degrees."},{"Start":"05:18.350 ","End":"05:24.785","Text":"But it should be set to radians so that\u0027s Pi over 4."},{"Start":"05:24.785 ","End":"05:32.220","Text":"If I take a 45 degree line and rotate it about the z-axis,"},{"Start":"05:32.220 ","End":"05:34.515","Text":"I\u0027ll get this cone."},{"Start":"05:34.515 ","End":"05:37.435","Text":"That\u0027s the cone part."},{"Start":"05:37.435 ","End":"05:45.405","Text":"The whole inside of this cone is on the border Phi is 45 degrees,"},{"Start":"05:45.405 ","End":"05:47.430","Text":"but inside Phi is less."},{"Start":"05:47.430 ","End":"05:48.810","Text":"Here Phi is 0."},{"Start":"05:48.810 ","End":"05:51.240","Text":"Here Phi is 45 degrees,"},{"Start":"05:51.240 ","End":"05:55.825","Text":"that\u0027s Pi over 4 or if you like it and call it the solid cone,"},{"Start":"05:55.825 ","End":"06:03.360","Text":"is given by Phi between 0 and 45 degrees."},{"Start":"06:03.360 ","End":"06:08.510","Text":"Now let\u0027s see if we can do the sphere."},{"Start":"06:09.180 ","End":"06:15.070","Text":"X squared plus y squared plus z squared is 16,"},{"Start":"06:15.070 ","End":"06:24.635","Text":"which is 4 squared and then r squared equals 4 squared."},{"Start":"06:24.635 ","End":"06:30.940","Text":"R is positive so we have that r equals 4."},{"Start":"06:30.940 ","End":"06:35.300","Text":"Maybe I\u0027ll try and give this a bit of a 3D look as well."},{"Start":"06:36.410 ","End":"06:39.360","Text":"But we want not the sphere,"},{"Start":"06:39.360 ","End":"06:45.820","Text":"but the solid sphere or the ball including the inside,"},{"Start":"06:45.820 ","End":"06:50.620","Text":"will be given by everything from 0 up to 4,"},{"Start":"06:50.620 ","End":"06:58.820","Text":"so I can write 0 less than or equal to r less than or equal to 4."},{"Start":"06:58.920 ","End":"07:05.930","Text":"As for Theta, Theta goes around here."},{"Start":"07:05.930 ","End":"07:08.130","Text":"Not quite clear where I start,"},{"Start":"07:08.130 ","End":"07:10.260","Text":"where is the x and where is the y-axis,"},{"Start":"07:10.260 ","End":"07:13.540","Text":"but I go a full circle here."},{"Start":"07:13.970 ","End":"07:17.010","Text":"Because of this full circularity,"},{"Start":"07:17.010 ","End":"07:25.700","Text":"Theta will go from 0-360 degrees, which is 2Pi."},{"Start":"07:25.700 ","End":"07:28.410","Text":"Now these 3 things together,"},{"Start":"07:28.410 ","End":"07:35.315","Text":"this inequality and this inequality and this inequality completely define the solid,"},{"Start":"07:35.315 ","End":"07:39.170","Text":"which is the section of the solid cone with the bowl."},{"Start":"07:39.170 ","End":"07:42.755","Text":"Now we want to start doing the integral."},{"Start":"07:42.755 ","End":"07:46.535","Text":"I need to give you the formula for the volume again."},{"Start":"07:46.535 ","End":"07:48.795","Text":"Let\u0027s say the body,"},{"Start":"07:48.795 ","End":"07:50.510","Text":"in our case or in general,"},{"Start":"07:50.510 ","End":"07:52.730","Text":"we\u0027ll just call it B for body."},{"Start":"07:52.730 ","End":"08:00.590","Text":"We have a general formula that the volume of a body B is the triple"},{"Start":"08:00.590 ","End":"08:10.220","Text":"integral over this body of the function 1 constant 1 dv."},{"Start":"08:10.220 ","End":"08:14.615","Text":"Now what we wanted to do was to convert this to"},{"Start":"08:14.615 ","End":"08:19.235","Text":"spherical coordinates and because of these 3,"},{"Start":"08:19.235 ","End":"08:22.425","Text":"what we get in our case,"},{"Start":"08:22.425 ","End":"08:32.220","Text":"Theta which goes from 0-2Pi and then we take Phi,"},{"Start":"08:32.220 ","End":"08:38.225","Text":"which we said goes from 0-45 degrees,"},{"Start":"08:38.225 ","End":"08:43.830","Text":"which in radians is Pi over 4 and then the r,"},{"Start":"08:43.830 ","End":"08:46.550","Text":"which is the distance to the origin,"},{"Start":"08:46.550 ","End":"08:49.910","Text":"goes from 0 up to 4."},{"Start":"08:49.910 ","End":"08:55.430","Text":"R squared sine Phi"},{"Start":"08:55.430 ","End":"09:02.170","Text":"dr d Phi d Theta."},{"Start":"09:02.170 ","End":"09:06.800","Text":"At this point, it\u0027s a totally technical computation."},{"Start":"09:06.800 ","End":"09:15.350","Text":"We don\u0027t need diagrams or explanations anymore and we\u0027ll start from the inside out."},{"Start":"09:15.590 ","End":"09:20.590","Text":"So we\u0027ll do the dr first."},{"Start":"09:20.850 ","End":"09:25.540","Text":"Actually I like to simplify things because if I\u0027m doing an integral dr,"},{"Start":"09:25.540 ","End":"09:27.625","Text":"and sine Phi is a constant,"},{"Start":"09:27.625 ","End":"09:31.875","Text":"let me just remove it from here and put it here."},{"Start":"09:31.875 ","End":"09:35.910","Text":"Let me put the sine Phi in front of the dr integral."},{"Start":"09:35.910 ","End":"09:39.795","Text":"Now, this 1 I\u0027d like to do at the side,"},{"Start":"09:39.795 ","End":"09:42.845","Text":"so I have r squared,"},{"Start":"09:42.845 ","End":"09:48.070","Text":"so it becomes 1/3 r cubed,"},{"Start":"09:48.070 ","End":"09:53.570","Text":"and I need to take this from 0 to 4."},{"Start":"09:53.610 ","End":"09:57.310","Text":"If I plug in 0,"},{"Start":"09:57.310 ","End":"09:59.545","Text":"I just get 0."},{"Start":"09:59.545 ","End":"10:03.670","Text":"If I plug in 4, I get 4 cubed, is 64/3,"},{"Start":"10:03.670 ","End":"10:08.230","Text":"so the answer is just 64/3."},{"Start":"10:08.230 ","End":"10:11.080","Text":"Now I can go back here, in fact,"},{"Start":"10:11.080 ","End":"10:14.575","Text":"I can put the 64/3 right in front,"},{"Start":"10:14.575 ","End":"10:19.780","Text":"and then we will get the integral from 0 to"},{"Start":"10:19.780 ","End":"10:26.260","Text":"2 Pi 64/3 in front,"},{"Start":"10:26.260 ","End":"10:35.500","Text":"and then the integral from 0 to Pi over 4 of"},{"Start":"10:35.500 ","End":"10:44.910","Text":"just sine of Phi d Phi d Theta."},{"Start":"10:44.910 ","End":"10:50.490","Text":"Once again, we work our way from the inside out,"},{"Start":"10:50.490 ","End":"10:56.200","Text":"and so we compute this integral."},{"Start":"10:56.790 ","End":"11:00.129","Text":"I\u0027ll also do this 1 at the side."},{"Start":"11:00.129 ","End":"11:06.385","Text":"The integral of sine Phi is minus cosine Phi,"},{"Start":"11:06.385 ","End":"11:13.930","Text":"which I take between 0 and Pi over 4."},{"Start":"11:13.930 ","End":"11:18.205","Text":"Now I often get confused with the minus and the subtraction,"},{"Start":"11:18.205 ","End":"11:21.550","Text":"so what I like to do is if this is a minus,"},{"Start":"11:21.550 ","End":"11:24.280","Text":"I\u0027ll reverse the upper and the lower."},{"Start":"11:24.280 ","End":"11:27.295","Text":"This is the same as cosine Phi,"},{"Start":"11:27.295 ","End":"11:34.030","Text":"but from 0 on the top and Pi over 4 at the bottom,"},{"Start":"11:34.030 ","End":"11:36.295","Text":"I\u0027m less likely to make a mistake."},{"Start":"11:36.295 ","End":"11:39.340","Text":"First of all, I put cosine of"},{"Start":"11:39.340 ","End":"11:48.640","Text":"0 and then minus cosine Pi over 4."},{"Start":"11:48.640 ","End":"11:52.165","Text":"We know that cosine 0 is 1,"},{"Start":"11:52.165 ","End":"12:00.820","Text":"we know that cosine of 45 degrees is 1 over root 2,"},{"Start":"12:00.820 ","End":"12:04.720","Text":"or sometimes we write it as root 2/2,"},{"Start":"12:04.720 ","End":"12:10.615","Text":"both are okay, and that would be the answer to that."},{"Start":"12:10.615 ","End":"12:13.165","Text":"I would put that back in here,"},{"Start":"12:13.165 ","End":"12:14.920","Text":"except when it\u0027s a constant,"},{"Start":"12:14.920 ","End":"12:17.450","Text":"I could bring it up front."},{"Start":"12:17.550 ","End":"12:26.240","Text":"While I get a 64/3 times"},{"Start":"12:26.550 ","End":"12:31.640","Text":"1 minus root 2/2."},{"Start":"12:32.700 ","End":"12:44.590","Text":"Then all we\u0027re left with here is the integral from 0 to 2 Pi of d Theta or 1 d Theta,"},{"Start":"12:44.590 ","End":"12:48.445","Text":"and the integral of 1 is just this minus this,"},{"Start":"12:48.445 ","End":"12:52.690","Text":"so we have 64/3,"},{"Start":"12:52.690 ","End":"12:58.195","Text":"1 minus root 2/2 times 2 Pi."},{"Start":"12:58.195 ","End":"12:59.890","Text":"Not much to simplify,"},{"Start":"12:59.890 ","End":"13:03.655","Text":"perhaps I\u0027ll take this 2 and multiply it in here,"},{"Start":"13:03.655 ","End":"13:06.520","Text":"and then I have 64 over 3,"},{"Start":"13:06.520 ","End":"13:08.245","Text":"if I double this it\u0027s 2,"},{"Start":"13:08.245 ","End":"13:10.780","Text":"I double this, it\u0027s just root 2,"},{"Start":"13:10.780 ","End":"13:13.059","Text":"then I only have a single Pi,"},{"Start":"13:13.059 ","End":"13:18.430","Text":"and that is the formula for the volume."},{"Start":"13:18.430 ","End":"13:21.910","Text":"That is the first part of the question answered."},{"Start":"13:21.910 ","End":"13:25.150","Text":"Next we have to go on to the centroid."},{"Start":"13:25.150 ","End":"13:30.565","Text":"The centroid is a point."},{"Start":"13:30.565 ","End":"13:33.399","Text":"It has an x and y and z,"},{"Start":"13:33.399 ","End":"13:35.740","Text":"but it\u0027s a special x, y and z,"},{"Start":"13:35.740 ","End":"13:38.095","Text":"so I\u0027ll call it x-bar, y-bar,"},{"Start":"13:38.095 ","End":"13:41.500","Text":"z-bar, and there are formulas."},{"Start":"13:41.500 ","End":"13:45.295","Text":"I\u0027ll just copy paste them from a previous exercise."},{"Start":"13:45.295 ","End":"13:52.130","Text":"Here are the 3 formulas for x-bar, y-bar, and z-bar."},{"Start":"13:52.500 ","End":"13:57.640","Text":"We\u0027ve used the concept of symmetry before because"},{"Start":"13:57.640 ","End":"14:03.400","Text":"the sphere and the cone are all symmetrical around the z-axis."},{"Start":"14:03.400 ","End":"14:06.745","Text":"The centroid is also going to be on the z-axis."},{"Start":"14:06.745 ","End":"14:10.570","Text":"In other words, these 2 are both going to be 0,"},{"Start":"14:10.570 ","End":"14:14.200","Text":"and basically for reasons of symmetry,"},{"Start":"14:14.200 ","End":"14:15.970","Text":"I\u0027ll just write shorthand,"},{"Start":"14:15.970 ","End":"14:22.610","Text":"I\u0027ll just write the word symmetry to explain why these are both 0."},{"Start":"14:22.610 ","End":"14:30.365","Text":"What we\u0027re left with is to compute the z coordinate of the centroid,"},{"Start":"14:30.365 ","End":"14:37.840","Text":"and this is a very similar integral to the volume integral."},{"Start":"14:37.840 ","End":"14:38.920","Text":"There\u0027s 2 differences,"},{"Start":"14:38.920 ","End":"14:44.720","Text":"we have to divide here by the volume and here instead of 1 we have a z."},{"Start":"14:44.730 ","End":"14:51.445","Text":"What I\u0027m going to do is just copy this integral."},{"Start":"14:51.445 ","End":"14:53.890","Text":"I copied that here,"},{"Start":"14:53.890 ","End":"14:58.510","Text":"and now I have the z-bar is equal to,"},{"Start":"14:58.510 ","End":"15:02.485","Text":"I need the 1 over v,"},{"Start":"15:02.485 ","End":"15:05.515","Text":"which later I\u0027ll substitute from here."},{"Start":"15:05.515 ","End":"15:11.650","Text":"The other thing is just like here I have a z,"},{"Start":"15:11.650 ","End":"15:13.780","Text":"I want to put a z here,"},{"Start":"15:13.780 ","End":"15:18.325","Text":"but we\u0027re working in spherical."},{"Start":"15:18.325 ","End":"15:22.375","Text":"If you remember or just go back and look,"},{"Start":"15:22.375 ","End":"15:28.465","Text":"z was equal to r cosine Phi,"},{"Start":"15:28.465 ","End":"15:34.090","Text":"and so I get rid of this and I put an extra r cosine Phi."},{"Start":"15:34.090 ","End":"15:42.445","Text":"So I\u0027ll put an extra r here and an extra cosine Phi here."},{"Start":"15:42.445 ","End":"15:46.735","Text":"Let me just clean up this mess,"},{"Start":"15:46.735 ","End":"15:50.560","Text":"so we write it nicely,"},{"Start":"15:50.560 ","End":"16:01.300","Text":"z-bar equals 1 over v. Integral Theta goes from 0 to 2 Pi,"},{"Start":"16:01.300 ","End":"16:07.405","Text":"and then Phi goes from 0 to Pi over 4."},{"Start":"16:07.405 ","End":"16:13.820","Text":"Here I have cosine Phi, sine Phi,"},{"Start":"16:16.680 ","End":"16:24.310","Text":"and then here I have the integral from 0 to 4,"},{"Start":"16:24.310 ","End":"16:31.150","Text":"that\u0027s r of r cubed dr,"},{"Start":"16:31.150 ","End":"16:35.515","Text":"d Phi, d Theta."