Double Integrals, Jacobian
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[{"Name":"Double Integrals, Jacobian","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Double Integrals, Change of Variables","Duration":"17m 18s","ChapterTopicVideoID":8451,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8451.jpeg","UploadDate":"2020-02-26T06:11:02.9370000","DurationForVideoObject":"PT17M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.680","Text":"In this clip, we\u0027re going to talk about change of variables in double integrals."},{"Start":"00:05.680 ","End":"00:09.045","Text":"Now we\u0027ve actually encountered such a thing."},{"Start":"00:09.045 ","End":"00:14.400","Text":"The change to polar coordinates is a specific example of change of"},{"Start":"00:14.400 ","End":"00:16.770","Text":"variables and I\u0027ll get back to"},{"Start":"00:16.770 ","End":"00:20.840","Text":"polar coordinates later because there\u0027s some debt I still owe you,"},{"Start":"00:20.840 ","End":"00:22.475","Text":"something called the Jacobian,"},{"Start":"00:22.475 ","End":"00:25.500","Text":"in fact, why don\u0027t I even write that word down."},{"Start":"00:25.610 ","End":"00:31.875","Text":"Jacobian. Let\u0027s start with an example."},{"Start":"00:31.875 ","End":"00:33.360","Text":"There\u0027s all kinds of scenarios."},{"Start":"00:33.360 ","End":"00:35.670","Text":"I\u0027m just going to take 1 simple example."},{"Start":"00:35.670 ","End":"00:40.550","Text":"There will be a lot of solved exercises where you learn most of it from."},{"Start":"00:40.550 ","End":"00:43.970","Text":"In this example, we\u0027re asked to evaluate the double"},{"Start":"00:43.970 ","End":"00:48.725","Text":"integral over the region R to be described in a moment,"},{"Start":"00:48.725 ","End":"00:52.885","Text":"of x plus y, dA."},{"Start":"00:52.885 ","End":"00:58.710","Text":"What we\u0027re told about R is that it\u0027s a region that\u0027s bounded by,"},{"Start":"00:58.710 ","End":"01:03.530","Text":"and I\u0027m going to give you some lines."},{"Start":"01:03.530 ","End":"01:07.480","Text":"The lines are y equals x,"},{"Start":"01:07.480 ","End":"01:13.110","Text":"y equals 3x and x plus y equals 4."},{"Start":"01:13.110 ","End":"01:14.955","Text":"These are 3 straight lines."},{"Start":"01:14.955 ","End":"01:18.790","Text":"It looks like ours is going to be a triangular region."},{"Start":"01:18.790 ","End":"01:25.210","Text":"I brought in a sketch of the region and the 3 lines."},{"Start":"01:25.210 ","End":"01:28.600","Text":"I guess I should\u0027ve written complete sentences,"},{"Start":"01:28.600 ","End":"01:30.655","Text":"I want to evaluate,"},{"Start":"01:30.655 ","End":"01:32.320","Text":"let me just write that."},{"Start":"01:32.320 ","End":"01:36.040","Text":"Evaluate this double integral."},{"Start":"01:36.040 ","End":"01:41.515","Text":"Then I say, where R is the region bounded by this,"},{"Start":"01:41.515 ","End":"01:45.610","Text":"and I\u0027m going to actually give you the substitution."},{"Start":"01:45.610 ","End":"01:49.029","Text":"It\u0027s not always given in the exercise,"},{"Start":"01:49.029 ","End":"01:50.425","Text":"sometimes we have to figure out,"},{"Start":"01:50.425 ","End":"01:56.300","Text":"but here we\u0027re going to give it using the substitution."},{"Start":"01:56.300 ","End":"02:00.870","Text":"The substitution is what x equals and y equals."},{"Start":"02:00.870 ","End":"02:04.590","Text":"Typically, we use the letters u and v. So x is going to be"},{"Start":"02:04.590 ","End":"02:12.090","Text":"u minus v. In this case y equals u plus v. This was given as to what to substitute."},{"Start":"02:12.130 ","End":"02:14.570","Text":"In the case of polar coordinates,"},{"Start":"02:14.570 ","End":"02:16.820","Text":"we don\u0027t use u and v, we use r and theta,"},{"Start":"02:16.820 ","End":"02:19.760","Text":"but these are typically the 2 letters we use,"},{"Start":"02:19.760 ","End":"02:22.620","Text":"just like x and y here."},{"Start":"02:23.720 ","End":"02:26.435","Text":"How do we go about this?"},{"Start":"02:26.435 ","End":"02:29.855","Text":"Now there are several steps to the solution and there is also a formula."},{"Start":"02:29.855 ","End":"02:31.370","Text":"I don\u0027t want to give it just yet."},{"Start":"02:31.370 ","End":"02:34.700","Text":"But one of the things we have to do is to figure out what"},{"Start":"02:34.700 ","End":"02:38.390","Text":"is this region that we know in terms of x and y,"},{"Start":"02:38.390 ","End":"02:41.120","Text":"what is it in terms of u and v?"},{"Start":"02:41.120 ","End":"02:44.750","Text":"I\u0027m going to look at these 3 lines and convert"},{"Start":"02:44.750 ","End":"02:49.000","Text":"them to u and v. Then we\u0027ll see where we are."},{"Start":"02:49.000 ","End":"02:50.900","Text":"I\u0027ll take them one at a time."},{"Start":"02:50.900 ","End":"02:53.195","Text":"If y equals x,"},{"Start":"02:53.195 ","End":"02:58.510","Text":"then because we know that y is u plus v and x is u minus v,"},{"Start":"02:58.510 ","End":"03:03.410","Text":"it gives us that u plus v equals"},{"Start":"03:03.410 ","End":"03:09.770","Text":"u minus v. If we bring everything to the left-hand side,"},{"Start":"03:09.770 ","End":"03:14.570","Text":"what we\u0027ll get is u minus u and v plus v. What we\u0027ll get is"},{"Start":"03:14.570 ","End":"03:21.525","Text":"2v equals 0 and hence v equals 0."},{"Start":"03:21.525 ","End":"03:23.970","Text":"That\u0027s one of the lines."},{"Start":"03:23.970 ","End":"03:28.380","Text":"Now the next one, y equals 3x."},{"Start":"03:28.380 ","End":"03:34.140","Text":"We get that u plus v equals 3 times"},{"Start":"03:34.140 ","End":"03:41.150","Text":"u minus v. v is on the left and u is on the right so I have here v,"},{"Start":"03:41.150 ","End":"03:42.530","Text":"and there\u0027s a minus 3v,"},{"Start":"03:42.530 ","End":"03:44.210","Text":"which is plus 3v."},{"Start":"03:44.210 ","End":"03:46.880","Text":"I get 4v on the left."},{"Start":"03:46.880 ","End":"03:47.895","Text":"Now u is on the right,"},{"Start":"03:47.895 ","End":"03:56.320","Text":"3u minus u is 2u and that gives us that v is equal to u over 2."},{"Start":"03:56.320 ","End":"04:00.315","Text":"Now the last one, x plus y equals 4."},{"Start":"04:00.315 ","End":"04:08.105","Text":"If I substitute, I get u minus v, which is x,"},{"Start":"04:08.105 ","End":"04:12.800","Text":"plus u plus v is y equals 4,"},{"Start":"04:12.800 ","End":"04:17.060","Text":"minus v and v cancel out,"},{"Start":"04:17.060 ","End":"04:20.750","Text":"and that gives us u equals 2."},{"Start":"04:20.750 ","End":"04:24.005","Text":"Now what we do next is draw"},{"Start":"04:24.005 ","End":"04:32.550","Text":"a new sketch for u and v. I didn\u0027t label the axis here."},{"Start":"04:32.550 ","End":"04:35.590","Text":"This is y, this is x. I want"},{"Start":"04:35.590 ","End":"04:40.720","Text":"a similar one where the vertical one is v and the horizontal one is u."},{"Start":"04:40.720 ","End":"04:46.890","Text":"Here\u0027s the sketch, v equals 0 is the u-axis."},{"Start":"04:46.890 ","End":"04:49.610","Text":"The horizontal one is u,"},{"Start":"04:49.610 ","End":"04:54.860","Text":"and the vertical one is v. This is v equals 0."},{"Start":"04:54.860 ","End":"04:59.330","Text":"The next one, v is u over 2 is a straight line through the origin and"},{"Start":"04:59.330 ","End":"05:04.070","Text":"u equals 2 is a vertical line through 2."},{"Start":"05:04.070 ","End":"05:05.480","Text":"We\u0027ve got 3 lines here."},{"Start":"05:05.480 ","End":"05:08.990","Text":"We also have a triangle and we have to give this a different name."},{"Start":"05:08.990 ","End":"05:10.670","Text":"Usually, this is R,"},{"Start":"05:10.670 ","End":"05:17.420","Text":"this is S. We\u0027ve got the step of converting the region from x,"},{"Start":"05:17.420 ","End":"05:22.520","Text":"y to u, v. Next I have to convert the function."},{"Start":"05:22.520 ","End":"05:27.025","Text":"But what we do is we just take a double integral."},{"Start":"05:27.025 ","End":"05:29.525","Text":"This time instead of R,"},{"Start":"05:29.525 ","End":"05:36.290","Text":"we take it over S. We substitute x and y for whatever they were."},{"Start":"05:36.290 ","End":"05:41.280","Text":"We\u0027ve got x is u minus v"},{"Start":"05:41.780 ","End":"05:49.705","Text":"plus y is u plus v. The a is a bit trickier."},{"Start":"05:49.705 ","End":"05:52.250","Text":"For one thing, we have to decide whether we\u0027re going to"},{"Start":"05:52.250 ","End":"05:54.755","Text":"do it in the type 1 or type 2 region,"},{"Start":"05:54.755 ","End":"05:56.750","Text":"vertical or horizontal slices."},{"Start":"05:56.750 ","End":"06:02.930","Text":"I suggest, let\u0027s do it with vertical slices as it seems the most natural type 1 region."},{"Start":"06:02.930 ","End":"06:07.455","Text":"That\u0027s going to be dvdu."},{"Start":"06:07.455 ","End":"06:11.860","Text":"We put here dvdu."},{"Start":"06:11.860 ","End":"06:14.420","Text":"Now I\u0027ve deliberately left a space here."},{"Start":"06:14.420 ","End":"06:16.820","Text":"The missing ingredient here,"},{"Start":"06:16.820 ","End":"06:19.130","Text":"if you think back to the polar coordinates,"},{"Start":"06:19.130 ","End":"06:23.540","Text":"we had a mysterious r in the rdrd theta."},{"Start":"06:23.540 ","End":"06:27.185","Text":"This mysterious thing I told you was called a Jacobian."},{"Start":"06:27.185 ","End":"06:32.795","Text":"So our missing thing here is the absolute value of the Jacobian."},{"Start":"06:32.795 ","End":"06:36.845","Text":"I\u0027m going to explain what this thing is."},{"Start":"06:36.845 ","End":"06:39.230","Text":"Before I explain this Jacobian,"},{"Start":"06:39.230 ","End":"06:42.420","Text":"perhaps I\u0027ll give you the general formula."},{"Start":"06:43.090 ","End":"06:49.880","Text":"We start off with a double integral over a region R in x,"},{"Start":"06:49.880 ","End":"06:57.270","Text":"y of some function of x and y and dA,"},{"Start":"06:57.270 ","End":"07:00.000","Text":"which could be dxdy or dydx."},{"Start":"07:00.000 ","End":"07:02.960","Text":"What we do is,"},{"Start":"07:02.960 ","End":"07:07.580","Text":"once we have the substitution for x and y in terms of u and v,"},{"Start":"07:07.580 ","End":"07:12.170","Text":"what we will say is that this is equal to the double integral over"},{"Start":"07:12.170 ","End":"07:18.365","Text":"S and S is the region described in terms of u and v,"},{"Start":"07:18.365 ","End":"07:20.000","Text":"just like here, x and y,"},{"Start":"07:20.000 ","End":"07:27.320","Text":"u and v. Then what we do is to replace x in terms of u and"},{"Start":"07:27.320 ","End":"07:34.925","Text":"v and y in terms of u and v. Then here the Jacobian,"},{"Start":"07:34.925 ","End":"07:39.080","Text":"which is actually a function of x and y in absolute value."},{"Start":"07:39.080 ","End":"07:41.615","Text":"We still haven\u0027t explained what it is yet."},{"Start":"07:41.615 ","End":"07:50.630","Text":"Then dvdu, though it could also be dudv depending on type 1 or type 2 region."},{"Start":"07:50.630 ","End":"07:55.140","Text":"But I\u0027m not going to be using this formula much more,"},{"Start":"07:55.140 ","End":"07:58.280","Text":"we\u0027ll learn how to do it the recipe or cookbook style."},{"Start":"07:58.280 ","End":"08:00.770","Text":"We did one of the main steps,"},{"Start":"08:00.770 ","End":"08:05.540","Text":"was to convert the region from R to S. The next step will be to do"},{"Start":"08:05.540 ","End":"08:10.670","Text":"the substitution in the function and the next step will be to compute the Jacobian."},{"Start":"08:10.670 ","End":"08:13.505","Text":"Then we\u0027ll have to actually evaluate the integral."},{"Start":"08:13.505 ","End":"08:16.960","Text":"Let\u0027s go and do the substitution."},{"Start":"08:16.960 ","End":"08:20.705","Text":"In our case, f of x, y was just x plus y."},{"Start":"08:20.705 ","End":"08:23.390","Text":"So x of u and v is this,"},{"Start":"08:23.390 ","End":"08:25.640","Text":"this was what we substituted for x,"},{"Start":"08:25.640 ","End":"08:29.070","Text":"and this is what we substituted for y."},{"Start":"08:29.330 ","End":"08:34.710","Text":"We get the double integral just simplifying S, let\u0027s see,"},{"Start":"08:34.710 ","End":"08:44.005","Text":"this plus this is just 2u and then absolute value of J, dvdu."},{"Start":"08:44.005 ","End":"08:46.320","Text":"Now I\u0027m going to tell you what J is."},{"Start":"08:46.320 ","End":"08:55.890","Text":"J is equal to a 2 by 2 determinant."},{"Start":"08:55.890 ","End":"09:05.415","Text":"Here I have the partial derivative dx by du, dx,"},{"Start":"09:05.415 ","End":"09:09.420","Text":"by dv, dy by"},{"Start":"09:09.420 ","End":"09:16.740","Text":"du and dy by dv."},{"Start":"09:16.740 ","End":"09:19.290","Text":"That\u0027s our J Jacobian,"},{"Start":"09:19.290 ","End":"09:21.510","Text":"and sometimes it\u0027s written,"},{"Start":"09:21.510 ","End":"09:23.880","Text":"you\u0027ll see it in books as,"},{"Start":"09:23.880 ","End":"09:28.005","Text":"here I put x and y with a partial derivative sign,"},{"Start":"09:28.005 ","End":"09:34.575","Text":"and here u and v with one of these d\u0027s."},{"Start":"09:34.575 ","End":"09:39.330","Text":"The next thing we\u0027re going to do is to compute the Jacobian."},{"Start":"09:39.330 ","End":"09:48.780","Text":"Now just to remind you that the substitution was x equals u minus v,"},{"Start":"09:48.780 ","End":"09:51.750","Text":"y equals u plus v."},{"Start":"09:51.750 ","End":"09:56.085","Text":"Sometimes this is called a transformation and not the substitution anyway."},{"Start":"09:56.085 ","End":"10:03.615","Text":"Now let\u0027s see J is equal to dx by du,"},{"Start":"10:03.615 ","End":"10:07.005","Text":"this is going to come out easy because they\u0027re all going to be constants,"},{"Start":"10:07.005 ","End":"10:08.565","Text":"so I chose the example."},{"Start":"10:08.565 ","End":"10:11.385","Text":"Derivative of x with respect to u is just 1,"},{"Start":"10:11.385 ","End":"10:14.730","Text":"derivative of x with respect to v is minus 1,"},{"Start":"10:14.730 ","End":"10:17.310","Text":"y with respect to u is 1,"},{"Start":"10:17.310 ","End":"10:20.595","Text":"y with respect to v is also 1."},{"Start":"10:20.595 ","End":"10:23.670","Text":"I hope you remember your determinants, but if not,"},{"Start":"10:23.670 ","End":"10:28.560","Text":"we take the products of this diagonal and subtract the products of this diagonal."},{"Start":"10:28.560 ","End":"10:31.365","Text":"This is equal to, 1 times 1 is 1,"},{"Start":"10:31.365 ","End":"10:36.525","Text":"minus minus 1, so this comes out to be 2."},{"Start":"10:36.525 ","End":"10:44.100","Text":"If this thing here is 2, where are we?"},{"Start":"10:44.100 ","End":"10:48.555","Text":"Here we are, we just computed to be 2,"},{"Start":"10:48.555 ","End":"10:52.650","Text":"and the absolute value of 2 is also 2."