},{"Start":"16:35.515 ","End":"16:41.240","Text":"I will start with the integral dr."},{"Start":"16:42.290 ","End":"16:46.140","Text":"I\u0027ll do this one at the side over here."},{"Start":"16:46.140 ","End":"16:51.660","Text":"The integral of r cubed from"},{"Start":"16:51.660 ","End":"17:01.530","Text":"0-4 is just 1/4r^4 from 0-4."},{"Start":"17:01.530 ","End":"17:03.660","Text":"At 0, I get nothing."},{"Start":"17:03.660 ","End":"17:06.135","Text":"I just need to plug in 4."},{"Start":"17:06.135 ","End":"17:12.225","Text":"4^4 over 4 is 4 cubed, which is 64."},{"Start":"17:12.225 ","End":"17:16.485","Text":"This thing all comes out to be 64,"},{"Start":"17:16.485 ","End":"17:19.575","Text":"and 64 is a constant,"},{"Start":"17:19.575 ","End":"17:22.020","Text":"so I can bring it up front."},{"Start":"17:22.020 ","End":"17:32.625","Text":"Then I have z bar is 64 over v integral from 0-2Pi,"},{"Start":"17:32.625 ","End":"17:38.895","Text":"integral from 0 to Pi over 4 of"},{"Start":"17:38.895 ","End":"17:48.570","Text":"cosine Phi sine Phi d Phi d Theta."},{"Start":"17:48.570 ","End":"17:51.960","Text":"Let\u0027s do this integral."},{"Start":"17:51.960 ","End":"17:56.920","Text":"For this one, I want to use a trigonometric identity."},{"Start":"17:58.040 ","End":"18:03.060","Text":"Remember the identity that the sine of twice an angle"},{"Start":"18:03.060 ","End":"18:08.895","Text":"is 2 sine of that angle times the cosine of the angle."},{"Start":"18:08.895 ","End":"18:13.990","Text":"Then I use it with Alpha as Phi here."},{"Start":"18:14.210 ","End":"18:17.280","Text":"I\u0027ll do this one at the side."},{"Start":"18:17.280 ","End":"18:22.695","Text":"What I get is the integral from 0 to Pi over 4."},{"Start":"18:22.695 ","End":"18:24.750","Text":"I don\u0027t have a 2 here,"},{"Start":"18:24.750 ","End":"18:30.720","Text":"so I\u0027ll just put a half and that will make up for it of"},{"Start":"18:30.720 ","End":"18:40.155","Text":"sine of 2Phi d Phi using this identity that the 2 on the other side,"},{"Start":"18:40.155 ","End":"18:46.110","Text":"this is equal to 1/2."},{"Start":"18:46.110 ","End":"18:51.540","Text":"The integral of sine 2Phi is"},{"Start":"18:51.540 ","End":"18:57.410","Text":"almost cosine 2Phi so"},{"Start":"18:57.410 ","End":"19:03.705","Text":"it has to be a minus because the integral of sine is minus cosine."},{"Start":"19:03.705 ","End":"19:06.570","Text":"But because of the 2 in front,"},{"Start":"19:06.570 ","End":"19:10.335","Text":"I also have to divide by 2,"},{"Start":"19:10.335 ","End":"19:14.550","Text":"so another half here."},{"Start":"19:14.550 ","End":"19:23.470","Text":"All this has to be taken between 0 and Pi over 4."},{"Start":"19:24.770 ","End":"19:28.830","Text":"Now, if I plug in,"},{"Start":"19:28.830 ","End":"19:31.260","Text":"I can just take the quarter out."},{"Start":"19:31.260 ","End":"19:34.560","Text":"This equals, you know what,"},{"Start":"19:34.560 ","End":"19:36.405","Text":"I\u0027m going to do 2 things."},{"Start":"19:36.405 ","End":"19:41.010","Text":"This half and this half I can combine as a quarter."},{"Start":"19:41.010 ","End":"19:47.625","Text":"The other trick I like to do is whenever I have a minus here,"},{"Start":"19:47.625 ","End":"19:50.460","Text":"I can make the minus into a plus."},{"Start":"19:50.460 ","End":"19:52.320","Text":"If I switched the order of these 2,"},{"Start":"19:52.320 ","End":"19:53.955","Text":"the order of the subtraction."},{"Start":"19:53.955 ","End":"19:59.355","Text":"I\u0027ll make it as plus cosine 2Phi,"},{"Start":"19:59.355 ","End":"20:01.080","Text":"but I\u0027ll reverse here,"},{"Start":"20:01.080 ","End":"20:05.310","Text":"0 here, Pi over 4."},{"Start":"20:05.310 ","End":"20:12.435","Text":"Let\u0027s see now, cosine of 0 is 1,"},{"Start":"20:12.435 ","End":"20:16.990","Text":"write the quarter, cosine of 0 is 1."},{"Start":"20:17.420 ","End":"20:25.920","Text":"Cosine of 2Phi, when Phi is Pi over 4 is cosine Pi over 2."},{"Start":"20:25.920 ","End":"20:31.620","Text":"Cosine of Pi over 2 is 0."},{"Start":"20:31.620 ","End":"20:38.070","Text":"This just comes out to be 1/4."},{"Start":"20:38.070 ","End":"20:41.100","Text":"I\u0027ll just write this 1/4 here."},{"Start":"20:41.100 ","End":"20:49.470","Text":"Now continuing, bring the 1/4 in front."},{"Start":"20:49.470 ","End":"20:52.680","Text":"But it cancels with the 64,"},{"Start":"20:52.680 ","End":"20:57.120","Text":"64 is 16 times 4,"},{"Start":"20:57.120 ","End":"21:07.050","Text":"so it\u0027s just 16 over V times the integral from 0 to 2Pi of d Theta."},{"Start":"21:07.050 ","End":"21:08.835","Text":"Write a 1 here."},{"Start":"21:08.835 ","End":"21:11.010","Text":"We know that the integral of 1 is"},{"Start":"21:11.010 ","End":"21:14.535","Text":"just the upper limit minus the lower limit, so that\u0027s 2Pi."},{"Start":"21:14.535 ","End":"21:24.780","Text":"I have 16 times 2Pi."},{"Start":"21:24.780 ","End":"21:34.035","Text":"Now the over V means that I can put V in the denominator."},{"Start":"21:34.035 ","End":"21:37.004","Text":"I mean, switch it the other way around."},{"Start":"21:37.004 ","End":"21:45.555","Text":"I\u0027ll write it first of all as 1 over V. Now let\u0027s see if we can simplify this."},{"Start":"21:45.555 ","End":"21:52.350","Text":"What I get is here I have 32Pi."},{"Start":"21:52.350 ","End":"21:55.840","Text":"Now I have to take V reciprocal."},{"Start":"21:56.030 ","End":"22:03.525","Text":"The 64 over 3 becomes 3 over 64."},{"Start":"22:03.525 ","End":"22:14.310","Text":"In the denominator, I have to put this Pi backwards."},{"Start":"22:14.310 ","End":"22:20.130","Text":"Doesn\u0027t matter. Pi times 2 minus root 2."},{"Start":"22:20.130 ","End":"22:23.890","Text":"Let\u0027s see what we can cancel."},{"Start":"22:24.080 ","End":"22:29.715","Text":"32 into 64 goes"},{"Start":"22:29.715 ","End":"22:37.300","Text":"twice and the Pi can slow."},{"Start":"22:37.850 ","End":"22:46.740","Text":"This is equal to 3 over twice 2 minus root 2."},{"Start":"22:46.740 ","End":"22:52.470","Text":"It is possible to simplify this further using the conjugate."},{"Start":"22:52.470 ","End":"22:54.960","Text":"I\u0027m not going to do all of that."},{"Start":"22:54.960 ","End":"22:58.920","Text":"We\u0027ll leave it like this, I\u0027m not even going to do a numerical evaluation."},{"Start":"22:58.920 ","End":"23:09.060","Text":"I just want to summarize because the centroid is x bar,"},{"Start":"23:09.060 ","End":"23:11.670","Text":"y bar, z bar."},{"Start":"23:11.670 ","End":"23:14.115","Text":"We know that these first 2 are 0,"},{"Start":"23:14.115 ","End":"23:25.140","Text":"so I just want to write at the end that the centroid is equal to 0,"},{"Start":"23:25.140 ","End":"23:34.710","Text":"0, 3 over twice 2 minus root 2."},{"Start":"23:34.710 ","End":"23:41.130","Text":"That\u0027s the other part of the question, volume centroid,"},{"Start":"23:41.130 ","End":"23:44.590","Text":"so now we are done."}],"ID":9576},{"Watched":false,"Name":"Exercise 6","Duration":"9m 14s","ChapterTopicVideoID":9247,"CourseChapterTopicPlaylistID":21193,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9247.jpeg","UploadDate":"2017-06-27T02:44:55.4270000","DurationForVideoObject":"PT9M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.869","Text":"In this exercise, we have to find the volume of a region"},{"Start":"00:03.869 ","End":"00:08.520","Text":"above the xy-plane and it\u0027s bounded by the paraboloid,"},{"Start":"00:08.520 ","End":"00:12.045","Text":"this and the cylinder this."},{"Start":"00:12.045 ","End":"00:18.390","Text":"Note that the xy-plane could be written as a surface z equals 0."},{"Start":"00:18.390 ","End":"00:24.955","Text":"I actually have the region sandwiched between 2 surfaces,"},{"Start":"00:24.955 ","End":"00:29.745","Text":"above and below and at the sides there\u0027s a cylinder."},{"Start":"00:29.745 ","End":"00:33.555","Text":"This general problem I\u0027ve got a sketch for,"},{"Start":"00:33.555 ","End":"00:37.560","Text":"is a picture I found on the Internet for when we have a type"},{"Start":"00:37.560 ","End":"00:41.580","Text":"of region or body you know what?"},{"Start":"00:41.580 ","End":"00:48.125","Text":"I\u0027ll change the letter V to the letter B for body or solid."},{"Start":"00:48.125 ","End":"00:52.115","Text":"When we have an upper surface in our case the paraboloid,"},{"Start":"00:52.115 ","End":"00:55.160","Text":"a lower surface, in this case the xy-plane."},{"Start":"00:55.160 ","End":"00:59.850","Text":"We know its projection onto the xy-plane."},{"Start":"00:59.850 ","End":"01:02.040","Text":"We can deduce this from here,"},{"Start":"01:02.040 ","End":"01:03.830","Text":"we\u0027ll do that in a moment."},{"Start":"01:03.830 ","End":"01:08.515","Text":"In fact, let\u0027s first of all see what D is."},{"Start":"01:08.515 ","End":"01:14.030","Text":"The D here is just a disc with radius a."},{"Start":"01:14.030 ","End":"01:21.335","Text":"The cylinders projection is x squared plus y squared equals a,"},{"Start":"01:21.335 ","End":"01:27.020","Text":"but actually we have the interior of it also so it\u0027s less than or"},{"Start":"01:27.020 ","End":"01:34.260","Text":"equal to a squared not equal,"},{"Start":"01:34.260 ","End":"01:37.710","Text":"less than or equal because it includes the interior."},{"Start":"01:37.710 ","End":"01:41.885","Text":"Now what we can do is,"},{"Start":"01:41.885 ","End":"01:45.380","Text":"first of all, I\u0027ll write the general formula."},{"Start":"01:45.380 ","End":"01:48.305","Text":"For the volume of a body,"},{"Start":"01:48.305 ","End":"01:55.220","Text":"we can say that the volume of B is just the triple"},{"Start":"01:55.220 ","End":"02:02.030","Text":"integral over that body of 1 dv."},{"Start":"02:02.030 ","End":"02:08.080","Text":"But here we can spell out more precisely what this body is."},{"Start":"02:08.080 ","End":"02:11.010","Text":"Now, what I\u0027m going to do is,"},{"Start":"02:11.010 ","End":"02:13.290","Text":"it\u0027s not going to be Cartesian,"},{"Start":"02:13.290 ","End":"02:18.170","Text":"and I\u0027m not going to do dx, dy, dz."},{"Start":"02:18.170 ","End":"02:25.395","Text":"I\u0027m thinking cylindrical coordinates because this has a very circular symmetry,"},{"Start":"02:25.395 ","End":"02:29.280","Text":"and also I see even x squared plus y squared here,"},{"Start":"02:29.280 ","End":"02:33.150","Text":"which is in 2-dimensionals polar and 3-dimensions cylindrical."},{"Start":"02:33.150 ","End":"02:38.690","Text":"What I\u0027m going to say is that v is a triple integral,"},{"Start":"02:38.690 ","End":"02:40.830","Text":"and I\u0027m going to first of all say,"},{"Start":"02:40.830 ","End":"02:45.575","Text":"the inside will be z,"},{"Start":"02:45.575 ","End":"02:50.885","Text":"and z will go from the lower surface to the upper surface."},{"Start":"02:50.885 ","End":"02:58.879","Text":"In this case I can say from 0 to x squared plus y squared."},{"Start":"02:58.879 ","End":"03:02.000","Text":"But I\u0027m not going to have here x and y."},{"Start":"03:02.000 ","End":"03:04.340","Text":"What I\u0027m going to do here is just first of all,"},{"Start":"03:04.340 ","End":"03:08.825","Text":"leave it in general and say it\u0027s the integral over"},{"Start":"03:08.825 ","End":"03:15.150","Text":"D. I\u0027m also going"},{"Start":"03:15.150 ","End":"03:20.610","Text":"to say that is 1 dv."},{"Start":"03:20.610 ","End":"03:26.650","Text":"But what I want to do is a conversion to cylindrical coordinates."},{"Start":"03:26.650 ","End":"03:34.525","Text":"Here I copy pasted all the formulas for cylindrical coordinates here."},{"Start":"03:34.525 ","End":"03:37.040","Text":"These are the 3 main ones."},{"Start":"03:37.040 ","End":"03:41.164","Text":"Well, x and y, just like in polar, z remains unchanged,"},{"Start":"03:41.164 ","End":"03:50.910","Text":"and we have the formula for how to change dv plus the useful last one,"},{"Start":"03:50.910 ","End":"03:53.265","Text":"which really comes out of the first 2."},{"Start":"03:53.265 ","End":"04:00.285","Text":"How do I express D in cylindrical coordinates."},{"Start":"04:00.285 ","End":"04:04.040","Text":"I need to know what r and Theta do."},{"Start":"04:04.040 ","End":"04:06.470","Text":"Well, we\u0027ve seen this many times before,"},{"Start":"04:06.470 ","End":"04:08.090","Text":"but I\u0027ll still give you a reminder."},{"Start":"04:08.090 ","End":"04:10.130","Text":"If I take a general Theta,"},{"Start":"04:10.130 ","End":"04:14.750","Text":"I could start from here and work my way in the positive direction,"},{"Start":"04:14.750 ","End":"04:19.145","Text":"which is counterclockwise all the way around up to here."},{"Start":"04:19.145 ","End":"04:23.360","Text":"In other words, I\u0027d be starting off at theta equals 0 and I\u0027d"},{"Start":"04:23.360 ","End":"04:27.990","Text":"be ending up at Theta equals to 2 Pi."},{"Start":"04:27.990 ","End":"04:34.220","Text":"For each Theta, r would start from 0 up to a,"},{"Start":"04:34.220 ","End":"04:36.725","Text":"that would be the same."},{"Start":"04:36.725 ","End":"04:41.330","Text":"The outer integral, instead of carving it up,"},{"Start":"04:41.330 ","End":"04:44.450","Text":"the type 1 or type 2 with x and y,"},{"Start":"04:44.450 ","End":"04:47.970","Text":"leave it as polar or cylindrical."},{"Start":"04:47.980 ","End":"04:54.950","Text":"The integral Theta goes from 0 to 2 Pi,"},{"Start":"04:54.950 ","End":"05:01.125","Text":"and then r goes from 0 to a."},{"Start":"05:01.125 ","End":"05:06.830","Text":"As for z, I can\u0027t leave it as x squared plus y squared,"},{"Start":"05:06.830 ","End":"05:12.550","Text":"but I can use this formula here to write r squared."},{"Start":"05:12.550 ","End":"05:16.485","Text":"Also, instead of dv,"},{"Start":"05:16.485 ","End":"05:18.435","Text":"I have to use this."},{"Start":"05:18.435 ","End":"05:26.745","Text":"I have r, dz, dr, dTheta."},{"Start":"05:26.745 ","End":"05:32.600","Text":"Now this expresses the volume and I have it in cylindrical coordinates,"},{"Start":"05:32.600 ","End":"05:35.395","Text":"and all we have to do with the computation."},{"Start":"05:35.395 ","End":"05:37.720","Text":"We start from the inside,"},{"Start":"05:37.720 ","End":"05:40.940","Text":"which will be the dz integral."},{"Start":"05:43.670 ","End":"05:47.920","Text":"There\u0027s something I often do like I could take this r,"},{"Start":"05:47.920 ","End":"05:53.755","Text":"which is not part of the dz and I could move it in front here,"},{"Start":"05:53.755 ","End":"05:57.340","Text":"but it\u0027s not really worth the bother."},{"Start":"05:57.340 ","End":"06:03.800","Text":"I\u0027ll leave it here and we\u0027ll do this integral at the side."},{"Start":"06:03.800 ","End":"06:06.480","Text":"I just mentioned about r as a possibility,"},{"Start":"06:06.480 ","End":"06:08.940","Text":"but either we don\u0027t have to do that."},{"Start":"06:08.940 ","End":"06:11.460","Text":"Let me do this over here."},{"Start":"06:11.460 ","End":"06:19.530","Text":"What I get, r is a constant so I get r times"},{"Start":"06:19.530 ","End":"06:25.790","Text":"z taken between z"},{"Start":"06:25.790 ","End":"06:31.010","Text":"equals 0 and z equals r squared."},{"Start":"06:31.010 ","End":"06:33.995","Text":"If I do this computation,"},{"Start":"06:33.995 ","End":"06:36.005","Text":"when z is r squared,"},{"Start":"06:36.005 ","End":"06:38.490","Text":"I get r cubed,"},{"Start":"06:38.490 ","End":"06:40.740","Text":"when z is 0,"},{"Start":"06:40.740 ","End":"06:44.225","Text":"I just get 0 so it\u0027s r cubed minus 0,"},{"Start":"06:44.225 ","End":"06:45.995","Text":"which is just r cubed."},{"Start":"06:45.995 ","End":"06:51.630","Text":"In other words, this middle bit comes down to r cubed."},{"Start":"06:51.630 ","End":"06:54.555","Text":"Now let me rewrite this."},{"Start":"06:54.555 ","End":"07:02.505","Text":"Integral of Theta from 0 to 2 Pi,"},{"Start":"07:02.505 ","End":"07:11.895","Text":"r from 0 to a of r cubed drdTheta."},{"Start":"07:11.895 ","End":"07:16.525","Text":"Next, we do the dr integral,"},{"Start":"07:16.525 ","End":"07:18.245","Text":"get a bit more space."},{"Start":"07:18.245 ","End":"07:21.110","Text":"Once again, I\u0027ll do it at the side."},{"Start":"07:21.110 ","End":"07:27.325","Text":"What I get for r cubed is 1/4 r^4,"},{"Start":"07:27.325 ","End":"07:34.360","Text":"and I have to do this between 0 and a."},{"Start":"07:34.360 ","End":"07:37.755","Text":"What I get is, if I put r equals a,"},{"Start":"07:37.755 ","End":"07:42.420","Text":"you get a^4 over 4 or 1/4 a^4,"},{"Start":"07:42.420 ","End":"07:45.735","Text":"put r equals 0, you just get 0."},{"Start":"07:45.735 ","End":"07:53.140","Text":"I put that back in here and I just write it as 1/4 a^4."},{"Start":"07:53.140 ","End":"07:57.035","Text":"This time I think I will take the constant in front."},{"Start":"07:57.035 ","End":"08:02.180","Text":"What we get is,"},{"Start":"08:02.180 ","End":"08:04.655","Text":"and I should have been writing these equals,"},{"Start":"08:04.655 ","End":"08:08.630","Text":"the 1/4, like I said,"},{"Start":"08:08.630 ","End":"08:13.110","Text":"and then the integral from 0 to 2 Pi."},{"Start":"08:13.370 ","End":"08:21.940","Text":"Silly me. The a is also a constant so I\u0027ll also put that in front,"},{"Start":"08:22.280 ","End":"08:28.810","Text":"1/4 a^4 and then all I\u0027m left with is I d Theta."},{"Start":"08:30.170 ","End":"08:33.210","Text":"We\u0027ll leave something there."},{"Start":"08:33.210 ","End":"08:38.820","Text":"Then we have the integral of 1,"},{"Start":"08:38.820 ","End":"08:41.360","Text":"we just take the upper limit minus the lower limit."},{"Start":"08:41.360 ","End":"08:43.580","Text":"I mean it\u0027s Theta, substitute these two,"},{"Start":"08:43.580 ","End":"08:45.380","Text":"we get 2 Pi minus 0,"},{"Start":"08:45.380 ","End":"08:53.600","Text":"so it\u0027s just 1/4 a^4 and then here 2 Pi."},{"Start":"08:53.700 ","End":"08:59.515","Text":"Just the 1/4 with the 2 so the answer is"},{"Start":"08:59.515 ","End":"09:07.770","Text":"1/2 a^4 times Pi."},{"Start":"09:07.770 ","End":"09:14.250","Text":"This is the volume that we\u0027re looking for and we are done."}],"ID":9577},{"Watched":false,"Name":"Exercise 7","Duration":"13m 4s","ChapterTopicVideoID":26760,"CourseChapterTopicPlaylistID":21193,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/26760.jpeg","UploadDate":"2021-12-11T12:04:39.8770000","DurationForVideoObject":"PT13M4S","Description":null,"VideoComments":[],"Subtitles":[],"ID":27776}],"Thumbnail":null,"ID":21193},{"Name":"Triple Integrals, Jacobian","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"14m 46s","ChapterTopicVideoID":8572,"CourseChapterTopicPlaylistID":21194,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8572.jpeg","UploadDate":"2017-02-13T03:36:48.7170000","DurationForVideoObject":"PT14M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.750","Text":"In this exercise we have to compute the following triple integral,"},{"Start":"00:04.750 ","End":"00:11.175","Text":"where B is the body bounded by these surfaces and there are 6 of them."},{"Start":"00:11.175 ","End":"00:15.870","Text":"It\u0027s very difficult to see what body this is,"},{"Start":"00:15.870 ","End":"00:18.225","Text":"what these surfaces are,"},{"Start":"00:18.225 ","End":"00:22.425","Text":"and what this calls for is a change of variables."},{"Start":"00:22.425 ","End":"00:29.190","Text":"For example, we see right away that we have an xy here and an xy here,"},{"Start":"00:29.190 ","End":"00:32.520","Text":"so this calls for a substitution,"},{"Start":"00:32.520 ","End":"00:36.840","Text":"u equals xy, and"},{"Start":"00:36.840 ","End":"00:44.165","Text":"then this will become just u equals 4,"},{"Start":"00:44.165 ","End":"00:49.630","Text":"and this 1 will become u equals 2."},{"Start":"00:49.630 ","End":"00:52.290","Text":"Continuing with the next 2,"},{"Start":"00:52.290 ","End":"00:55.035","Text":"we see a z here and a y here,"},{"Start":"00:55.035 ","End":"01:00.060","Text":"this leads me to the idea that I could slightly"},{"Start":"01:00.060 ","End":"01:06.255","Text":"rewrite this 2 as z minus y equals 1,"},{"Start":"01:06.255 ","End":"01:12.185","Text":"and this 1 as z minus y equals 0."},{"Start":"01:12.185 ","End":"01:15.095","Text":"Then I have z minus y here and here,"},{"Start":"01:15.095 ","End":"01:18.420","Text":"and they\u0027re both equal to some constant."},{"Start":"01:18.560 ","End":"01:21.675","Text":"The next substitution we\u0027ll make,"},{"Start":"01:21.675 ","End":"01:23.894","Text":"and after u comes v,"},{"Start":"01:23.894 ","End":"01:28.650","Text":"we\u0027ll let v equals z minus y,"},{"Start":"01:28.650 ","End":"01:38.880","Text":"and then these 2 I can write as v equals 1 and v equals 0."},{"Start":"01:38.880 ","End":"01:41.235","Text":"The last 2 are okay,"},{"Start":"01:41.235 ","End":"01:43.130","Text":"but when we\u0027re substituting,"},{"Start":"01:43.130 ","End":"01:44.720","Text":"we usually go the whole way,"},{"Start":"01:44.720 ","End":"01:47.585","Text":"instead of x, y, z we want u, v, and w,"},{"Start":"01:47.585 ","End":"01:51.760","Text":"so we\u0027ll just rename x as w,"},{"Start":"01:51.760 ","End":"01:53.940","Text":"it is a substitution,"},{"Start":"01:53.940 ","End":"01:57.640","Text":"just let w equals x, and then the last 2,"},{"Start":"01:57.640 ","End":"02:04.435","Text":"we can write as w equals 3 and w equals 1."},{"Start":"02:04.435 ","End":"02:07.755","Text":"Now, each of these comes in pairs,"},{"Start":"02:07.755 ","End":"02:12.360","Text":"u equals on the 1 hand 4, on the other hand 2,"},{"Start":"02:12.360 ","End":"02:16.890","Text":"these are now planes in the u, v,"},{"Start":"02:16.890 ","End":"02:20.310","Text":"w space, between 4 and 2,"},{"Start":"02:20.310 ","End":"02:21.690","Text":"between 1 and 0,"},{"Start":"02:21.690 ","End":"02:24.040","Text":"between 3 and 1."},{"Start":"02:24.770 ","End":"02:29.565","Text":"We can see that this integral,"},{"Start":"02:29.565 ","End":"02:31.530","Text":"at least in the u, v,"},{"Start":"02:31.530 ","End":"02:39.585","Text":"w space, I can write it now as an iterated integral,"},{"Start":"02:39.585 ","End":"02:42.490","Text":"let\u0027s say we\u0027ll take it in the order u,"},{"Start":"02:42.490 ","End":"02:44.980","Text":"and then v, and then w,"},{"Start":"02:44.980 ","End":"02:46.720","Text":"it doesn\u0027t really matter."},{"Start":"02:46.720 ","End":"02:52.065","Text":"U is sandwiched between 4 and 2,"},{"Start":"02:52.065 ","End":"02:55.710","Text":"obviously, it\u0027s 2 below and 4 above,"},{"Start":"02:55.710 ","End":"03:00.130","Text":"v between 0 and 1,"},{"Start":"03:03.860 ","End":"03:09.580","Text":"w between 1 and 3."},{"Start":"03:09.950 ","End":"03:12.915","Text":"We can continue."},{"Start":"03:12.915 ","End":"03:18.770","Text":"The exercise is so setup that this has a nice expression in terms of u, v,"},{"Start":"03:18.770 ","End":"03:25.385","Text":"w. Z minus y from here is v,"},{"Start":"03:25.385 ","End":"03:28.840","Text":"so this is just v squared,"},{"Start":"03:28.840 ","End":"03:33.870","Text":"xy I read off here is u."},{"Start":"03:33.870 ","End":"03:37.845","Text":"The thing is, we have to convert dV."},{"Start":"03:37.845 ","End":"03:39.830","Text":"DV is normally like dx,"},{"Start":"03:39.830 ","End":"03:42.035","Text":"dy, dz in some order."},{"Start":"03:42.035 ","End":"03:46.280","Text":"Here\u0027s where we use the theory that when you"},{"Start":"03:46.280 ","End":"03:51.030","Text":"replace or when you make a change of variables from say,"},{"Start":"03:51.030 ","End":"03:52.995","Text":"x, y, z to u, v, w,"},{"Start":"03:52.995 ","End":"03:58.100","Text":"we don\u0027t just put what you would expect, du, dv,"},{"Start":"03:58.100 ","End":"04:04.845","Text":"dw in some order or in this case it would have to be dw, dv,"},{"Start":"04:04.845 ","End":"04:07.500","Text":"du, but there\u0027s a missing piece,"},{"Start":"04:07.500 ","End":"04:09.910","Text":"and this is the important thing not to forget,"},{"Start":"04:09.910 ","End":"04:13.025","Text":"the absolute value of the Jacobian."},{"Start":"04:13.025 ","End":"04:16.310","Text":"This Jacobian J I\u0027ll remind you what it is,"},{"Start":"04:16.310 ","End":"04:23.645","Text":"but that\u0027s the important piece that you wouldn\u0027t think of when you are converting."},{"Start":"04:23.645 ","End":"04:26.420","Text":"Now, in general, the Jacobian,"},{"Start":"04:26.420 ","End":"04:30.450","Text":"whenever we have such a change of variables from x,"},{"Start":"04:30.450 ","End":"04:32.579","Text":"y, z to u, v, w,"},{"Start":"04:32.579 ","End":"04:36.900","Text":"is given as a 3 by 3 determinant,"},{"Start":"04:36.900 ","End":"04:43.265","Text":"and here these are all partial derivatives,"},{"Start":"04:43.265 ","End":"04:45.350","Text":"x with respect to v,"},{"Start":"04:45.350 ","End":"04:47.120","Text":"x with respect to w,"},{"Start":"04:47.120 ","End":"04:48.350","Text":"that\u0027s the first row."