},{"Start":"10:52.650 ","End":"11:00.405","Text":"What we get is the double integral over s"},{"Start":"11:00.405 ","End":"11:11.190","Text":"of 2u times 2dv du."},{"Start":"11:11.190 ","End":"11:17.715","Text":"We said that we\u0027re going to take it as a type 1 region."},{"Start":"11:17.715 ","End":"11:22.590","Text":"Perhaps I\u0027ll just bring the picture in again."},{"Start":"11:22.590 ","End":"11:25.530","Text":"Yeah, I just copied it again."},{"Start":"11:25.530 ","End":"11:27.690","Text":"What we get, well,"},{"Start":"11:27.690 ","End":"11:31.810","Text":"this thing is going to be simplified to just 4u."},{"Start":"11:32.570 ","End":"11:36.180","Text":"Look u goes from 0-2."},{"Start":"11:36.180 ","End":"11:38.230","Text":"That\u0027s the outer loop."},{"Start":"11:42.730 ","End":"11:46.550","Text":"What confused me is v is not on the u axis."},{"Start":"11:46.550 ","End":"11:51.210","Text":"In fact, I can even write it that,"},{"Start":"11:51.210 ","End":"11:54.600","Text":"it\u0027s u that goes from 0-2,"},{"Start":"11:54.600 ","End":"11:56.775","Text":"that\u0027s this bit here,"},{"Start":"11:56.775 ","End":"12:00.340","Text":"and for each particular u,"},{"Start":"12:01.280 ","End":"12:04.335","Text":"if I have a particular u here,"},{"Start":"12:04.335 ","End":"12:07.410","Text":"I\u0027ll take this vertical slice or arrow,"},{"Start":"12:07.410 ","End":"12:10.840","Text":"and it enters here and leaves here."},{"Start":"12:11.390 ","End":"12:21.135","Text":"This v goes from 0 up to u over 2."},{"Start":"12:21.135 ","End":"12:25.980","Text":"The outer loop is du,"},{"Start":"12:25.980 ","End":"12:31.695","Text":"that\u0027s for this, an inside v goes from 0 to u over 2dv."},{"Start":"12:31.695 ","End":"12:36.550","Text":"Now I have this bit which we said was 4u,"},{"Start":"12:36.620 ","End":"12:41.205","Text":"and so we\u0027ve got it written as an iterated integral."},{"Start":"12:41.205 ","End":"12:43.785","Text":"Now we just have to compute it,"},{"Start":"12:43.785 ","End":"12:46.170","Text":"which is the least important part,"},{"Start":"12:46.170 ","End":"12:49.510","Text":"but I might as well go on to the end."},{"Start":"12:51.260 ","End":"12:58.245","Text":"We get the integral from u equals naught to 2 du."},{"Start":"12:58.245 ","End":"13:00.000","Text":"Now remember we work from inside out,"},{"Start":"13:00.000 ","End":"13:03.000","Text":"I\u0027m going to do this bit first."},{"Start":"13:03.000 ","End":"13:06.480","Text":"4u is a constant as far as v is concerned,"},{"Start":"13:06.480 ","End":"13:16.515","Text":"so the integral is 4u times v. But we have to evaluate this between 0 and u over 2."},{"Start":"13:16.515 ","End":"13:20.235","Text":"If I substitute that v equals 0,"},{"Start":"13:20.235 ","End":"13:23.550","Text":"emphasize, this is the limit for v. V is 0,"},{"Start":"13:23.550 ","End":"13:27.870","Text":"I get 0 if I let v equals u over 2,"},{"Start":"13:27.870 ","End":"13:34.755","Text":"so 4u times u over 2 will equal 2u squared."},{"Start":"13:34.755 ","End":"13:37.620","Text":"Here I have the integral from"},{"Start":"13:37.620 ","End":"13:45.390","Text":"0-2 of 2u squared du."},{"Start":"13:45.390 ","End":"13:52.050","Text":"Now this is equal to 2u squared polynomial raise the power by 1, 3,"},{"Start":"13:52.050 ","End":"14:00.660","Text":"divide by it, so I\u0027ve got 2/3 u cubed from 0-2."},{"Start":"14:00.660 ","End":"14:03.840","Text":"When I put in u equals 0, it\u0027s nothing,"},{"Start":"14:03.840 ","End":"14:10.080","Text":"if put in u equals 2 it\u0027s 2/3 times 2 cubed."},{"Start":"14:10.080 ","End":"14:12.990","Text":"2 cubed is 88 times 2 is 16,"},{"Start":"14:12.990 ","End":"14:16.705","Text":"so the answer is 16 over 3,"},{"Start":"14:16.705 ","End":"14:20.224","Text":"and we are done with the example,"},{"Start":"14:20.224 ","End":"14:24.130","Text":"but don\u0027t go, I just wanted to say something about the polar coordinates."},{"Start":"14:24.130 ","End":"14:26.670","Text":"When we did polar coordinates,"},{"Start":"14:26.670 ","End":"14:28.320","Text":"we changed from x,"},{"Start":"14:28.320 ","End":"14:30.075","Text":"y to r Theta."},{"Start":"14:30.075 ","End":"14:32.070","Text":"We didn\u0027t use uv,"},{"Start":"14:32.070 ","End":"14:34.560","Text":"we used r and Theta, but the same idea."},{"Start":"14:34.560 ","End":"14:38.340","Text":"We had this mysterious r here,"},{"Start":"14:38.340 ","End":"14:42.630","Text":"which I\u0027m now going to show you how it ties in with the Jacobian."},{"Start":"14:42.630 ","End":"14:46.020","Text":"The substitution was this,"},{"Start":"14:46.020 ","End":"14:48.795","Text":"this was the standard substitution for polar,"},{"Start":"14:48.795 ","End":"14:52.830","Text":"and just to be more compatible, the x,"},{"Start":"14:52.830 ","End":"14:58.380","Text":"y should be r, this should be s. R is the region described in terms of x,"},{"Start":"14:58.380 ","End":"15:02.070","Text":"y, and s the region described in terms of u and v,"},{"Start":"15:02.070 ","End":"15:04.660","Text":"or in our case r and Theta."},{"Start":"15:05.840 ","End":"15:14.219","Text":"Let\u0027s see what is the Jacobian of u,"},{"Start":"15:14.219 ","End":"15:20.880","Text":"v. Well, it\u0027s not u,"},{"Start":"15:20.880 ","End":"15:23.670","Text":"v, in our case,"},{"Start":"15:23.670 ","End":"15:27.435","Text":"it\u0027s r, Theta, so we just modify the rule slightly."},{"Start":"15:27.435 ","End":"15:30.015","Text":"We have dx by,"},{"Start":"15:30.015 ","End":"15:37.455","Text":"instead of du we put dr and then dx by d Theta,"},{"Start":"15:37.455 ","End":"15:41.190","Text":"then dy by dr,"},{"Start":"15:41.190 ","End":"15:44.730","Text":"dy by d Theta."},{"Start":"15:44.730 ","End":"15:52.710","Text":"Let\u0027s see what this comes out to be dx by dr. Theta\u0027s a constant and so is cosine Theta,"},{"Start":"15:52.710 ","End":"15:57.075","Text":"so the derivative of x with respect to r is just cosine Theta."},{"Start":"15:57.075 ","End":"16:02.775","Text":"Dy by dr is just going to be sine Theta."},{"Start":"16:02.775 ","End":"16:05.505","Text":"Now with respect to Theta,"},{"Start":"16:05.505 ","End":"16:10.590","Text":"the derivative of cosine Theta is minus sine Theta and r is a constant,"},{"Start":"16:10.590 ","End":"16:14.025","Text":"so here we have minus r sine Theta,"},{"Start":"16:14.025 ","End":"16:18.060","Text":"and here sine Theta gives us cosine Theta and r is a constant,"},{"Start":"16:18.060 ","End":"16:21.240","Text":"so it just stays cosine Theta."},{"Start":"16:21.240 ","End":"16:27.164","Text":"The determinant means this diagonal product minus the product of this diagonal,"},{"Start":"16:27.164 ","End":"16:31.720","Text":"so this diagonal gives me r cosine squared Theta,"},{"Start":"16:32.510 ","End":"16:38.805","Text":"and this diagonal gives me r sine squared Theta,"},{"Start":"16:38.805 ","End":"16:40.860","Text":"but it\u0027s minus, so minus,"},{"Start":"16:40.860 ","End":"16:45.975","Text":"minus is plus, so it\u0027s r sine squared Theta."},{"Start":"16:45.975 ","End":"16:52.395","Text":"This is just equal to r cosine squared Theta plus sine squared Theta,"},{"Start":"16:52.395 ","End":"16:59.220","Text":"which is just equal to r. Since r is never negative in"},{"Start":"16:59.220 ","End":"17:06.270","Text":"polar so the absolute value of J is the absolute value of r,"},{"Start":"17:06.270 ","End":"17:09.750","Text":"which is just r, and so it really ties"},{"Start":"17:09.750 ","End":"17:14.040","Text":"in that this is the absolute value of the Jacobian,"},{"Start":"17:14.040 ","End":"17:19.030","Text":"and that explains that mystery. Now we\u0027re done."