},{"Start":"04:48.350 ","End":"04:52.260","Text":"Second row, y with respect to u and"},{"Start":"04:52.260 ","End":"04:59.020","Text":"v and w. You probably get the last row z with respect to u,"},{"Start":"04:59.020 ","End":"05:01.670","Text":"with respect to v,"},{"Start":"05:01.670 ","End":"05:08.585","Text":"and with respect to w. The problem is at the moment,"},{"Start":"05:08.585 ","End":"05:11.315","Text":"is that if you look at this line,"},{"Start":"05:11.315 ","End":"05:13.280","Text":"we see that we have u, v,"},{"Start":"05:13.280 ","End":"05:16.550","Text":"and w in terms of x, y, and z."},{"Start":"05:16.550 ","End":"05:18.335","Text":"But we would like the opposite."},{"Start":"05:18.335 ","End":"05:20.090","Text":"I would like x, y,"},{"Start":"05:20.090 ","End":"05:21.920","Text":"and z in terms of u,"},{"Start":"05:21.920 ","End":"05:26.335","Text":"v, and w, and then I can do these calculations."},{"Start":"05:26.335 ","End":"05:28.650","Text":"The first 1 is easy."},{"Start":"05:28.650 ","End":"05:32.465","Text":"X is equal to w. We can get that off here."},{"Start":"05:32.465 ","End":"05:37.165","Text":"Now, what about y?"},{"Start":"05:37.165 ","End":"05:39.550","Text":"Well, y appears here and here,"},{"Start":"05:39.550 ","End":"05:41.765","Text":"it suits me to look at y here."},{"Start":"05:41.765 ","End":"05:43.925","Text":"If I take y from there,"},{"Start":"05:43.925 ","End":"05:47.015","Text":"I would write it as u over x,"},{"Start":"05:47.015 ","End":"05:49.460","Text":"but x is equal to w,"},{"Start":"05:49.460 ","End":"05:55.115","Text":"so from here I can get that y equals u over w,"},{"Start":"05:55.115 ","End":"06:00.465","Text":"and then finally z is equal to,"},{"Start":"06:00.465 ","End":"06:03.700","Text":"from here, v plus y."},{"Start":"06:03.700 ","End":"06:08.225","Text":"But instead of v plus y, we only have y,"},{"Start":"06:08.225 ","End":"06:12.545","Text":"so that\u0027s u over w. Now,"},{"Start":"06:12.545 ","End":"06:16.480","Text":"I have the reverse of this over here."},{"Start":"06:16.480 ","End":"06:21.280","Text":"This is good for making these computations."},{"Start":"06:21.350 ","End":"06:26.015","Text":"Let\u0027s continue and we\u0027ll take 9 partial derivatives."},{"Start":"06:26.015 ","End":"06:29.570","Text":"Let\u0027s see if I can make a big determinant sign"},{"Start":"06:29.570 ","End":"06:33.980","Text":"here and hope that it will be enough room to fit all 9 of them in."},{"Start":"06:33.980 ","End":"06:36.670","Text":"X with respect to u,"},{"Start":"06:36.670 ","End":"06:39.935","Text":"that\u0027s nothing because there is no u here,"},{"Start":"06:39.935 ","End":"06:42.964","Text":"x with respect to v, also nothing,"},{"Start":"06:42.964 ","End":"06:47.065","Text":"x with respect to w, just 1."},{"Start":"06:47.065 ","End":"06:52.080","Text":"Next 1, y, first with respect to u,"},{"Start":"06:52.080 ","End":"06:58.140","Text":"that makes w a constant so it\u0027s just 1 over w,"},{"Start":"06:58.140 ","End":"07:01.115","Text":"then with respect to v,"},{"Start":"07:01.115 ","End":"07:06.295","Text":"there is no v, so that\u0027s 0, and with respect to w,"},{"Start":"07:06.295 ","End":"07:11.010","Text":"1 over w gives minus 1 over w squared and the u sticks,"},{"Start":"07:11.010 ","End":"07:16.155","Text":"so it\u0027s minus u over w squared."},{"Start":"07:16.155 ","End":"07:19.710","Text":"Then the z, well,"},{"Start":"07:19.710 ","End":"07:21.120","Text":"with respect to u,"},{"Start":"07:21.120 ","End":"07:24.340","Text":"it\u0027s just 1 over w,"},{"Start":"07:24.470 ","End":"07:27.165","Text":"with respect to v,"},{"Start":"07:27.165 ","End":"07:33.450","Text":"it\u0027s just 1, and with respect to w,"},{"Start":"07:33.450 ","End":"07:40.670","Text":"it\u0027s minus u over w squared."},{"Start":"07:40.670 ","End":"07:43.945","Text":"I\u0027m assuming you know about determinants"},{"Start":"07:43.945 ","End":"07:48.819","Text":"and in particular how to expand according to a row or column."},{"Start":"07:48.819 ","End":"07:52.390","Text":"I noticed that this row has couple of 0s in it."},{"Start":"07:52.390 ","End":"07:54.400","Text":"I could have also used this middle column,"},{"Start":"07:54.400 ","End":"07:55.885","Text":"also has a couple of 0s."},{"Start":"07:55.885 ","End":"07:58.975","Text":"Let\u0027s expand by this row."},{"Start":"07:58.975 ","End":"08:06.085","Text":"What we do is, we multiply each of these by its co-factor and add them up."},{"Start":"08:06.085 ","End":"08:07.930","Text":"But there is only 1 non 0,"},{"Start":"08:07.930 ","End":"08:10.930","Text":"1, which is this 1,"},{"Start":"08:10.930 ","End":"08:21.460","Text":"and it\u0027s minor, is this 2 by 2 matrix."},{"Start":"08:21.460 ","End":"08:26.860","Text":"The co-factor means that we take the determinant"},{"Start":"08:26.860 ","End":"08:32.140","Text":"of this and multiply it also by plus or minus 1."},{"Start":"08:32.140 ","End":"08:36.490","Text":"Basically, what we do is we multiply, first of all,"},{"Start":"08:36.490 ","End":"08:38.425","Text":"the 1 from here,"},{"Start":"08:38.425 ","End":"08:43.000","Text":"and then the determinant of this."},{"Start":"08:43.000 ","End":"08:49.270","Text":"That\u0027s the minor of this entry."},{"Start":"08:49.270 ","End":"08:52.975","Text":"The difference between a minor and a co-factor is"},{"Start":"08:52.975 ","End":"08:56.365","Text":"that the co-factor has an extra plus or minus in it."},{"Start":"08:56.365 ","End":"08:58.495","Text":"We have to multiply this,"},{"Start":"08:58.495 ","End":"09:03.820","Text":"it\u0027s minus 1 to the power of the row plus column."},{"Start":"09:03.820 ","End":"09:05.110","Text":"But there\u0027s an easier way."},{"Start":"09:05.110 ","End":"09:07.465","Text":"Just imagine it like a checkerboard"},{"Start":"09:07.465 ","End":"09:10.750","Text":"that starts with a plus here and then alternate of plus,"},{"Start":"09:10.750 ","End":"09:14.680","Text":"minus, plus, minus, plus, and so on."},{"Start":"09:14.680 ","End":"09:17.410","Text":"Might as well through the whole thing, plus, minus, plus,"},{"Start":"09:17.410 ","End":"09:21.475","Text":"and this entry has a plus next to it."},{"Start":"09:21.475 ","End":"09:26.530","Text":"I\u0027m just going to write plus here to show you that I remembered."},{"Start":"09:26.530 ","End":"09:28.630","Text":"Because if it was this entry,"},{"Start":"09:28.630 ","End":"09:30.430","Text":"or this entry, for example,"},{"Start":"09:30.430 ","End":"09:32.770","Text":"it would be a minus."},{"Start":"09:32.770 ","End":"09:35.590","Text":"Now, 2 by 2 determinants are easy."},{"Start":"09:35.590 ","End":"09:39.730","Text":"It\u0027s just the product of this diagonal minus the product of this diagonal."},{"Start":"09:39.730 ","End":"09:43.600","Text":"This diagonal is the only 1 that counts because this 1 has a 0 in it."},{"Start":"09:43.600 ","End":"09:46.675","Text":"It\u0027s 1 over w times the 1,"},{"Start":"09:46.675 ","End":"09:53.170","Text":"so we just get 1 over w. In this integral,"},{"Start":"09:53.170 ","End":"09:57.910","Text":"I have to replace the J by 1 over w. Since"},{"Start":"09:57.910 ","End":"10:03.190","Text":"w is between 1 and 3 and therefore 1 over w is also positive."},{"Start":"10:03.190 ","End":"10:10.975","Text":"The whole of the absolute value of the Jacobian could be replaced by 1 over w,"},{"Start":"10:10.975 ","End":"10:14.200","Text":"and now we\u0027ve got a new integral,"},{"Start":"10:14.200 ","End":"10:16.735","Text":"and I\u0027ll rewrite it over here."},{"Start":"10:16.735 ","End":"10:22.675","Text":"The integral for you going from 2-4."},{"Start":"10:22.675 ","End":"10:25.210","Text":"Now, there\u0027s something I sometimes do, not always,"},{"Start":"10:25.210 ","End":"10:30.250","Text":"I find it often convenient to bring variables further to the front."},{"Start":"10:30.250 ","End":"10:35.110","Text":"For example, u doesn\u0027t depend on w or v, it\u0027s a constant."},{"Start":"10:35.110 ","End":"10:42.355","Text":"I can bring it all the way here and then I have the integral v goes from 0-1,"},{"Start":"10:42.355 ","End":"10:45.940","Text":"and this v squared is a constant as far as w goes,"},{"Start":"10:45.940 ","End":"10:47.890","Text":"so I can pull this in front,"},{"Start":"10:47.890 ","End":"10:53.305","Text":"v squared, integral w goes from 1-3."},{"Start":"10:53.305 ","End":"11:01.885","Text":"The 1 over w has to stay in here and then it\u0027s dw dv du."},{"Start":"11:01.885 ","End":"11:07.885","Text":"This just makes things slightly neater to compute if I bring things further upfront,"},{"Start":"11:07.885 ","End":"11:10.375","Text":"it\u0027s optional, but I like it."},{"Start":"11:10.375 ","End":"11:13.000","Text":"As always, we start from the inside,"},{"Start":"11:13.000 ","End":"11:15.940","Text":"so that would be the dw integral,"},{"Start":"11:15.940 ","End":"11:19.430","Text":"and I can use some more space."},{"Start":"11:20.220 ","End":"11:23.905","Text":"I\u0027ll do this as a side exercise over here,"},{"Start":"11:23.905 ","End":"11:29.785","Text":"so the integral from 1-3 of 1 over w dw."},{"Start":"11:29.785 ","End":"11:37.030","Text":"This is the natural log of w taken from 1-3,"},{"Start":"11:37.030 ","End":"11:47.545","Text":"and this equals natural log of 3 minus natural log of 1."},{"Start":"11:47.545 ","End":"11:50.485","Text":"But natural log of 1 is 0,"},{"Start":"11:50.485 ","End":"11:53.140","Text":"so it\u0027s just natural log of 3."},{"Start":"11:53.140 ","End":"11:55.150","Text":"I guess I could write minus 0,"},{"Start":"11:55.150 ","End":"11:57.980","Text":"just to show you I didn\u0027t forget."},{"Start":"11:58.170 ","End":"12:01.810","Text":"We get natural log of 3 here,"},{"Start":"12:01.810 ","End":"12:03.700","Text":"I\u0027ll just write that down here."},{"Start":"12:03.700 ","End":"12:06.100","Text":"Natural log of 3 for this whole thing,"},{"Start":"12:06.100 ","End":"12:09.425","Text":"but that\u0027s a constant so I can pull it out front,"},{"Start":"12:09.425 ","End":"12:15.900","Text":"and so now I get the integral from 2-4,"},{"Start":"12:19.350 ","End":"12:26.060","Text":"silly me after saying it I forgot to put the natural log of 3 in front."},{"Start":"12:26.100 ","End":"12:31.900","Text":"From 2-4 and then from 0-1,"},{"Start":"12:31.900 ","End":"12:34.045","Text":"this was the u,"},{"Start":"12:34.045 ","End":"12:36.400","Text":"and this was the v,"},{"Start":"12:36.400 ","End":"12:39.505","Text":"and here I have v squared."},{"Start":"12:39.505 ","End":"12:45.065","Text":"This whole thing was taken up front, then dv du."},{"Start":"12:45.065 ","End":"12:48.070","Text":"Next is this integral."},{"Start":"12:48.070 ","End":"12:50.095","Text":"We work from the inside,"},{"Start":"12:50.095 ","End":"12:53.635","Text":"and I like to do these as a side exercise."},{"Start":"12:53.635 ","End":"13:00.250","Text":"The integral from 0-1 of v squared dv."},{"Start":"13:00.250 ","End":"13:06.655","Text":"This is 1/3 v cubed from 0-1."},{"Start":"13:06.655 ","End":"13:08.785","Text":"0 gives me nothing,"},{"Start":"13:08.785 ","End":"13:11.410","Text":"1 gives me 1/3 times 1 cubed,"},{"Start":"13:11.410 ","End":"13:14.215","Text":"so I end up with just 1/3."},{"Start":"13:14.215 ","End":"13:17.425","Text":"Back here, I\u0027ll just make a note to that, this is 1/3."},{"Start":"13:17.425 ","End":"13:18.850","Text":"Again, it\u0027s a constant,"},{"Start":"13:18.850 ","End":"13:21.620","Text":"so I can bring it in front."},{"Start":"13:21.630 ","End":"13:23.710","Text":"What we get is,"},{"Start":"13:23.710 ","End":"13:28.510","Text":"let\u0027s see, 1/3, then natural log of 3."},{"Start":"13:28.510 ","End":"13:36.500","Text":"Now, all we have is the integral for u from 2-4 of u du."},{"Start":"13:38.760 ","End":"13:42.040","Text":"There\u0027s just this integral to compute,"},{"Start":"13:42.040 ","End":"13:45.685","Text":"just for consistency, I\u0027ll do this 1 at the side also."},{"Start":"13:45.685 ","End":"13:52.600","Text":"I get the integral from 2-4 of u du is 1/2 u"},{"Start":"13:52.600 ","End":"14:00.355","Text":"squared from 2-4 and this equals,"},{"Start":"14:00.355 ","End":"14:09.700","Text":"let\u0027s see, 1/2, now 4 squared is 16,"},{"Start":"14:09.700 ","End":"14:12.040","Text":"2 squared is 4,"},{"Start":"14:12.040 ","End":"14:17.485","Text":"16 minus 4, 1/2 of that it\u0027s 1/2 of 12."},{"Start":"14:17.485 ","End":"14:20.005","Text":"That gives me 6."},{"Start":"14:20.005 ","End":"14:26.260","Text":"This comes out to 6 and so this whole thing is just equal"},{"Start":"14:26.260 ","End":"14:32.605","Text":"to 1/3 times natural log of 3 times 6."