}],"ID":8667},{"Watched":false,"Name":"Exercise 1","Duration":"11m 52s","ChapterTopicVideoID":8452,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8452.jpeg","UploadDate":"2017-02-13T01:39:15.1100000","DurationForVideoObject":"PT11M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.775","Text":"In this exercise, we have to compute this double integral over the region R,"},{"Start":"00:05.775 ","End":"00:08.490","Text":"where R is bounded by 1,"},{"Start":"00:08.490 ","End":"00:10.260","Text":"2, 3, 4 lines."},{"Start":"00:10.260 ","End":"00:17.595","Text":"I\u0027d like to start with sketching the region R. I\u0027ve already brought a coordinate axis in."},{"Start":"00:17.595 ","End":"00:23.955","Text":"y equals x is familiar to 45 Degree line through the origin."},{"Start":"00:23.955 ","End":"00:26.880","Text":"y equals x minus 1,"},{"Start":"00:26.880 ","End":"00:30.270","Text":"you can just make some substitutions, for example,"},{"Start":"00:30.270 ","End":"00:34.965","Text":"when x is 0,"},{"Start":"00:34.965 ","End":"00:36.840","Text":"y is minus 1,"},{"Start":"00:36.840 ","End":"00:39.335","Text":"and when x is 1, y is 0."},{"Start":"00:39.335 ","End":"00:45.405","Text":"It\u0027s a parallel line to the above line, something like this."},{"Start":"00:45.405 ","End":"00:48.420","Text":"This is 0, this is 1."},{"Start":"00:48.420 ","End":"00:50.265","Text":"Now let\u0027s take the next one."},{"Start":"00:50.265 ","End":"00:53.370","Text":"y equals 1 minus x."},{"Start":"00:53.370 ","End":"00:56.880","Text":"When x is 0, y is 1,"},{"Start":"00:56.880 ","End":"00:59.935","Text":"when x is 1, y is 0."},{"Start":"00:59.935 ","End":"01:01.565","Text":"It goes through."},{"Start":"01:01.565 ","End":"01:03.980","Text":"If I mark the point 1 here,"},{"Start":"01:03.980 ","End":"01:09.270","Text":"it\u0027ll go through like this."},{"Start":"01:09.620 ","End":"01:13.005","Text":"y equals 3 minus x,"},{"Start":"01:13.005 ","End":"01:17.540","Text":"intersection with the axis shows us that it\u0027s a 3 in both cases,"},{"Start":"01:17.540 ","End":"01:20.590","Text":"1, 2, say this is 3."},{"Start":"01:20.590 ","End":"01:23.415","Text":"Let\u0027s say this is 3,"},{"Start":"01:23.415 ","End":"01:26.670","Text":"and the line through here."},{"Start":"01:26.670 ","End":"01:28.755","Text":"This really doesn\u0027t have to be precise."},{"Start":"01:28.755 ","End":"01:34.835","Text":"Our region R, and I\u0027ve shaded our region."},{"Start":"01:34.835 ","End":"01:39.410","Text":"Now, it would be quite difficult to do the double integral over the region as is,"},{"Start":"01:39.410 ","End":"01:41.900","Text":"we\u0027d have to separate it into 3 parts."},{"Start":"01:41.900 ","End":"01:46.500","Text":"We\u0027d have to go here, and here,"},{"Start":"01:46.500 ","End":"01:50.085","Text":"and here, and here,"},{"Start":"01:50.085 ","End":"01:52.400","Text":"and we\u0027d have to take 3 separate integrals."},{"Start":"01:52.400 ","End":"01:56.240","Text":"We\u0027d have to go like this part A, this part B,"},{"Start":"01:56.240 ","End":"02:00.200","Text":"this part C. Compute the intersection points."},{"Start":"02:00.200 ","End":"02:03.050","Text":"Well, some of them we have, some of them we don\u0027t."},{"Start":"02:03.050 ","End":"02:08.645","Text":"Messy. The idea is to use a change of variables."},{"Start":"02:08.645 ","End":"02:11.030","Text":"If you look at these equations a while,"},{"Start":"02:11.030 ","End":"02:12.520","Text":"you\u0027ll see the first 2 are similar,"},{"Start":"02:12.520 ","End":"02:14.255","Text":"y equals x something,"},{"Start":"02:14.255 ","End":"02:15.890","Text":"and the second 2 are similar,"},{"Start":"02:15.890 ","End":"02:17.800","Text":"y equals something minus x."},{"Start":"02:17.800 ","End":"02:20.600","Text":"If we just mess with them a bit,"},{"Start":"02:20.600 ","End":"02:27.585","Text":"we can get the first one in the form y minus x equals 0,"},{"Start":"02:27.585 ","End":"02:30.255","Text":"the second one would be"},{"Start":"02:30.255 ","End":"02:40.035","Text":"y minus x equals minus 1."},{"Start":"02:40.035 ","End":"02:41.280","Text":"You know what? I\u0027ve changed my mind."},{"Start":"02:41.280 ","End":"02:42.360","Text":"I don\u0027t want negatives,"},{"Start":"02:42.360 ","End":"02:44.820","Text":"let\u0027s do x minus y instead."},{"Start":"02:44.820 ","End":"02:48.155","Text":"I just changed the order and I\u0027ve got a plus 1 here."},{"Start":"02:48.155 ","End":"02:49.580","Text":"Now the next equation,"},{"Start":"02:49.580 ","End":"02:53.670","Text":"I can write as x plus y equals 1,"},{"Start":"02:54.400 ","End":"02:59.520","Text":"and this one as x plus y equals 3."},{"Start":"02:59.980 ","End":"03:04.700","Text":"Notice that here and here I have x minus y,"},{"Start":"03:04.700 ","End":"03:07.130","Text":"and here and here I have x plus y."},{"Start":"03:07.130 ","End":"03:09.970","Text":"This naturally leads to the substitution."},{"Start":"03:09.970 ","End":"03:14.880","Text":"We let u equals x minus y,"},{"Start":"03:14.880 ","End":"03:19.950","Text":"and we let v equals x plus y,"},{"Start":"03:19.950 ","End":"03:23.935","Text":"and then these equations will become,"},{"Start":"03:23.935 ","End":"03:27.755","Text":"this one will be u equals 0,"},{"Start":"03:27.755 ","End":"03:30.260","Text":"and then u equals 1,"},{"Start":"03:30.260 ","End":"03:33.380","Text":"and then v equals 1,"},{"Start":"03:33.380 ","End":"03:36.505","Text":"and here v equals 3."},{"Start":"03:36.505 ","End":"03:38.520","Text":"That\u0027s a lot easier to deal with."},{"Start":"03:38.520 ","End":"03:41.815","Text":"If I now draw a second graph."},{"Start":"03:41.815 ","End":"03:44.705","Text":"Here I have some axes for u and v,"},{"Start":"03:44.705 ","End":"03:47.885","Text":"so now these 4 are just all parallel to the axis."},{"Start":"03:47.885 ","End":"03:53.895","Text":"u equals 0 is the v axis,"},{"Start":"03:53.895 ","End":"03:59.130","Text":"u equals 1 goes through 1."},{"Start":"03:59.130 ","End":"04:03.180","Text":"This is 1, this is 0. v equals 1,"},{"Start":"04:03.180 ","End":"04:07.320","Text":"this horizontal line here through 1,"},{"Start":"04:07.320 ","End":"04:13.120","Text":"and v equals 3 might be somewhere up here,"},{"Start":"04:13.460 ","End":"04:22.970","Text":"and so we end up getting a rectangle that\u0027s parallel to the axis."},{"Start":"04:22.970 ","End":"04:24.725","Text":"Let me give it a name,"},{"Start":"04:24.725 ","End":"04:31.455","Text":"after R comes S. I\u0027ve shaded it, highlighted."},{"Start":"04:31.455 ","End":"04:36.560","Text":"Now we\u0027d make a change of variables and substitute in the integral."},{"Start":"04:36.560 ","End":"04:38.420","Text":"Here\u0027s how it goes."},{"Start":"04:38.420 ","End":"04:41.950","Text":"I take the double integral,"},{"Start":"04:41.950 ","End":"04:46.245","Text":"instead of the old region I write the new region S."},{"Start":"04:46.245 ","End":"04:50.450","Text":"I substitute x minus y and x plus y here."},{"Start":"04:50.450 ","End":"04:54.020","Text":"Let\u0027s see, x minus y is u,"},{"Start":"04:54.020 ","End":"05:02.525","Text":"and x plus y is v. Here\u0027s the thing."},{"Start":"05:02.525 ","End":"05:07.220","Text":"We have to multiply by the absolute value of something called the Jacobian."