},{"Start":"14:32.605 ","End":"14:35.755","Text":"But 1/3 of 6 is 2."},{"Start":"14:35.755 ","End":"14:41.905","Text":"This is just equal to 2 natural log of 3,"},{"Start":"14:41.905 ","End":"14:43.960","Text":"and I\u0027ll highlight this,"},{"Start":"14:43.960 ","End":"14:47.060","Text":"and that\u0027s our answer. We\u0027re done."}],"ID":8747},{"Watched":false,"Name":"Exercise 2","Duration":"8m 17s","ChapterTopicVideoID":8573,"CourseChapterTopicPlaylistID":21194,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8573.jpeg","UploadDate":"2017-02-13T03:37:46.8200000","DurationForVideoObject":"PT8M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.790","Text":"In this exercise, we\u0027re asked to compute the volume of the ellipsoid,"},{"Start":"00:04.790 ","End":"00:08.115","Text":"and this is the equation of an ellipsoid."},{"Start":"00:08.115 ","End":"00:13.170","Text":"There\u0027s something not quite precise about this question because the ellipsoid,"},{"Start":"00:13.170 ","End":"00:16.425","Text":"this equation is just the shell of the ellipsoid."},{"Start":"00:16.425 ","End":"00:22.210","Text":"We mean the volume of the body enclosed by the ellipsoid or the solid ellipsoid."},{"Start":"00:22.210 ","End":"00:26.730","Text":"Really, this should be less than or equal to 1,"},{"Start":"00:26.730 ","End":"00:29.730","Text":"and then it would include the interior as well."},{"Start":"00:29.730 ","End":"00:32.800","Text":"Well, that\u0027s just being precise."},{"Start":"00:36.560 ","End":"00:39.300","Text":"Let\u0027s give it a name, the ellipsoid,"},{"Start":"00:39.300 ","End":"00:41.610","Text":"I\u0027ll call it E for ellipsoid."},{"Start":"00:41.610 ","End":"00:45.970","Text":"In general, the volume of a body,"},{"Start":"00:45.970 ","End":"00:48.800","Text":"whatever lateral it is of E,"},{"Start":"00:48.800 ","End":"00:58.800","Text":"is just the triple integral over that solid body E of 1 dV."},{"Start":"00:58.800 ","End":"01:04.160","Text":"The thing is that this E is quite difficult to describe"},{"Start":"01:04.160 ","End":"01:09.395","Text":"if you try to do it with Cartesian coordinates just as is."},{"Start":"01:09.395 ","End":"01:13.280","Text":"It\u0027s quite awkward to say what x goes from and to and to"},{"Start":"01:13.280 ","End":"01:17.820","Text":"find y as a function of x, and so on."},{"Start":"01:17.820 ","End":"01:24.415","Text":"What I suggest is a change of variables to make this shape a bit friendlier."},{"Start":"01:24.415 ","End":"01:30.420","Text":"In fact, an ellipsoid is nothing but a stretched sphere or"},{"Start":"01:30.420 ","End":"01:37.990","Text":"a ball scaled in each direction like a football or a blimp or whatever."},{"Start":"01:37.990 ","End":"01:41.305","Text":"Anyway, if we just do a change of scale,"},{"Start":"01:41.305 ","End":"01:43.780","Text":"let me tell you specifically what I\u0027m suggesting."},{"Start":"01:43.780 ","End":"01:50.149","Text":"I\u0027m suggesting that we substitute u equals x over a,"},{"Start":"01:50.149 ","End":"01:54.595","Text":"that will let v equals y over b,"},{"Start":"01:54.595 ","End":"01:59.660","Text":"that will let w equals z over c like x,"},{"Start":"01:59.660 ","End":"02:02.975","Text":"y and z multiplied or divided by a constant,"},{"Start":"02:02.975 ","End":"02:04.985","Text":"and if we do that,"},{"Start":"02:04.985 ","End":"02:09.750","Text":"this is exactly u squared, this first term."},{"Start":"02:09.790 ","End":"02:12.095","Text":"Well, I could write it again."},{"Start":"02:12.095 ","End":"02:16.700","Text":"Let me just write it again as x over a squared plus y"},{"Start":"02:16.700 ","End":"02:21.774","Text":"over b squared plus z over c squared,"},{"Start":"02:21.774 ","End":"02:26.195","Text":"and I\u0027ll do the solid ellipsoid less than or equal to 1."},{"Start":"02:26.195 ","End":"02:29.095","Text":"Then after this substitution,"},{"Start":"02:29.095 ","End":"02:32.010","Text":"this was our ellipsoid E,"},{"Start":"02:32.010 ","End":"02:34.770","Text":"and after the substitution,"},{"Start":"02:34.770 ","End":"02:39.890","Text":"we\u0027ll get that u squared plus v"},{"Start":"02:39.890 ","End":"02:46.340","Text":"squared plus w squared is less than or equal to 1,"},{"Start":"02:46.340 ","End":"02:51.750","Text":"and that\u0027s the equation of a ball in the u, v, w plane."},{"Start":"02:51.750 ","End":"02:56.920","Text":"I say ball, it\u0027s a sphere including the interior."},{"Start":"02:56.920 ","End":"02:59.530","Text":"Let\u0027s give this a name,"},{"Start":"02:59.530 ","End":"03:03.595","Text":"let\u0027s call this B for body or ball."},{"Start":"03:03.595 ","End":"03:10.530","Text":"This is a much easier shape to work with because we\u0027re familiar with this."},{"Start":"03:10.530 ","End":"03:15.370","Text":"What we have is if we make this conversion,"},{"Start":"03:15.370 ","End":"03:26.505","Text":"we now have that the volume here is going to equal the triple integral,"},{"Start":"03:26.505 ","End":"03:32.260","Text":"E, the ellipsoid changes into B, the ball,"},{"Start":"03:32.260 ","End":"03:36.780","Text":"but dV, which is like dx, dy,"},{"Start":"03:36.780 ","End":"03:38.750","Text":"dz in some order,"},{"Start":"03:38.750 ","End":"03:42.980","Text":"does not just become du, dv, dw."},{"Start":"03:42.980 ","End":"03:49.125","Text":"It becomes the absolute value of the Jacobian,"},{"Start":"03:49.125 ","End":"03:56.265","Text":"remember the Jacobian, then times in whatever order."},{"Start":"03:56.265 ","End":"04:06.390","Text":"Let\u0027s just write it as dwdvdu."},{"Start":"04:06.390 ","End":"04:15.085","Text":"I copy-pasted the definition of the Jacobian for 3 variables from our previous exercise."},{"Start":"04:15.085 ","End":"04:18.620","Text":"If you remember or even if not,"},{"Start":"04:18.620 ","End":"04:22.010","Text":"the thing is that we have u,"},{"Start":"04:22.010 ","End":"04:25.550","Text":"v, and w in terms of x, y, and z."},{"Start":"04:25.550 ","End":"04:27.650","Text":"To do these partial derivatives,"},{"Start":"04:27.650 ","End":"04:29.149","Text":"we need the opposite."},{"Start":"04:29.149 ","End":"04:37.190","Text":"I want to reverse the formula and have x,"},{"Start":"04:37.190 ","End":"04:38.960","Text":"y, and z in terms of u, v,"},{"Start":"04:38.960 ","End":"04:46.774","Text":"and w. Well, that\u0027s fairly straightforward because I have from here x equals a times u,"},{"Start":"04:46.774 ","End":"04:51.310","Text":"y equals b times v,"},{"Start":"04:51.310 ","End":"04:55.940","Text":"and z equals c times w. Now,"},{"Start":"04:55.940 ","End":"04:58.715","Text":"I can compute this fairly easily."},{"Start":"04:58.715 ","End":"05:01.580","Text":"Let\u0027s see. We get that J equals,"},{"Start":"05:01.580 ","End":"05:07.370","Text":"now leave room for a 3 by 3. Let\u0027s see."},{"Start":"05:07.370 ","End":"05:10.220","Text":"Take each variable x, y,"},{"Start":"05:10.220 ","End":"05:12.500","Text":"z with respect to each variable u,"},{"Start":"05:12.500 ","End":"05:14.720","Text":"v, w. Let\u0027s start with the x."},{"Start":"05:14.720 ","End":"05:19.325","Text":"With respect to u, the derivative is a,"},{"Start":"05:19.325 ","End":"05:22.025","Text":"but with respect to v and w,"},{"Start":"05:22.025 ","End":"05:25.115","Text":"it\u0027s 0 because there is no v and w here."},{"Start":"05:25.115 ","End":"05:32.820","Text":"Similarly for y, it\u0027s 0 as far as u and w go,"},{"Start":"05:32.820 ","End":"05:37.230","Text":"and it\u0027s b with respect to v and for the last 1,"},{"Start":"05:37.230 ","End":"05:42.530","Text":"z only depends on w and its derivative is c. Now,"},{"Start":"05:42.530 ","End":"05:47.580","Text":"this is an easy determinant to compute."},{"Start":"05:48.140 ","End":"05:50.460","Text":"When we have a matrix,"},{"Start":"05:50.460 ","End":"05:54.660","Text":"which just has a main diagonal and nothing else,"},{"Start":"05:54.660 ","End":"06:00.125","Text":"then the Jacobian turns out to be just the product of the entries on the diagonal."},{"Start":"06:00.125 ","End":"06:03.480","Text":"This happens to be a, b,"},{"Start":"06:03.480 ","End":"06:08.595","Text":"c. Now, this is a constant."},{"Start":"06:08.595 ","End":"06:11.520","Text":"If I put instead of J here, a,"},{"Start":"06:11.520 ","End":"06:14.300","Text":"b, and c are assumed to be positive."},{"Start":"06:14.300 ","End":"06:16.485","Text":"I should have written that a, b,"},{"Start":"06:16.485 ","End":"06:21.400","Text":"and c are all positive."},{"Start":"06:21.890 ","End":"06:25.110","Text":"Then a, b, c is also positive,"},{"Start":"06:25.110 ","End":"06:27.515","Text":"so the absolute value of the Jacobian,"},{"Start":"06:27.515 ","End":"06:30.020","Text":"this bit, is just a, b,"},{"Start":"06:30.020 ","End":"06:36.740","Text":"c. Just copy the words,"},{"Start":"06:36.740 ","End":"06:39.980","Text":"the volume of E,"},{"Start":"06:39.980 ","End":"06:45.900","Text":"the ellipsoid is, I can bring this out front, is a, b,"},{"Start":"06:45.900 ","End":"06:51.229","Text":"c times the triple integral over the ball,"},{"Start":"06:51.229 ","End":"06:53.825","Text":"that\u0027s the sphere with the inside,"},{"Start":"06:53.825 ","End":"07:02.900","Text":"given by this, of dwdvdu,"},{"Start":"07:02.900 ","End":"07:05.425","Text":"and I\u0027ll put a 1 here."},{"Start":"07:05.425 ","End":"07:12.485","Text":"Now, the integral of 1 over a region is just the volume of the region."},{"Start":"07:12.485 ","End":"07:18.090","Text":"What we get here is a, b, c times,"},{"Start":"07:18.090 ","End":"07:19.485","Text":"I\u0027ll write it in words,"},{"Start":"07:19.485 ","End":"07:25.420","Text":"the volume of B."},{"Start":"07:25.520 ","End":"07:32.290","Text":"Now, there\u0027s a standard formula for the unit ball."},{"Start":"07:36.050 ","End":"07:41.400","Text":"The formula says 4/3 Pi r cubed,"},{"Start":"07:41.400 ","End":"07:43.095","Text":"where r is the radius,"},{"Start":"07:43.095 ","End":"07:45.570","Text":"but here, r equals 1,"},{"Start":"07:45.570 ","End":"07:49.240","Text":"and so the volume of the unit ball,"},{"Start":"07:49.240 ","End":"07:56.700","Text":"this bit here, is just 4/3 Pi because r is 1."},{"Start":"07:56.700 ","End":"08:07.590","Text":"Altogether, what we get is 4/3 Pi abc,"},{"Start":"08:07.590 ","End":"08:09.735","Text":"or in some other order."},{"Start":"08:09.735 ","End":"08:12.000","Text":"That is the answer,"},{"Start":"08:12.000 ","End":"08:14.730","Text":"and this is the volume of the ellipsoid."},{"Start":"08:14.730 ","End":"08:17.320","Text":"We are done."}],"ID":8748},{"Watched":false,"Name":"Exercise 3","Duration":"15m 39s","ChapterTopicVideoID":8574,"CourseChapterTopicPlaylistID":21194,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8574.jpeg","UploadDate":"2017-02-13T03:41:17.4030000","DurationForVideoObject":"PT15M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.240","Text":"In this exercise, we have to compute the triple integral of x squared over the region E,"},{"Start":"00:09.240 ","End":"00:14.685","Text":"where E is the ellipsoid given here."},{"Start":"00:14.685 ","End":"00:18.660","Text":"Strictly speaking, this is just the shell of the ellipsoid,"},{"Start":"00:18.660 ","End":"00:23.190","Text":"it really should have been written as less than or equal to,"},{"Start":"00:23.190 ","End":"00:27.610","Text":"and then it includes the interior too, the solid ellipsoid."},{"Start":"00:27.610 ","End":"00:31.070","Text":"Sometimes we mix the 2,"},{"Start":"00:31.070 ","End":"00:33.920","Text":"like a sphere and a ball which we\u0027ll see,"},{"Start":"00:33.920 ","End":"00:38.400","Text":"the sphere is just the shell and the ball is the interior."},{"Start":"00:38.410 ","End":"00:44.840","Text":"We had a similar 1 like this where actually we did do a change of variables,"},{"Start":"00:44.840 ","End":"00:49.130","Text":"and we squashed the ellipsoid to a ball."},{"Start":"00:49.130 ","End":"00:55.055","Text":"The substitution that works is to let u equals x over a,"},{"Start":"00:55.055 ","End":"00:59.299","Text":"to let v equals y over b,"},{"Start":"00:59.299 ","End":"01:07.650","Text":"and to let w to equal z over c,"},{"Start":"01:07.650 ","End":"01:17.385","Text":"then this thing becomes u squared plus v squared plus w squared."},{"Start":"01:17.385 ","End":"01:22.445","Text":"Because u squared is x over a squared which is x squared over a squared, and so on,"},{"Start":"01:22.445 ","End":"01:26.340","Text":"equals 1 or really less than or equal to"},{"Start":"01:26.340 ","End":"01:30.830","Text":"1 because equals 1 would give me a sphere in the u,"},{"Start":"01:30.830 ","End":"01:33.740","Text":"v, w, but I want the solid sphere,"},{"Start":"01:33.740 ","End":"01:35.390","Text":"which is also called the ball,"},{"Start":"01:35.390 ","End":"01:37.040","Text":"the unit ball, in fact,"},{"Start":"01:37.040 ","End":"01:39.305","Text":"because this is 1 squared."},{"Start":"01:39.305 ","End":"01:43.585","Text":"In fact, let\u0027s name this region as B,"},{"Start":"01:43.585 ","End":"01:47.085","Text":"B for ball or B for body."},{"Start":"01:47.085 ","End":"01:51.985","Text":"What we would get if we did this conversion,"},{"Start":"01:51.985 ","End":"01:56.220","Text":"is we would get the triple integral"},{"Start":"01:56.480 ","End":"02:05.095","Text":"over the unit ball B of x squared."},{"Start":"02:05.095 ","End":"02:14.900","Text":"Now, x squared, you can see from here that x is a u. You know what else?"},{"Start":"02:14.900 ","End":"02:16.505","Text":"While we\u0027re at it,"},{"Start":"02:16.505 ","End":"02:18.680","Text":"let\u0027s do the inverse transformations."},{"Start":"02:18.680 ","End":"02:21.160","Text":"X is equal to au,"},{"Start":"02:21.160 ","End":"02:24.870","Text":"y is equal to bv,"},{"Start":"02:24.870 ","End":"02:27.870","Text":"and z equals cw."},{"Start":"02:27.870 ","End":"02:30.465","Text":"We\u0027ll need these inverse transformations."},{"Start":"02:30.465 ","End":"02:33.975","Text":"Well, here we see that x is au,"},{"Start":"02:33.975 ","End":"02:38.670","Text":"so we have au squared,"},{"Start":"02:38.670 ","End":"02:48.060","Text":"and then dv is not going to be just du, dv,"},{"Start":"02:48.060 ","End":"02:54.675","Text":"dw, actually, I prefer to do it as dw,"},{"Start":"02:54.675 ","End":"03:00.305","Text":"dv, du, but there\u0027s also an extra piece."},{"Start":"03:00.305 ","End":"03:01.940","Text":"When you do such a transformation,"},{"Start":"03:01.940 ","End":"03:06.020","Text":"you need the absolute value of the Jacobian."},{"Start":"03:06.020 ","End":"03:10.120","Text":"I\u0027ll remind you in a moment what the Jacobian is,"},{"Start":"03:10.120 ","End":"03:12.980","Text":"I\u0027ll do it at the side here."},{"Start":"03:12.980 ","End":"03:16.085","Text":"The Jacobian is the determinant,"},{"Start":"03:16.085 ","End":"03:17.480","Text":"this is also bars,"},{"Start":"03:17.480 ","End":"03:19.085","Text":"but it\u0027s not absolute value,"},{"Start":"03:19.085 ","End":"03:23.239","Text":"of partial derivatives, x with respect to u,"},{"Start":"03:23.239 ","End":"03:25.250","Text":"x with respect to v,"},{"Start":"03:25.250 ","End":"03:27.500","Text":"x with respect to w, and so on."},{"Start":"03:27.500 ","End":"03:30.649","Text":"Y with respect to u, and v,"},{"Start":"03:30.649 ","End":"03:37.340","Text":"and w, and the partial derivatives of z with respect to u, v,"},{"Start":"03:37.340 ","End":"03:46.530","Text":"and w. This is another reason why we needed to get the inverse transformation from u,"},{"Start":"03:46.530 ","End":"03:48.750","Text":"v, w back to x, y, z."},{"Start":"03:48.750 ","End":"03:53.640","Text":"Now, we can compute this determinant c,"},{"Start":"03:53.640 ","End":"03:55.015","Text":"it\u0027s a 3 by 3."},{"Start":"03:55.015 ","End":"03:57.890","Text":"Now, x with respect to u,"},{"Start":"03:57.890 ","End":"04:02.840","Text":"the derivative is a and for the other 2 variables is 0."},{"Start":"04:02.840 ","End":"04:08.000","Text":"Similarly, partial derivatives of y with respect to v only,"},{"Start":"04:08.000 ","End":"04:13.270","Text":"is it something, and here we\u0027ll have a c and the rest of them will be 0s."},{"Start":"04:13.270 ","End":"04:17.114","Text":"When you have a diagonal matrix,"},{"Start":"04:17.114 ","End":"04:23.215","Text":"then the determinant is just the product of the entries along the diagonal,"},{"Start":"04:23.215 ","End":"04:26.530","Text":"so this is a, b, c. Now, a,"},{"Start":"04:26.530 ","End":"04:29.970","Text":"b and c, we presume are positive numbers."},{"Start":"04:29.970 ","End":"04:32.470","Text":"If they\u0027re not, just make them positive,"},{"Start":"04:32.470 ","End":"04:34.915","Text":"take the absolute value, it will make no difference."},{"Start":"04:34.915 ","End":"04:40.615","Text":"I\u0027m assuming that they\u0027re all bigger than 0."},{"Start":"04:40.615 ","End":"04:47.650","Text":"Our integral becomes, instead of this Jacobian,"},{"Start":"04:47.650 ","End":"04:50.020","Text":"I can put a, b, c,"},{"Start":"04:50.020 ","End":"04:53.110","Text":"and then I can take constants upfront,"},{"Start":"04:53.110 ","End":"04:55.105","Text":"so I have an a squared from here,"},{"Start":"04:55.105 ","End":"04:58.530","Text":"so what I get is a squared times a,"},{"Start":"04:58.530 ","End":"05:03.265","Text":"so it\u0027s a cubed times b times c"},{"Start":"05:03.265 ","End":"05:09.490","Text":"times the triple integral over the unit ball,"},{"Start":"05:09.490 ","End":"05:15.150","Text":"but in u, v, w coordinates of u"},{"Start":"05:15.150 ","End":"05:21.415","Text":"squared dw, dv, du."},{"Start":"05:21.415 ","End":"05:27.859","Text":"Now, the unit ball in Cartesian coordinates is a bit awkward,"},{"Start":"05:27.859 ","End":"05:31.250","Text":"so we\u0027re going to move to spherical coordinates."},{"Start":"05:31.250 ","End":"05:35.285","Text":"I mean, we are inside the unit sphere, the unit ball."},{"Start":"05:35.285 ","End":"05:42.370","Text":"The substitution, I\u0027ll just copy paste the formulas."},{"Start":"05:42.370 ","End":"05:48.370","Text":"Well, I copy pasted but this is not right because we\u0027re working not in x,"},{"Start":"05:48.370 ","End":"05:50.140","Text":"y, z, we\u0027re already in u, v,"},{"Start":"05:50.140 ","End":"05:52.690","Text":"w. Let me replace all the x, y,"},{"Start":"05:52.690 ","End":"05:56.685","Text":"z by u, v,"},{"Start":"05:56.685 ","End":"06:02.780","Text":"w, and here u, v, w. Also,"},{"Start":"06:02.780 ","End":"06:07.505","Text":"here the element of volume is the dw,"},{"Start":"06:07.505 ","End":"06:12.930","Text":"dv, du, what would have been dz,"},{"Start":"06:12.930 ","End":"06:15.370","Text":"dy, dx or whatever order."},{"Start":"06:15.370 ","End":"06:19.645","Text":"These are the substitutions that we have to make here."},{"Start":"06:19.645 ","End":"06:25.820","Text":"Returning here and doing the ball in spherical coordinates,"},{"Start":"06:25.820 ","End":"06:28.005","Text":"we will get, well,"},{"Start":"06:28.005 ","End":"06:32.820","Text":"a cubed bc stays."},{"Start":"06:32.820 ","End":"06:40.220","Text":"Now, this integral, we are going to have to write the ball in spherical coordinates."},{"Start":"06:40.220 ","End":"06:45.165","Text":"We\u0027ll see what does Theta go from and to,"},{"Start":"06:45.165 ","End":"06:48.210","Text":"what does Phi go from and to,"},{"Start":"06:48.210 ","End":"06:51.975","Text":"and what does r go from and to."},{"Start":"06:51.975 ","End":"06:55.560","Text":"I\u0027ll write these in a minute, let\u0027s just continue."},{"Start":"06:55.560 ","End":"07:02.205","Text":"The u squared would be r squared"},{"Start":"07:02.205 ","End":"07:11.205","Text":"sine squared Phi cosine squared Theta,"},{"Start":"07:11.205 ","End":"07:14.460","Text":"and then this is"},{"Start":"07:14.460 ","End":"07:22.380","Text":"another r squared sine Phi,"},{"Start":"07:22.380 ","End":"07:29.160","Text":"and then dr, dPhi, dTheta."},{"Start":"07:29.160 ","End":"07:33.360","Text":"As for the limits, well,"},{"Start":"07:33.360 ","End":"07:35.595","Text":"we know what the unit ball is,"},{"Start":"07:35.595 ","End":"07:37.470","Text":"I\u0027m not going to draw a sketch,"},{"Start":"07:37.470 ","End":"07:41.370","Text":"but Theta goes all the way round from 0 to 2 Pi."},{"Start":"07:41.370 ","End":"07:46.295","Text":"Phi, which is like the angle from the North Pole,"},{"Start":"07:46.295 ","End":"07:49.490","Text":"it only goes from the North Pole to the South Pole,"},{"Start":"07:49.490 ","End":"07:53.495","Text":"it only goes 180 degrees from 0 to Pi,"},{"Start":"07:53.495 ","End":"07:59.650","Text":"and the radius, because it\u0027s a unit sphere or a ball, goes from 0-1."},{"Start":"08:00.170 ","End":"08:04.985","Text":"Now, we just have this technical integral to do."},{"Start":"08:04.985 ","End":"08:07.205","Text":"What I\u0027m going to do is,"},{"Start":"08:07.205 ","End":"08:11.660","Text":"I like to pull variables out in front where they\u0027re not needed."},{"Start":"08:11.660 ","End":"08:16.490","Text":"For example, here I only need the stuff relating to r,"},{"Start":"08:16.490 ","End":"08:18.875","Text":"at least stuff from the inside."},{"Start":"08:18.875 ","End":"08:23.940","Text":"I have the integral r goes from 0-1."},{"Start":"08:23.940 ","End":"08:28.290","Text":"Now, the stuff with r, I\u0027ll take the r squared and the r squared,"},{"Start":"08:28.290 ","End":"08:34.140","Text":"and that will give me r^4 dr."},{"Start":"08:34.140 ","End":"08:38.260","Text":"I\u0027m going to pull the others further up front,"},{"Start":"08:38.260 ","End":"08:40.765","Text":"all these Phi and Theta."},{"Start":"08:40.765 ","End":"08:45.680","Text":"But you don\u0027t have to do this, but I find it works out neater."},{"Start":"08:46.410 ","End":"08:48.610","Text":"In front of this,"},{"Start":"08:48.610 ","End":"08:55.915","Text":"I can take the sine squared Phi with sine Phi,"},{"Start":"08:55.915 ","End":"09:00.955","Text":"so that becomes sine cubed Phi,"},{"Start":"09:00.955 ","End":"09:03.595","Text":"and then it\u0027s d Phi."},{"Start":"09:03.595 ","End":"09:07.885","Text":"The stuff with Theta can even be pulled out still further,"},{"Start":"09:07.885 ","End":"09:13.120","Text":"because it\u0027s a constant as far as r and Phi go."},{"Start":"09:13.120 ","End":"09:18.910","Text":"I just have this cosine squared Theta,"},{"Start":"09:18.910 ","End":"09:20.455","Text":"I\u0027ll put that here."},{"Start":"09:20.455 ","End":"09:27.445","Text":"Cosine squared Theta, d Theta."},{"Start":"09:27.445 ","End":"09:32.575","Text":"I still have the a cubed bc here."},{"Start":"09:32.575 ","End":"09:35.215","Text":"I have to copy the limits."},{"Start":"09:35.215 ","End":"09:38.680","Text":"This Theta goes from 0-2 Pi."},{"Start":"09:38.680 ","End":"09:43.359","Text":"Phi goes just from 0 to Pi,"},{"Start":"09:43.359 ","End":"09:45.985","Text":"and r we said here, from 0-1."},{"Start":"09:45.985 ","End":"09:50.560","Text":"Now, you know how to do these from the inside out."},{"Start":"09:50.560 ","End":"09:55.495","Text":"First I\u0027ll do the dr integral,"},{"Start":"09:55.495 ","End":"09:58.300","Text":"and I\u0027ll do this at the side."},{"Start":"09:58.300 ","End":"10:01.090","Text":"The integral from 0-1,"},{"Start":"10:01.090 ","End":"10:10.750","Text":"r^4 dr is just r^5 over 5, from 0-1."},{"Start":"10:10.750 ","End":"10:13.240","Text":"When it\u0027s 0, it just gives 0."},{"Start":"10:13.240 ","End":"10:15.250","Text":"When it\u0027s 1, it gives me a 1/5."},{"Start":"10:15.250 ","End":"10:17.035","Text":"The answer is just a 1/5."},{"Start":"10:17.035 ","End":"10:20.125","Text":"I\u0027ll make a note, this is 1/5."},{"Start":"10:20.125 ","End":"10:24.235","Text":"Since it\u0027s a constant, I can pull this also to the front."},{"Start":"10:24.235 ","End":"10:29.920","Text":"I now have 1/5 a cubed bc,"},{"Start":"10:29.920 ","End":"10:36.385","Text":"integral from 0-2 Pi of cosine squared Theta,"},{"Start":"10:36.385 ","End":"10:40.840","Text":"and then integral from 0 to Pi of"},{"Start":"10:40.840 ","End":"10:49.735","Text":"sine cubed Phi d Phi d Theta."},{"Start":"10:49.735 ","End":"10:54.970","Text":"Next, the innermost is this one."},{"Start":"10:54.970 ","End":"10:59.800","Text":"Once again, I\u0027ll do it at the side over here. What do I have?"},{"Start":"10:59.800 ","End":"11:09.610","Text":"The integral from 0 to Pi sine cubed Phi d Phi."},{"Start":"11:09.610 ","End":"11:13.000","Text":"This integral, I\u0027m going to look up in the table of integrals,"},{"Start":"11:13.000 ","End":"11:15.535","Text":"not going to compute it from scratch here."},{"Start":"11:15.535 ","End":"11:22.975","Text":"I looked it up and it comes out to be 1/3 cosine cubed."},{"Start":"11:22.975 ","End":"11:24.580","Text":"Well, when I looked it up,"},{"Start":"11:24.580 ","End":"11:28.299","Text":"it was with x, but I\u0027ve got it with Phi,"},{"Start":"11:28.299 ","End":"11:35.680","Text":"minus cosine of Phi,"},{"Start":"11:35.680 ","End":"11:42.835","Text":"squeeze it in, from 0 to Pi."