},{"Start":"05:07.220 ","End":"05:10.400","Text":"I\u0027ll remind you in a moment what that is."},{"Start":"05:10.400 ","End":"05:14.015","Text":"Then instead of set dA, we put,"},{"Start":"05:14.015 ","End":"05:15.980","Text":"well, I don\u0027t want to use the same letter,"},{"Start":"05:15.980 ","End":"05:17.870","Text":"either du,dv or dv,du."},{"Start":"05:17.870 ","End":"05:20.395","Text":"I\u0027ll write it at the moment as du,dv,"},{"Start":"05:20.395 ","End":"05:23.810","Text":"but we still have to decide which one to do first."},{"Start":"05:23.810 ","End":"05:26.210","Text":"They could be the other way around."},{"Start":"05:26.210 ","End":"05:29.135","Text":"That\u0027s how it works."},{"Start":"05:29.135 ","End":"05:36.425","Text":"Now this Jacobian is the determinant of a 2 by 2 matrix,"},{"Start":"05:36.425 ","End":"05:40.605","Text":"which is u with respect to x."},{"Start":"05:40.605 ","End":"05:43.910","Text":"Sorry, x with respect to u,"},{"Start":"05:43.910 ","End":"05:45.680","Text":"x with respect to v,"},{"Start":"05:45.680 ","End":"05:49.415","Text":"partial derivatives, and y with respect to u,"},{"Start":"05:49.415 ","End":"05:56.760","Text":"y with respect to v. If you don\u0027t know what the determinant is,"},{"Start":"05:56.760 ","End":"06:01.910","Text":"it\u0027s just this times this minus this times this."},{"Start":"06:01.910 ","End":"06:03.830","Text":"I\u0027ll write it up here. In general,"},{"Start":"06:03.830 ","End":"06:06.170","Text":"the determinant of a, b, c,"},{"Start":"06:06.170 ","End":"06:13.465","Text":"d is ad, it\u0027s one diagonal minus the other diagonal bc."},{"Start":"06:13.465 ","End":"06:16.265","Text":"Let\u0027s see what we have in our case."},{"Start":"06:16.265 ","End":"06:20.750","Text":"Our problem is that we have from these 2 equations,"},{"Start":"06:20.750 ","End":"06:24.170","Text":"u and v in terms of x and y, but for this,"},{"Start":"06:24.170 ","End":"06:28.310","Text":"we\u0027d like to have x and y in terms of u and v. I just want to take"},{"Start":"06:28.310 ","End":"06:33.530","Text":"this pair of equations and solve it for x and y to get the inverse equations."},{"Start":"06:33.530 ","End":"06:36.500","Text":"Let me continue over here."},{"Start":"06:36.500 ","End":"06:44.960","Text":"What I can do is add these 2 equations and get that u plus v. If I add the 2 of them,"},{"Start":"06:44.960 ","End":"06:48.120","Text":"then I\u0027ll just get 2x,"},{"Start":"06:48.120 ","End":"06:52.815","Text":"because the y will cancel with the minus y, so that\u0027s 2x."},{"Start":"06:52.815 ","End":"06:58.110","Text":"If I subtract, I\u0027d like a plus,"},{"Start":"06:58.110 ","End":"07:03.070","Text":"so I\u0027ll do this one minus this one,"},{"Start":"07:03.070 ","End":"07:04.735","Text":"the bottom minus the top,"},{"Start":"07:04.735 ","End":"07:10.205","Text":"and I\u0027ll get that v minus u equals,"},{"Start":"07:10.205 ","End":"07:14.920","Text":"x minus x cancels and y minus minus y is 2y."},{"Start":"07:14.920 ","End":"07:21.205","Text":"Altogether, what I get from these is the following."},{"Start":"07:21.205 ","End":"07:28.875","Text":"I get that x equals 1/2 of u plus v,"},{"Start":"07:28.875 ","End":"07:37.605","Text":"and that y equals 1/2 of v minus u,"},{"Start":"07:37.605 ","End":"07:43.620","Text":"or minus u plus v. Now I can continue over here."},{"Start":"07:43.620 ","End":"07:46.200","Text":"This equals, let\u0027s see,"},{"Start":"07:46.200 ","End":"07:47.700","Text":"I need 4 things."},{"Start":"07:47.700 ","End":"07:52.275","Text":"x with respect to u is 1/2, the constant."},{"Start":"07:52.275 ","End":"07:56.145","Text":"x with respect to v, also 1/2."},{"Start":"07:56.145 ","End":"07:59.065","Text":"y with respect to u,"},{"Start":"07:59.065 ","End":"08:02.665","Text":"that will come out to be minus 1/2,"},{"Start":"08:02.665 ","End":"08:06.760","Text":"and y with respect to v is 1/2."},{"Start":"08:06.760 ","End":"08:11.755","Text":"Then using this formula for the determinant,"},{"Start":"08:11.755 ","End":"08:16.210","Text":"it\u0027s this diagonal minus this diagonal, 1/4 minus,"},{"Start":"08:16.210 ","End":"08:19.690","Text":"minus 1/4, which is 1/4 plus 1/4,"},{"Start":"08:19.690 ","End":"08:26.620","Text":"this is equal to just 1/2, so that\u0027s our J."},{"Start":"08:26.620 ","End":"08:30.070","Text":"The next step is to decide whether we want to slice"},{"Start":"08:30.070 ","End":"08:34.075","Text":"this region up vertically or horizontally."},{"Start":"08:34.075 ","End":"08:39.055","Text":"Type 1 or type 2, du,dv or dv,du."},{"Start":"08:39.055 ","End":"08:41.095","Text":"Let\u0027s leave it as a du,dv,"},{"Start":"08:41.095 ","End":"08:44.115","Text":"which means that the outward loop is on v,"},{"Start":"08:44.115 ","End":"08:50.130","Text":"so we\u0027ll will take on the outside dv ."},{"Start":"08:50.130 ","End":"08:58.440","Text":"We\u0027ll take v from 1-3, and that\u0027s dv."},{"Start":"08:58.440 ","End":"09:05.280","Text":"Then inside that, for each v,"},{"Start":"09:05.280 ","End":"09:07.275","Text":"let\u0027s say this is the typical v,"},{"Start":"09:07.275 ","End":"09:15.165","Text":"then it cuts at u equals 0 and u equals 1."},{"Start":"09:15.165 ","End":"09:20.630","Text":"We got u equals 0 to 1du."},{"Start":"09:20.630 ","End":"09:29.850","Text":"Then we have u over v. The absolute value of J,"},{"Start":"09:29.850 ","End":"09:38.280","Text":"well, it\u0027s positive, so it\u0027s just J is 1/2du,dv."},{"Start":"09:38.630 ","End":"09:42.100","Text":"From this point on, it\u0027s just purely computational."},{"Start":"09:42.100 ","End":"09:46.330","Text":"Let me get a bit more space. There we go."},{"Start":"09:46.330 ","End":"09:50.350","Text":"The 1/2 I\u0027ll take completely out front."},{"Start":"09:50.350 ","End":"09:55.880","Text":"Then I\u0027ve got the integral v from 1-3."},{"Start":"09:55.880 ","End":"09:58.340","Text":"Now this inner integral is with respect to u,"},{"Start":"09:58.340 ","End":"10:00.020","Text":"so the v is a constant."},{"Start":"10:00.020 ","End":"10:05.155","Text":"I can actually pull the 1 over v in front of the integral sign,"},{"Start":"10:05.155 ","End":"10:10.030","Text":"and just get the integral from 0-1 of"},{"Start":"10:10.030 ","End":"10:16.775","Text":"just u,du and then dv."},{"Start":"10:16.775 ","End":"10:20.645","Text":"What I want to do first is the inner integral."},{"Start":"10:20.645 ","End":"10:22.430","Text":"I mean this one,"},{"Start":"10:22.430 ","End":"10:24.730","Text":"the one that\u0027s du."},{"Start":"10:24.730 ","End":"10:29.090","Text":"I think it\u0027s simple enough to do mentally,"},{"Start":"10:29.090 ","End":"10:32.015","Text":"or maybe I\u0027ll just jot something down at the side."},{"Start":"10:32.015 ","End":"10:36.455","Text":"This integral of u is just 1/2u squared,"},{"Start":"10:36.455 ","End":"10:39.590","Text":"and I\u0027m taking it between 0 and 1."},{"Start":"10:39.590 ","End":"10:40.790","Text":"At 0 I get nothing,"},{"Start":"10:40.790 ","End":"10:42.560","Text":"at 1 I get 1/2."},{"Start":"10:42.560 ","End":"10:49.800","Text":"This whole thing that I\u0027ve shaded just comes out to be 1/2."},{"Start":"10:50.210 ","End":"10:54.180","Text":"I can pull that also out in front,"},{"Start":"10:54.180 ","End":"11:02.150","Text":"and then what we get is 1/4 from this half,"},{"Start":"11:02.150 ","End":"11:03.590","Text":"which is the answer to this."},{"Start":"11:03.590 ","End":"11:12.060","Text":"Then we\u0027ve got the integral from 1-3 of 1 over v,dv."},{"Start":"11:13.420 ","End":"11:20.480","Text":"Now we know the integral of 1 over v is the natural log of v,"},{"Start":"11:20.480 ","End":"11:25.700","Text":"I\u0027ll continue over here because I like to keep the picture, is 1/4."},{"Start":"11:25.700 ","End":"11:32.170","Text":"Then we have natural log of v from 1-3."},{"Start":"11:32.170 ","End":"11:35.910","Text":"Remember, natural log of 1 is 0,"},{"Start":"11:35.910 ","End":"11:40.680","Text":"so we just get natural log of 3 times the 1/4."},{"Start":"11:40.680 ","End":"11:47.285","Text":"We get 1/4 natural log of 3,"},{"Start":"11:47.285 ","End":"11:50.640","Text":"and that is our answer."}],"ID":8668},{"Watched":false,"Name":"Exercise 2","Duration":"24m 5s","ChapterTopicVideoID":8453,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8453.jpeg","UploadDate":"2017-02-13T01:46:00.5530000","DurationForVideoObject":"PT24M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this exercise, we have to compute the double integral of this"},{"Start":"00:04.200 ","End":"00:10.080","Text":"and R is the region bounded by 4 functions as given here."},{"Start":"00:10.080 ","End":"00:12.390","Text":"I want to focus on the region first,"},{"Start":"00:12.390 ","End":"00:15.750","Text":"and I want to sketch just briefly,"},{"Start":"00:15.750 ","End":"00:19.455","Text":"each of these 4 curves or lines."},{"Start":"00:19.455 ","End":"00:22.730","Text":"y equals x we\u0027ve seen before,"},{"Start":"00:22.730 ","End":"00:24.830","Text":"45 degree line through the origin."},{"Start":"00:24.830 ","End":"00:28.205","Text":"Maybe I\u0027ll mark a few points on and it goes through 0, 0."},{"Start":"00:28.205 ","End":"00:32.160","Text":"When x is 1, y is 1, say 1,"},{"Start":"00:32.160 ","End":"00:35.175","Text":"1 and when x is 2,"},{"Start":"00:35.175 ","End":"00:39.195","Text":"y is 2 say here."},{"Start":"00:39.195 ","End":"00:44.515","Text":"Not to scale and put a line through these."},{"Start":"00:44.515 ","End":"00:48.750","Text":"y equals 1/2 x is going to have a smaller slope,"},{"Start":"00:48.750 ","End":"00:50.625","Text":"slope of 1/2, for example,"},{"Start":"00:50.625 ","End":"00:52.305","Text":"when x is 2,"},{"Start":"00:52.305 ","End":"00:54.360","Text":"y will be 1."},{"Start":"00:54.360 ","End":"00:58.035","Text":"When x is 1, y will be 1/2 and so on"},{"Start":"00:58.035 ","End":"01:02.225","Text":"so that I will get another line also through the origin,"},{"Start":"01:02.225 ","End":"01:06.015","Text":"this time, through this point here."},{"Start":"01:06.015 ","End":"01:08.305","Text":"That\u0027s the second curve."},{"Start":"01:08.305 ","End":"01:11.255","Text":"Maybe I\u0027ll label them y equals x,"},{"Start":"01:11.255 ","End":"01:14.790","Text":"y equals 1/2 x."},{"Start":"01:14.790 ","End":"01:17.285","Text":"Let\u0027s see, y equals 1/x."},{"Start":"01:17.285 ","End":"01:22.695","Text":"When x is 1, y is 1 so it will also go through this point."},{"Start":"01:22.695 ","End":"01:26.400","Text":"When x is 2,"},{"Start":"01:26.400 ","End":"01:28.245","Text":"y will be 1/2."},{"Start":"01:28.245 ","End":"01:36.180","Text":"When x is 1/2, y will be 2 so something through here like this."},{"Start":"01:36.180 ","End":"01:42.720","Text":"We know the hyperbola y equals 1/x and 2/x, well,"},{"Start":"01:42.720 ","End":"01:48.480","Text":"when x is 1, y will be 2 so maybe here,"},{"Start":"01:48.480 ","End":"01:52.690","Text":"when x is 2, y will be 1."},{"Start":"01:55.520 ","End":"02:04.260","Text":"It\u0027ll go through something like this and I didn\u0027t label them."},{"Start":"02:04.260 ","End":"02:13.590","Text":"This 1 is where y equals 1/x and this 1 is where y equals 2/x."},{"Start":"02:13.700 ","End":"02:18.130","Text":"The region R is what\u0027s bounded by them."},{"Start":"02:18.130 ","End":"02:22.430","Text":"I\u0027ve shaded it. This is an awful region to do an integral over."},{"Start":"02:22.430 ","End":"02:26.540","Text":"Never mind what we\u0027re integrating because you have to break it up into pieces."},{"Start":"02:26.540 ","End":"02:30.185","Text":"It\u0027s not quite clear if this point and this point are above each other."},{"Start":"02:30.185 ","End":"02:33.920","Text":"Actually, I computed it and they are but it doesn\u0027t look like it so in principle,"},{"Start":"02:33.920 ","End":"02:38.450","Text":"you might have to divide into several pieces."},{"Start":"02:38.450 ","End":"02:41.315","Text":"You might get region A, region B,"},{"Start":"02:41.315 ","End":"02:48.870","Text":"region C. It\u0027s a mess so we need a change of variables."},{"Start":"02:48.870 ","End":"02:51.730","Text":"If we rewrite these equations a bit, look,"},{"Start":"02:51.730 ","End":"02:53.050","Text":"the first 2 are quite similar,"},{"Start":"02:53.050 ","End":"02:55.255","Text":"y equals something times x."},{"Start":"02:55.255 ","End":"03:00.790","Text":"The first 2 equations could be written as y over x equals something."},{"Start":"03:00.790 ","End":"03:03.370","Text":"The first 1 is y over x equals 1."},{"Start":"03:03.370 ","End":"03:09.045","Text":"The second 1, y over x equals 0.5, I prefer 1/2."},{"Start":"03:09.045 ","End":"03:13.635","Text":"The third 1, y equals something over x so we multiply by x,"},{"Start":"03:13.635 ","End":"03:15.150","Text":"this 1 becomes x,"},{"Start":"03:15.150 ","End":"03:24.465","Text":"y equals 1 and this 1 becomes x, y equals 2."},{"Start":"03:24.465 ","End":"03:27.210","Text":"In both these, we have y/x,"},{"Start":"03:27.210 ","End":"03:28.410","Text":"in both these we have x,"},{"Start":"03:28.410 ","End":"03:31.965","Text":"y so it seems natural to make a substitution."},{"Start":"03:31.965 ","End":"03:37.995","Text":"We\u0027ll let u equals y/x and we let v"},{"Start":"03:37.995 ","End":"03:45.740","Text":"equals x times y and then we can rewrite these equations as,"},{"Start":"03:45.740 ","End":"03:48.500","Text":"here we have u equals 1,"},{"Start":"03:48.500 ","End":"03:51.455","Text":"here u equals 1/2,"},{"Start":"03:51.455 ","End":"03:54.140","Text":"here v equals 1,"},{"Start":"03:54.140 ","End":"03:57.415","Text":"and here v equals 2."},{"Start":"03:57.415 ","End":"04:01.170","Text":"If I now sketch these lines,"},{"Start":"04:01.170 ","End":"04:06.060","Text":"let\u0027s call this u axis, v axis."},{"Start":"04:06.060 ","End":"04:10.080","Text":"Then this 4 will form a simple rectangle."},{"Start":"04:10.080 ","End":"04:11.430","Text":"Look, u equals 1,"},{"Start":"04:11.430 ","End":"04:15.930","Text":"let\u0027s say this is 1, that\u0027s a vertical line through 1."},{"Start":"04:15.930 ","End":"04:22.560","Text":"u equals 1/2 so we have a vertical line through 1/2."},{"Start":"04:22.630 ","End":"04:30.410","Text":"Here, we have v equals 1 so that\u0027s horizontal line through"},{"Start":"04:30.410 ","End":"04:39.240","Text":"1 and v equals 2 horizontal line through 2."},{"Start":"04:39.240 ","End":"04:43.120","Text":"Just extend these a bit, sorry."},{"Start":"04:43.850 ","End":"04:47.540","Text":"I\u0027ll give a name to this rectangle,"},{"Start":"04:47.540 ","End":"04:51.120","Text":"S, and I\u0027ll shade it and now,"},{"Start":"04:51.120 ","End":"04:54.410","Text":"we can rewrite this double integral using"},{"Start":"04:54.410 ","End":"04:59.990","Text":"the transformation formula for change of variables as the double"},{"Start":"04:59.990 ","End":"05:09.720","Text":"integral over region S. Then we have each of the the power of x,"},{"Start":"05:09.720 ","End":"05:18.665","Text":"y is just v. Well, I won\u0027t call it DA."