},{"Start":"11:42.835 ","End":"11:46.960","Text":"In case you\u0027ve forgotten table of cosines,"},{"Start":"11:46.960 ","End":"11:50.890","Text":"cosine of 0 is 1,"},{"Start":"11:50.890 ","End":"11:56.290","Text":"and cosine of 180 or Pi is minus 1,"},{"Start":"11:56.290 ","End":"11:58.975","Text":"so I put that in here."},{"Start":"11:58.975 ","End":"12:02.125","Text":"If I put in Pi,"},{"Start":"12:02.125 ","End":"12:12.940","Text":"then I get 1/3 of minus 1 cubed minus minus 1."},{"Start":"12:12.940 ","End":"12:21.380","Text":"If I put in 0, I get 1/3 1 cubed minus 1."},{"Start":"12:22.680 ","End":"12:26.365","Text":"Here I\u0027ve got altogether,"},{"Start":"12:26.365 ","End":"12:30.039","Text":"minus 1/3 plus 1,"},{"Start":"12:30.039 ","End":"12:33.205","Text":"that makes that 2/3."},{"Start":"12:33.205 ","End":"12:39.865","Text":"This one is 1/3 minus 1 is minus 2/3,"},{"Start":"12:39.865 ","End":"12:45.310","Text":"but 2/3 minus minus 2/3 is 4/3."},{"Start":"12:45.310 ","End":"12:51.355","Text":"This integral comes out to be 4/3."},{"Start":"12:51.355 ","End":"12:56.320","Text":"Continuing, the 4/3 is a constant, comes out front."},{"Start":"12:56.320 ","End":"12:59.970","Text":"I have got altogether now, 4/3 times 5;"},{"Start":"12:59.970 ","End":"13:07.120","Text":"4/15 a cubed bc times the integral from"},{"Start":"13:07.120 ","End":"13:14.814","Text":"0-2 Pi of cosine squared Theta d Theta."},{"Start":"13:14.814 ","End":"13:18.040","Text":"I\u0027d like to do this integral a side also."},{"Start":"13:18.040 ","End":"13:19.945","Text":"I could use a table of integrals."},{"Start":"13:19.945 ","End":"13:25.195","Text":"Instead, I would prefer to use trigonometric identities,"},{"Start":"13:25.195 ","End":"13:30.265","Text":"and use the fact that the cosine squared"},{"Start":"13:30.265 ","End":"13:36.480","Text":"is 1/2 of 1 plus cosine 2 Theta."},{"Start":"13:36.480 ","End":"13:39.780","Text":"That\u0027s a trigonometrical identity element."},{"Start":"13:39.780 ","End":"13:43.390","Text":"I looked it up, it had Alpha there, but doesn\u0027t matter."},{"Start":"13:44.030 ","End":"13:46.920","Text":"That is the Theta."},{"Start":"13:46.920 ","End":"13:49.340","Text":"That\u0027s the indefinite integral."},{"Start":"13:49.340 ","End":"13:53.710","Text":"But we want it definite from 0-2 Pi."},{"Start":"13:53.710 ","End":"13:56.305","Text":"Let\u0027s see, what do we get?"},{"Start":"13:56.305 ","End":"14:07.400","Text":"This would be, the integral of 1/2 is just 1/2 Theta."},{"Start":"14:08.880 ","End":"14:11.380","Text":"I\u0027ll multiply by 1/2 in a moment."},{"Start":"14:11.380 ","End":"14:17.050","Text":"The integral of cosine 2 Theta is not quite sine 2 Theta."},{"Start":"14:17.050 ","End":"14:18.550","Text":"I have to divide by 2,"},{"Start":"14:18.550 ","End":"14:20.215","Text":"but I also have an 1/2 here,"},{"Start":"14:20.215 ","End":"14:23.860","Text":"so it\u0027s 1/4 sine of 2 Theta."},{"Start":"14:23.860 ","End":"14:26.995","Text":"If you\u0027re in doubt, just differentiate this,"},{"Start":"14:26.995 ","End":"14:32.330","Text":"you\u0027ll get 1/4 times 2 sine 2 Theta, which is 1/2."},{"Start":"14:32.430 ","End":"14:34.675","Text":"I meant cosine, anyway."},{"Start":"14:34.675 ","End":"14:36.910","Text":"This is what it is."},{"Start":"14:36.910 ","End":"14:43.825","Text":"I don\u0027t have to write the plus c because it\u0027s a definite integral from 0-2 Pi."},{"Start":"14:43.825 ","End":"14:54.535","Text":"Now, the sine of 0 is 0 and actually sine of 4 Pi is also 0,"},{"Start":"14:54.535 ","End":"14:56.740","Text":"because 4 Pi is 2 whole circles."},{"Start":"14:56.740 ","End":"14:59.215","Text":"I only get something from here."},{"Start":"14:59.215 ","End":"15:02.920","Text":"I get a 1/2 of 2 Pi minus 0."},{"Start":"15:02.920 ","End":"15:06.370","Text":"In short, it comes out to be just Pi."},{"Start":"15:06.370 ","End":"15:08.230","Text":"Comes out to be 1/2 of 2 Pi,"},{"Start":"15:08.230 ","End":"15:10.510","Text":"which is just Pi."},{"Start":"15:10.510 ","End":"15:13.705","Text":"Now that I\u0027ve got that this integral is Pi,"},{"Start":"15:13.705 ","End":"15:18.370","Text":"combining everything together, and so this equals,"},{"Start":"15:18.370 ","End":"15:20.740","Text":"actually I forgot to put the equal signs."},{"Start":"15:20.740 ","End":"15:23.245","Text":"Never mind, I will do it now."},{"Start":"15:23.245 ","End":"15:34.850","Text":"We get just 4 Pi over 15 a cubed bc."},{"Start":"15:34.950 ","End":"15:40.400","Text":"This is the answer. We are done."}],"ID":8749},{"Watched":false,"Name":"Exercise 4","Duration":"18m 20s","ChapterTopicVideoID":8575,"CourseChapterTopicPlaylistID":21194,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8575.jpeg","UploadDate":"2017-02-13T03:45:35.5270000","DurationForVideoObject":"PT18M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.825","Text":"In this exercise, we have to compute the volume of the region bounded by the surfaces,"},{"Start":"00:06.825 ","End":"00:08.805","Text":"the 6 of them."},{"Start":"00:08.805 ","End":"00:12.090","Text":"Let\u0027s give the region a name,"},{"Start":"00:12.090 ","End":"00:17.315","Text":"perhaps B for body, solid body,"},{"Start":"00:17.315 ","End":"00:25.860","Text":"and we have a general formula that the volume of a body B in"},{"Start":"00:25.860 ","End":"00:35.580","Text":"three-dimensions is just the triple integral over that body B of just 1 dv,"},{"Start":"00:35.580 ","End":"00:37.650","Text":"where dv would be like,"},{"Start":"00:37.650 ","End":"00:40.500","Text":"dx dy dz in some order."},{"Start":"00:40.500 ","End":"00:44.390","Text":"Now, in our particular case,"},{"Start":"00:44.390 ","End":"00:48.680","Text":"it\u0027s really not clear what the surfaces are."},{"Start":"00:48.680 ","End":"00:52.580","Text":"It\u0027s not a standard box or sphere or something."},{"Start":"00:52.580 ","End":"00:54.680","Text":"It\u0027s a total mess in fact."},{"Start":"00:54.680 ","End":"00:58.400","Text":"But if we look at it closely and you look at them in pairs,"},{"Start":"00:58.400 ","End":"01:00.680","Text":"you see that this pair is very similar."},{"Start":"01:00.680 ","End":"01:03.395","Text":"Y equals something z squared,"},{"Start":"01:03.395 ","End":"01:07.285","Text":"and then the other pair also have similarities."},{"Start":"01:07.285 ","End":"01:10.579","Text":"We\u0027re in the chapter on change of variable,"},{"Start":"01:10.579 ","End":"01:12.950","Text":"so that\u0027s what\u0027s called for."},{"Start":"01:12.950 ","End":"01:14.899","Text":"The first two equations,"},{"Start":"01:14.899 ","End":"01:16.055","Text":"if you think about it,"},{"Start":"01:16.055 ","End":"01:24.405","Text":"could be written as y over z squared equals 1 here,"},{"Start":"01:24.405 ","End":"01:31.300","Text":"and the other one would be y over z squared equals 4."},{"Start":"01:31.300 ","End":"01:36.005","Text":"We have the same y over z squared in both of these."},{"Start":"01:36.005 ","End":"01:39.815","Text":"Continuing with these two,"},{"Start":"01:39.815 ","End":"01:42.800","Text":"I\u0027d probably go for y minus 4x."},{"Start":"01:42.800 ","End":"01:44.825","Text":"But maybe you don\u0027t want a negative."},{"Start":"01:44.825 ","End":"01:47.240","Text":"Let\u0027s go with 4x minus y,"},{"Start":"01:47.240 ","End":"01:53.820","Text":"so this one I\u0027ll say is 4x minus y equals 0."},{"Start":"01:53.820 ","End":"01:57.900","Text":"Here, if I bring the y here and the 12 here,"},{"Start":"01:57.900 ","End":"02:02.655","Text":"I get 4x minus y equals 12,"},{"Start":"02:02.655 ","End":"02:04.170","Text":"and in both cases,"},{"Start":"02:04.170 ","End":"02:07.515","Text":"I have 4x minus y and 4x minus y."},{"Start":"02:07.515 ","End":"02:10.365","Text":"As for the last pair,"},{"Start":"02:10.365 ","End":"02:15.770","Text":"it seems obvious that what I should do is divide both sides by z,"},{"Start":"02:15.770 ","End":"02:18.275","Text":"get y over z equals 1,"},{"Start":"02:18.275 ","End":"02:22.520","Text":"and y over z equals 2."},{"Start":"02:22.520 ","End":"02:24.350","Text":"I have y over z,"},{"Start":"02:24.350 ","End":"02:26.135","Text":"and I have y over z."},{"Start":"02:26.135 ","End":"02:30.900","Text":"This suggests the change of variables that we need to perform."},{"Start":"02:30.900 ","End":"02:33.365","Text":"Let\u0027s make the substitution,"},{"Start":"02:33.365 ","End":"02:38.150","Text":"u will be y over z squared,"},{"Start":"02:38.150 ","End":"02:43.400","Text":"v equals 4x minus y,"},{"Start":"02:43.400 ","End":"02:49.240","Text":"and w equals y over z."},{"Start":"02:49.240 ","End":"02:57.335","Text":"Then this integral will become much simpler because we see that u goes from 1-4,"},{"Start":"02:57.335 ","End":"03:05.390","Text":"so we\u0027ll get the integral where u goes from 1-4."},{"Start":"03:05.390 ","End":"03:11.660","Text":"Then this, which is v goes from 0-12,"},{"Start":"03:11.660 ","End":"03:15.605","Text":"v goes from 0-12,"},{"Start":"03:15.605 ","End":"03:22.940","Text":"and w goes from 1-2."},{"Start":"03:22.940 ","End":"03:29.070","Text":"Then we\u0027ll have the 1,"},{"Start":"03:29.070 ","End":"03:33.890","Text":"and the thing is that when we replace dv,"},{"Start":"03:33.890 ","End":"03:37.565","Text":"which is dx dy dz in some order,"},{"Start":"03:37.565 ","End":"03:41.960","Text":"we have to add this extra thing called the Jacobian,"},{"Start":"03:41.960 ","End":"03:49.415","Text":"so it\u0027s the absolute value of the Jacobian, emphasize this J."},{"Start":"03:49.415 ","End":"03:53.360","Text":"Then we have du dv dw but it has to be in the right order."},{"Start":"03:53.360 ","End":"03:56.675","Text":"We need to close the w so it\u0027s dw,"},{"Start":"03:56.675 ","End":"04:02.165","Text":"and then dv, and then du."},{"Start":"04:02.165 ","End":"04:06.325","Text":"I will remind you what the Jacobian is."},{"Start":"04:06.325 ","End":"04:15.170","Text":"The Jacobian J is the determinant of a three-by-three matrix,"},{"Start":"04:15.170 ","End":"04:21.230","Text":"which is the partial derivatives of x with respect to u,"},{"Start":"04:21.230 ","End":"04:23.180","Text":"with respect to v,"},{"Start":"04:23.180 ","End":"04:24.950","Text":"x with respect to w,"},{"Start":"04:24.950 ","End":"04:26.030","Text":"and then so on,"},{"Start":"04:26.030 ","End":"04:28.385","Text":"y with respect to u,"},{"Start":"04:28.385 ","End":"04:35.585","Text":"v and w, and partial derivative of z with respect to u, v,"},{"Start":"04:35.585 ","End":"04:44.820","Text":"and w. The problem here is that we want x, y,"},{"Start":"04:44.820 ","End":"04:46.430","Text":"and z in terms of u, v,"},{"Start":"04:46.430 ","End":"04:49.085","Text":"and w. But what we have is u,"},{"Start":"04:49.085 ","End":"04:51.020","Text":"v, and w in terms of x, y,"},{"Start":"04:51.020 ","End":"04:54.080","Text":"and z. I need to solve this for x, y,"},{"Start":"04:54.080 ","End":"04:58.670","Text":"and z, like three equations in three unknowns."},{"Start":"04:58.670 ","End":"05:01.880","Text":"Now, looking at this,"},{"Start":"05:01.880 ","End":"05:05.090","Text":"I see that these two are very similar."},{"Start":"05:05.090 ","End":"05:08.150","Text":"Sorry, I meant yeah."},{"Start":"05:08.150 ","End":"05:15.020","Text":"I mean, u and w are very similar except that there\u0027s an extra z in the denominator here."},{"Start":"05:15.020 ","End":"05:21.815","Text":"So look, if I divide w divided by u,"},{"Start":"05:21.815 ","End":"05:32.590","Text":"I\u0027ll get that w over u is equal to y over z."},{"Start":"05:32.590 ","End":"05:35.335","Text":"Instead of dividing by u,"},{"Start":"05:35.335 ","End":"05:38.750","Text":"I can multiply by the inverse of u,"},{"Start":"05:38.750 ","End":"05:42.735","Text":"which is z squared over y."},{"Start":"05:42.735 ","End":"05:46.760","Text":"The y is cancel and z into z squared goes z times."},{"Start":"05:46.760 ","End":"05:50.630","Text":"This will give me z in terms of w and u."},{"Start":"05:50.630 ","End":"05:52.680","Text":"That\u0027s one of them."},{"Start":"05:53.170 ","End":"06:03.850","Text":"Next, I\u0027ll go for this equation which I can write as y equals w times z."},{"Start":"06:03.850 ","End":"06:07.330","Text":"But z we found is w over u."},{"Start":"06:07.330 ","End":"06:13.315","Text":"So this is w times w over u is w squared over u."},{"Start":"06:13.315 ","End":"06:15.925","Text":"Now we have z and we have y."