},{"Start":"05:18.665 ","End":"05:20.000","Text":"It\u0027ll be either du,"},{"Start":"05:20.000 ","End":"05:22.535","Text":"dv or dv, du."},{"Start":"05:22.535 ","End":"05:23.960","Text":"Well, actually when we write this,"},{"Start":"05:23.960 ","End":"05:26.285","Text":"we\u0027re not committing to 1 way or the other,"},{"Start":"05:26.285 ","End":"05:30.720","Text":"we\u0027ll decide whether we want type 1 or type 2 region."},{"Start":"05:32.930 ","End":"05:38.735","Text":"I almost forgot, the formula calls for the absolute value of J,"},{"Start":"05:38.735 ","End":"05:43.645","Text":"where J is the Jacobian and I\u0027ll remind you"},{"Start":"05:43.645 ","End":"05:51.035","Text":"that J is equal to the determinant of x with respect to u,"},{"Start":"05:51.035 ","End":"05:53.820","Text":"x with respect to v,"},{"Start":"05:53.840 ","End":"05:57.705","Text":"then y with respect to u,"},{"Start":"05:57.705 ","End":"06:00.360","Text":"y with respect to v,"},{"Start":"06:00.360 ","End":"06:04.610","Text":"and I\u0027ll even remind you what a determinant is."},{"Start":"06:04.610 ","End":"06:08.600","Text":"The determinant in general of 4 numbers, A, B, C,"},{"Start":"06:08.600 ","End":"06:14.350","Text":"D is just AD minus BC."},{"Start":"06:14.350 ","End":"06:21.600","Text":"Okay. We want to compute the Jacobian but we don\u0027t have x and y in terms of u and v,"},{"Start":"06:21.600 ","End":"06:22.640","Text":"we have the opposite."},{"Start":"06:22.640 ","End":"06:28.290","Text":"We have u and v in terms of x and y. I\u0027d like to look at these 2,"},{"Start":"06:28.290 ","End":"06:30.890","Text":"as 2 equations in 2 unknowns x and"},{"Start":"06:30.890 ","End":"06:35.630","Text":"y and I want from these to extract what x equals and y equals."},{"Start":"06:35.630 ","End":"06:37.670","Text":"What I propose is the following."},{"Start":"06:37.670 ","End":"06:40.039","Text":"I have y here and here in the numerator,"},{"Start":"06:40.039 ","End":"06:42.080","Text":"if I divide 1 equation by the other,"},{"Start":"06:42.080 ","End":"06:47.985","Text":"I\u0027ll get rid of y so if I take u over v,"},{"Start":"06:47.985 ","End":"06:53.630","Text":"this will equal y/x divided by x,"},{"Start":"06:53.630 ","End":"06:59.185","Text":"y and so we get 1 over x squared."},{"Start":"06:59.185 ","End":"07:01.640","Text":"If I divided the other way,"},{"Start":"07:01.640 ","End":"07:02.960","Text":"well, it doesn\u0027t matter."},{"Start":"07:02.960 ","End":"07:04.580","Text":"I can invert it, VO,"},{"Start":"07:04.580 ","End":"07:05.720","Text":"VU is x squared,"},{"Start":"07:05.720 ","End":"07:06.800","Text":"I like that better."},{"Start":"07:06.800 ","End":"07:15.605","Text":"Now, I can see that x is equal to the square root of v/u."},{"Start":"07:15.605 ","End":"07:17.180","Text":"This gave me this,"},{"Start":"07:17.180 ","End":"07:18.815","Text":"which gave me this."},{"Start":"07:18.815 ","End":"07:22.450","Text":"Now, how do we find y?"},{"Start":"07:22.450 ","End":"07:24.765","Text":"Well, I want to get rid of x."},{"Start":"07:24.765 ","End":"07:26.630","Text":"Now look, here\u0027s x in the denominator,"},{"Start":"07:26.630 ","End":"07:30.050","Text":"here in the numerator then let\u0027s try multiplying these 2 this time so I"},{"Start":"07:30.050 ","End":"07:34.550","Text":"get u times v is y/x times XY."},{"Start":"07:34.550 ","End":"07:39.870","Text":"This just gives me y squared and this gives me that y"},{"Start":"07:39.870 ","End":"07:45.780","Text":"is equal to the square root of u times v. Here,"},{"Start":"07:45.780 ","End":"07:50.905","Text":"I now have x and y in terms of v and u."},{"Start":"07:50.905 ","End":"07:54.035","Text":"Next we want to compute the Jacobian,"},{"Start":"07:54.035 ","End":"07:59.060","Text":"which means that we have to figure out all 4 partial derivatives."},{"Start":"07:59.060 ","End":"08:01.190","Text":"We need x with respect to u and v,"},{"Start":"08:01.190 ","End":"08:05.700","Text":"and y with respect to u and v. Let\u0027s compute the 4 of them."},{"Start":"08:05.990 ","End":"08:12.735","Text":"Now, I\u0027m going to need x with respect to u and then the other 3."},{"Start":"08:12.735 ","End":"08:15.540","Text":"I noticed that I have a lot of square roots."},{"Start":"08:15.540 ","End":"08:16.620","Text":"Let me, at the side,"},{"Start":"08:16.620 ","End":"08:22.775","Text":"show you a formula just a reminder that if I have the square root of something,"},{"Start":"08:22.775 ","End":"08:26.450","Text":"I\u0027ll call it box and I take the derivative,"},{"Start":"08:26.450 ","End":"08:28.970","Text":"the derivative of square root is 1"},{"Start":"08:28.970 ","End":"08:34.010","Text":"over twice the square root but if that something is a function of x,"},{"Start":"08:34.010 ","End":"08:37.910","Text":"then we need to take the inner derivative as well so it\u0027s box prime."},{"Start":"08:37.910 ","End":"08:42.140","Text":"I\u0027m going to use this 4 times actually in each of the derivatives."},{"Start":"08:42.140 ","End":"08:45.775","Text":"Let\u0027s start with x by u."},{"Start":"08:45.775 ","End":"08:48.620","Text":"We start off with something"},{"Start":"08:48.620 ","End":"08:56.910","Text":"over twice the square root of v/u and then the inner derivative."},{"Start":"08:56.910 ","End":"09:00.220","Text":"Now, the inner derivative of v/u,"},{"Start":"09:00.220 ","End":"09:08.640","Text":"we\u0027re differentiating with respect to u so v is a constant and the derivative of v/u is"},{"Start":"09:08.640 ","End":"09:12.840","Text":"just going to be minus v/u squared because"},{"Start":"09:12.840 ","End":"09:18.100","Text":"the derivative of 1/u is minus 1/u squared and the v just stays."},{"Start":"09:18.930 ","End":"09:27.820","Text":"I want x_v, and this will equal, again,"},{"Start":"09:27.820 ","End":"09:34.600","Text":"we\u0027re going to have on the denominator the square root of v over u, but this time,"},{"Start":"09:34.600 ","End":"09:37.660","Text":"the inner derivative is with respect to v,"},{"Start":"09:37.660 ","End":"09:39.385","Text":"and now u is the constant,"},{"Start":"09:39.385 ","End":"09:42.530","Text":"so it\u0027s just 1 over u."},{"Start":"09:42.530 ","End":"09:44.950","Text":"Then the other 2, let\u0027s see."},{"Start":"09:44.950 ","End":"09:48.400","Text":"y with respect to u, well,"},{"Start":"09:48.400 ","End":"09:53.530","Text":"here we\u0027re going to have in the denominator twice the square root of uv."},{"Start":"09:53.530 ","End":"09:56.440","Text":"I\u0027m already preparing myself for the next one."},{"Start":"09:56.440 ","End":"10:00.265","Text":"It\u0027s also going to be twice the square root of uv."},{"Start":"10:00.265 ","End":"10:03.835","Text":"Here I need the inner derivative with respect to u,"},{"Start":"10:03.835 ","End":"10:10.070","Text":"and that\u0027s v, and here the inner derivative with respect to v, which is u."},{"Start":"10:10.530 ","End":"10:14.860","Text":"We\u0027re about ready to compute that Jacobian now."},{"Start":"10:14.860 ","End":"10:18.850","Text":"The definition of a determinant has going off the board so I\u0027ll just remind"},{"Start":"10:18.850 ","End":"10:22.750","Text":"you that the determinant is the product of these 2 minus the product of these 2."