},{"Start":"06:15.925 ","End":"06:21.310","Text":"What remains is to find x. I"},{"Start":"06:21.310 ","End":"06:27.460","Text":"can extract x from here by bringing the y over and dividing by 4,"},{"Start":"06:27.460 ","End":"06:33.875","Text":"so I\u0027ve got x equals dividing by 4 means a quarter of v plus y."},{"Start":"06:33.875 ","End":"06:36.675","Text":"But I have y from here,"},{"Start":"06:36.675 ","End":"06:44.525","Text":"so that\u0027s one quarter of v plus w squared over u."},{"Start":"06:44.525 ","End":"06:51.280","Text":"Let me just write them out again neatly that I have,"},{"Start":"06:51.280 ","End":"06:53.870","Text":"let me just write it as x."},{"Start":"06:53.870 ","End":"06:57.435","Text":"Sorry. Yeah, that\u0027s right."},{"Start":"06:57.435 ","End":"06:59.445","Text":"Yeah, x, y, and z is what I want,"},{"Start":"06:59.445 ","End":"07:02.545","Text":"x, y, and z."},{"Start":"07:02.545 ","End":"07:09.840","Text":"So x from here is one-quarter of v plus w squared over u,"},{"Start":"07:09.840 ","End":"07:14.865","Text":"y is w squared over u,"},{"Start":"07:14.865 ","End":"07:23.500","Text":"and z is w over u."},{"Start":"07:24.380 ","End":"07:34.440","Text":"Now I can get back to computing the Jacobian J equals,"},{"Start":"07:34.440 ","End":"07:37.625","Text":"let me give it a lot of space here."},{"Start":"07:37.625 ","End":"07:40.445","Text":"I\u0027m going to have 9 entries here."},{"Start":"07:40.445 ","End":"07:45.590","Text":"Let\u0027s start with x and take its derivatives with respect to u, v,"},{"Start":"07:45.590 ","End":"07:50.400","Text":"and w. With respect to u,"},{"Start":"07:50.950 ","End":"07:54.330","Text":"I\u0027ll get the quarter."},{"Start":"07:55.310 ","End":"07:57.900","Text":"This thing with respect to u,"},{"Start":"07:57.900 ","End":"08:01.485","Text":"the v disappears, w squared is a constant,"},{"Start":"08:01.485 ","End":"08:05.050","Text":"so it\u0027s going to be minus,"},{"Start":"08:05.300 ","End":"08:08.265","Text":"I\u0027ll put the minus in front,"},{"Start":"08:08.265 ","End":"08:11.980","Text":"w squared over u squared."},{"Start":"08:13.160 ","End":"08:16.890","Text":"Now with respect to v,"},{"Start":"08:16.890 ","End":"08:18.240","Text":"this is a constant,"},{"Start":"08:18.240 ","End":"08:24.510","Text":"so I just get a 1/4 and x with respect to w,"},{"Start":"08:24.510 ","End":"08:33.750","Text":"I\u0027ll get 1/4 of 2w of u."},{"Start":"08:33.750 ","End":"08:39.270","Text":"It\u0027ll be 1/2 of w over u,"},{"Start":"08:39.270 ","End":"08:43.590","Text":"because the 2 and the 4 will partially cancel."},{"Start":"08:43.590 ","End":"08:52.470","Text":"Now with y, y with respect to u it\u0027ll will be minus w squared over u squared,"},{"Start":"08:52.470 ","End":"08:59.460","Text":"with respect to v it would be 0 and with respect to w,"},{"Start":"08:59.460 ","End":"09:06.090","Text":"it\u0027ll just be 2w over u and then for"},{"Start":"09:06.090 ","End":"09:13.230","Text":"z with respect to u it will be minus w over u squared,"},{"Start":"09:13.230 ","End":"09:15.825","Text":"with respect to v, 0,"},{"Start":"09:15.825 ","End":"09:22.559","Text":"with respect to w that will be just 1 over u."},{"Start":"09:22.559 ","End":"09:31.500","Text":"Now what I\u0027ll do is expand using this column because it has a couple of 0s in it."},{"Start":"09:31.500 ","End":"09:34.679","Text":"If I expand using this column,"},{"Start":"09:34.679 ","End":"09:42.375","Text":"I just need to take care of this entry and multiply it by the cofactor,"},{"Start":"09:42.375 ","End":"09:46.305","Text":"which is like the minor with a plus or minus sign."},{"Start":"09:46.305 ","End":"09:54.030","Text":"The minor is the determinant of this 2 by 2 matrix."},{"Start":"09:54.030 ","End":"09:58.680","Text":"But because of its position,"},{"Start":"09:58.680 ","End":"10:00.840","Text":"remember it\u0027s like a plus, minus,"},{"Start":"10:00.840 ","End":"10:04.635","Text":"plus, minus, plus, minus checkerboard."},{"Start":"10:04.635 ","End":"10:07.380","Text":"Let me just get some more space here."},{"Start":"10:07.380 ","End":"10:12.705","Text":"What we get is that this equals the entry,"},{"Start":"10:12.705 ","End":"10:19.620","Text":"which is 1/4 and then times the minus 1 because of the plus, minus, plus,"},{"Start":"10:19.620 ","End":"10:25.500","Text":"minus checkerboard thing and then the determinant of the minor which"},{"Start":"10:25.500 ","End":"10:32.010","Text":"is the 2 by 2 which is w squared over u squared with a minus,"},{"Start":"10:32.010 ","End":"10:41.970","Text":"2w over u minus w over u squared and 1 over u,"},{"Start":"10:41.970 ","End":"10:44.475","Text":"and let\u0027s see what we get."},{"Start":"10:44.475 ","End":"10:48.045","Text":"We get minus 1/4."},{"Start":"10:48.045 ","End":"10:51.660","Text":"Now, for a 2 by 2 determinant it\u0027s"},{"Start":"10:51.660 ","End":"10:55.845","Text":"just this diagonal product minus this diagonal product."},{"Start":"10:55.845 ","End":"11:04.140","Text":"This times this is minus w squared over u cubed."},{"Start":"11:04.140 ","End":"11:08.100","Text":"The 2 denominators give me a u cubed minus,"},{"Start":"11:08.100 ","End":"11:10.155","Text":"but this is going to be a minus."},{"Start":"11:10.155 ","End":"11:12.660","Text":"It\u0027s plus, minus, minus."},{"Start":"11:12.660 ","End":"11:18.270","Text":"Let\u0027s see, 2w times w is 2w"},{"Start":"11:18.270 ","End":"11:25.870","Text":"squared and the denominator is u cubed."},{"Start":"11:26.540 ","End":"11:29.790","Text":"What we end up with,"},{"Start":"11:29.790 ","End":"11:33.105","Text":"this is the same kind as this,"},{"Start":"11:33.105 ","End":"11:36.570","Text":"2 minus 1 is just 1."},{"Start":"11:36.570 ","End":"11:45.720","Text":"What I get is w squared over u cubed,"},{"Start":"11:45.720 ","End":"11:50.505","Text":"but there\u0027s also a minus 1/4."},{"Start":"11:50.505 ","End":"12:00.285","Text":"It\u0027s minus w squared over 4u cubed."},{"Start":"12:00.285 ","End":"12:02.190","Text":"I\u0027ll just highlight this,"},{"Start":"12:02.190 ","End":"12:08.820","Text":"that\u0027s the Jacobian which we needed over here."},{"Start":"12:08.820 ","End":"12:13.155","Text":"Let\u0027s continue with this integral down here,"},{"Start":"12:13.155 ","End":"12:18.999","Text":"and this integral I forgot to say was the volume I should have said V equals,"},{"Start":"12:20.480 ","End":"12:26.355","Text":"so the integral u goes from 1-4,"},{"Start":"12:26.355 ","End":"12:32.170","Text":"and then v goes from"},{"Start":"12:32.390 ","End":"12:41.925","Text":"0-12 and then w goes from 1-2."},{"Start":"12:41.925 ","End":"12:44.940","Text":"We don\u0027t need the 1 don\u0027t do anything."},{"Start":"12:44.940 ","End":"12:48.180","Text":"Absolute value of J is the same as J"},{"Start":"12:48.180 ","End":"12:54.939","Text":"because all these variables in particular w and u are positive."},{"Start":"12:56.420 ","End":"12:59.940","Text":"Yeah, w and u they go,"},{"Start":"12:59.940 ","End":"13:03.435","Text":"1-2, 1-4 is positive I don\u0027t need the absolute value."},{"Start":"13:03.435 ","End":"13:10.050","Text":"I can just put w squared over 4u cubed without the minus."},{"Start":"13:10.050 ","End":"13:14.350","Text":"But I\u0027d like to take the constant in front."},{"Start":"13:14.720 ","End":"13:20.220","Text":"I\u0027ll take this w squared."},{"Start":"13:20.220 ","End":"13:21.570","Text":"I want to break it up a bit."},{"Start":"13:21.570 ","End":"13:23.940","Text":"Well, I tell you what."},{"Start":"13:23.940 ","End":"13:26.324","Text":"First of all I\u0027ll write w squared"},{"Start":"13:26.324 ","End":"13:34.680","Text":"over 4u cubed dw dv du."},{"Start":"13:34.680 ","End":"13:42.450","Text":"But I\u0027d like to put the 4 upfront here as 1\u00274 and in fact,"},{"Start":"13:42.450 ","End":"13:47.700","Text":"the first 2 integrals don\u0027t involve u. I can"},{"Start":"13:47.700 ","End":"13:54.255","Text":"actually take this u cubed also in front and put 1 over u cubed here."},{"Start":"13:54.255 ","End":"13:57.600","Text":"It\u0027ll just be neater not to have to deal with it until"},{"Start":"13:57.600 ","End":"14:02.970","Text":"the final stage and all I have here is the w squared."},{"Start":"14:02.970 ","End":"14:07.980","Text":"This is something I like to do and what everyone likes to do it is to take variables"},{"Start":"14:07.980 ","End":"14:12.990","Text":"that pull them up to the front as much as possible."},{"Start":"14:12.990 ","End":"14:15.150","Text":"Let\u0027s continue."},{"Start":"14:15.150 ","End":"14:18.300","Text":"We start as usual from the inside."},{"Start":"14:18.300 ","End":"14:23.050","Text":"That would be the w integral."},{"Start":"14:23.510 ","End":"14:27.330","Text":"Let me do this as a side computation."},{"Start":"14:27.330 ","End":"14:33.930","Text":"I have the integral from 1-2 of w squared dw,"},{"Start":"14:33.930 ","End":"14:41.595","Text":"and this is equal to 1/3w cubed."},{"Start":"14:41.595 ","End":"14:45.340","Text":"I have to take this from 1-2."},{"Start":"14:45.500 ","End":"14:47.610","Text":"What do I get?"},{"Start":"14:47.610 ","End":"14:48.870","Text":"If I put in 2,"},{"Start":"14:48.870 ","End":"14:57.060","Text":"it\u0027s 2 cubed is 8 over 3 minus 1 over 3,"},{"Start":"14:57.060 ","End":"14:59.745","Text":"so that 7 over 3."},{"Start":"14:59.745 ","End":"15:07.755","Text":"Altogether this is 7 over 3 and I can actually pull this to the front."},{"Start":"15:07.755 ","End":"15:15.405","Text":"The next line is 1/4 times 7 over 3 which is 7 over 12."},{"Start":"15:15.405 ","End":"15:20.969","Text":"The integral from 1-4 of u,"},{"Start":"15:20.969 ","End":"15:22.875","Text":"1 over u cubed,"},{"Start":"15:22.875 ","End":"15:27.870","Text":"and then integral from 0-12,"},{"Start":"15:27.870 ","End":"15:32.295","Text":"that\u0027s for v of just dv,"},{"Start":"15:32.295 ","End":"15:37.230","Text":"which I prefer to write it as 1dv and then du."},{"Start":"15:37.230 ","End":"15:42.660","Text":"Next in line from the inside is dv integral."},{"Start":"15:42.660 ","End":"15:48.915","Text":"But the integral of 1 is just v from 0-12 which is 12 minus 0."},{"Start":"15:48.915 ","End":"15:53.190","Text":"This just comes out to be 12 and the 12 if I pull it up in"},{"Start":"15:53.190 ","End":"15:57.390","Text":"front will cancel with this 12 and so we get,"},{"Start":"15:57.390 ","End":"15:59.745","Text":"since this 12 and this 12 cancel,"},{"Start":"15:59.745 ","End":"16:04.695","Text":"7 times the integral from"},{"Start":"16:04.695 ","End":"16:13.810","Text":"1-4 of 1 over u cubed du."},{"Start":"16:16.310 ","End":"16:22.380","Text":"This 1 I\u0027ll also just do it at the side here."},{"Start":"16:22.380 ","End":"16:29.545","Text":"I have the integral from 1- 4 of u^minus 3 du."},{"Start":"16:29.545 ","End":"16:32.810","Text":"Which means if I raise the power by 1,"},{"Start":"16:32.810 ","End":"16:39.829","Text":"it\u0027s u^minus 2 and then I have to divide by the minus"},{"Start":"16:39.829 ","End":"16:49.445","Text":"2 and take this from 1 - 4 and there\u0027s a trick I always use."},{"Start":"16:49.445 ","End":"16:51.980","Text":"I don\u0027t like the minuses in here,"},{"Start":"16:51.980 ","End":"16:58.760","Text":"so what I do is I get rid of the minus and I change the upper and lower limits."},{"Start":"16:58.760 ","End":"17:07.050","Text":"This is u to the minus 2 over 2."},{"Start":"17:07.050 ","End":"17:14.535","Text":"In fact, u to the minus 2 is 1 over u squared. Rewrite this."},{"Start":"17:14.535 ","End":"17:19.910","Text":"It\u0027s 1 over 2u squared and also the minus will"},{"Start":"17:19.910 ","End":"17:26.960","Text":"disappear because I\u0027m going to switch the 1 and the 4 and that gets rid of this minus."},{"Start":"17:26.960 ","End":"17:35.185","Text":"This gives me, if I put in 1 I get 1/2,"},{"Start":"17:35.185 ","End":"17:37.800","Text":"and if I put in 4,"},{"Start":"17:37.800 ","End":"17:40.970","Text":"4 squared is 16 times 2 is 32,"},{"Start":"17:40.970 ","End":"17:43.860","Text":"minus 1 over 32,"},{"Start":"17:44.320 ","End":"17:47.785","Text":"1/2 is 16 over 32."},{"Start":"17:47.785 ","End":"17:52.500","Text":"This comes out to be 15 over 32."},{"Start":"17:52.500 ","End":"17:55.384","Text":"Now if I go back here,"},{"Start":"17:55.384 ","End":"18:02.240","Text":"we get 7 times 15 over 32,"},{"Start":"18:02.240 ","End":"18:03.995","Text":"or in other words,"},{"Start":"18:03.995 ","End":"18:12.460","Text":"105 over 32 and"},{"Start":"18:12.460 ","End":"18:20.530","Text":"that is the final answer to the volume and we are done."}],"ID":8750}],"Thumbnail":null,"ID":21194}]

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