},{"Start":"10:22.750 ","End":"10:30.730","Text":"In other words, x_uy_v minus the other diagonal, x_vy_u."},{"Start":"10:31.770 ","End":"10:37.195","Text":"In our case, that\u0027s in general,"},{"Start":"10:37.195 ","End":"10:40.135","Text":"here we\u0027ll get that J is equal to,"},{"Start":"10:40.135 ","End":"10:42.670","Text":"now x with respect to u,"},{"Start":"10:42.670 ","End":"10:45.550","Text":"I\u0027d like to simplify this a bit."},{"Start":"10:45.550 ","End":"10:47.710","Text":"I like to work with the fractions."},{"Start":"10:47.710 ","End":"10:52.675","Text":"I don\u0027t like to have fractions in the numerator and in the denominator."},{"Start":"10:52.675 ","End":"10:58.075","Text":"The denominator of the numerator goes downstairs and this goes upstairs."},{"Start":"10:58.075 ","End":"10:59.290","Text":"Basically, what I\u0027m saying is,"},{"Start":"10:59.290 ","End":"11:04.315","Text":"here we have on the numerator minus v,"},{"Start":"11:04.315 ","End":"11:13.340","Text":"what was, and also the square root of u from the denominator can go up to the numerator."},{"Start":"11:13.590 ","End":"11:16.780","Text":"The u squared here goes downstairs."},{"Start":"11:16.780 ","End":"11:18.850","Text":"We still have a 2, and that\u0027s a u squared,"},{"Start":"11:18.850 ","End":"11:21.445","Text":"and the square root of v stays."},{"Start":"11:21.445 ","End":"11:23.830","Text":"All this is x_u."},{"Start":"11:23.830 ","End":"11:27.790","Text":"Now, y_v, that\u0027s this one."},{"Start":"11:27.790 ","End":"11:29.170","Text":"I don\u0027t have a problem with that."},{"Start":"11:29.170 ","End":"11:31.930","Text":"There\u0027s no fractions within fractions."},{"Start":"11:31.930 ","End":"11:36.490","Text":"This is u over twice square root of"},{"Start":"11:36.490 ","End":"11:42.985","Text":"uv minus x with respect to v. That\u0027s this."},{"Start":"11:42.985 ","End":"11:46.015","Text":"I want to do some messing around here."},{"Start":"11:46.015 ","End":"11:51.850","Text":"The u here will go into the denominator,"},{"Start":"11:51.850 ","End":"11:54.860","Text":"so we\u0027ll get 2u,"},{"Start":"11:55.650 ","End":"11:59.665","Text":"then here we\u0027ll have the square root of v,"},{"Start":"11:59.665 ","End":"12:02.815","Text":"but the square root of u here goes up to the numerator,"},{"Start":"12:02.815 ","End":"12:06.775","Text":"square root of u, times,"},{"Start":"12:06.775 ","End":"12:10.540","Text":"then what do I need still?"},{"Start":"12:10.540 ","End":"12:14.800","Text":"y with respect to u."},{"Start":"12:14.800 ","End":"12:17.170","Text":"That\u0027s this one. That\u0027s no problem."},{"Start":"12:17.170 ","End":"12:25.405","Text":"That is, v over 2 root uv."},{"Start":"12:25.405 ","End":"12:27.400","Text":"Things are getting a bit cramped."},{"Start":"12:27.400 ","End":"12:29.470","Text":"I want to make some more space."},{"Start":"12:29.470 ","End":"12:32.575","Text":"Better and let\u0027s scroll down a bit."},{"Start":"12:32.575 ","End":"12:38.785","Text":"Let\u0027s continue with this computation of J. Let\u0027s see."},{"Start":"12:38.785 ","End":"12:46.060","Text":"I want to collect together all the powers of u and all the powers of v. Let\u0027s see."},{"Start":"12:46.060 ","End":"12:47.620","Text":"What do we have here?"},{"Start":"12:47.620 ","End":"12:50.755","Text":"Let\u0027s just collect from here all the u\u0027s,"},{"Start":"12:50.755 ","End":"12:52.570","Text":"or maybe I\u0027ll write it."},{"Start":"12:52.570 ","End":"12:54.730","Text":"On the denominator, well,"},{"Start":"12:54.730 ","End":"13:02.035","Text":"I have 1/4 or even minus 1/4. Now, let\u0027s see."},{"Start":"13:02.035 ","End":"13:05.200","Text":"That takes care of the minus and the 2 and the 2."},{"Start":"13:05.200 ","End":"13:07.585","Text":"Now, what do I have as far as u goes?"},{"Start":"13:07.585 ","End":"13:11.755","Text":"On the numerator, I have u^1 and 1/2,"},{"Start":"13:11.755 ","End":"13:19.900","Text":"and on denominator I have u^2 and 1/2 because I have 2 from here and 1/2 from here."},{"Start":"13:19.900 ","End":"13:26.920","Text":"In the numerator, I have 1 and 1/2,"},{"Start":"13:26.920 ","End":"13:31.450","Text":"in the denominator, I have u^2 and 1/2."},{"Start":"13:31.450 ","End":"13:37.165","Text":"Let\u0027s see what we have for v. In the numerator, I have v,"},{"Start":"13:37.165 ","End":"13:42.205","Text":"in the denominator, I have root v. Again,"},{"Start":"13:42.205 ","End":"13:45.460","Text":"root v is v^1/2 plus 1/2."},{"Start":"13:45.460 ","End":"13:50.950","Text":"That\u0027s just v. In the next one,"},{"Start":"13:50.950 ","End":"13:53.150","Text":"I also have minus1/4."},{"Start":"13:53.610 ","End":"13:57.895","Text":"I\u0027ve got the minus and I\u0027ve got the 2 and the 2. Now let\u0027s see."},{"Start":"13:57.895 ","End":"14:07.015","Text":"On the numerator, I have u^1/2 times v,"},{"Start":"14:07.015 ","End":"14:09.040","Text":"and on the denominator,"},{"Start":"14:09.040 ","End":"14:16.120","Text":"I have u^1 and 1/2."},{"Start":"14:16.120 ","End":"14:24.010","Text":"Again, root v times root v just gives me v. I see that the v cancels everywhere."},{"Start":"14:24.010 ","End":"14:26.440","Text":"We\u0027re going to be left with just u."},{"Start":"14:26.440 ","End":"14:28.150","Text":"Let\u0027s see how many u we have."},{"Start":"14:28.150 ","End":"14:29.410","Text":"Let me do side computation."},{"Start":"14:29.410 ","End":"14:31.990","Text":"When we divide, we subtract the exponents,"},{"Start":"14:31.990 ","End":"14:35.440","Text":"we get 1 and 1/2 minus 2 and 1/2,"},{"Start":"14:35.440 ","End":"14:38.365","Text":"and that is minus 1."},{"Start":"14:38.365 ","End":"14:41.590","Text":"We\u0027re just left with 1 over u."},{"Start":"14:41.590 ","End":"14:46.735","Text":"The first term is minus 1 over 4u."},{"Start":"14:46.735 ","End":"14:49.780","Text":"Let\u0027s see what the second term is."},{"Start":"14:49.780 ","End":"14:54.880","Text":"The power of u is going to be 1/2 minus 1 and 1/2."},{"Start":"14:54.880 ","End":"14:59.560","Text":"It\u0027s also minus 1.That means it\u0027s u^minus 1,"},{"Start":"14:59.560 ","End":"15:01.030","Text":"which is 1 over u."},{"Start":"15:01.030 ","End":"15:04.180","Text":"Again, we have minus 1 over 4u."},{"Start":"15:04.180 ","End":"15:08.905","Text":"Together, we have minus 1/4 minus 1/4."},{"Start":"15:08.905 ","End":"15:11.155","Text":"This becomes minus 1/2."},{"Start":"15:11.155 ","End":"15:14.035","Text":"It\u0027s minus 1 over 2u,"},{"Start":"15:14.035 ","End":"15:16.240","Text":"and that is the Jacobian."},{"Start":"15:16.240 ","End":"15:19.075","Text":"I\u0027ll just circle it. That\u0027s important."},{"Start":"15:19.075 ","End":"15:21.490","Text":"Now I want to go back here,"},{"Start":"15:21.490 ","End":"15:26.155","Text":"but we take the absolute value of the Jacobian."},{"Start":"15:26.155 ","End":"15:34.585","Text":"This absolute value of J from here is just 1 over to 2u. We\u0027re getting there."},{"Start":"15:34.585 ","End":"15:35.920","Text":"The next thing we have to do,"},{"Start":"15:35.920 ","End":"15:38.410","Text":"and I\u0027m just returning to the picture for the moment,"},{"Start":"15:38.410 ","End":"15:44.890","Text":"is to decide whether we want this to be a type 1 or type 2 region."},{"Start":"15:44.890 ","End":"15:47.080","Text":"It doesn\u0027t really matter. They\u0027ll both work whether you want to"},{"Start":"15:47.080 ","End":"15:49.675","Text":"slice it horizontally or vertically,"},{"Start":"15:49.675 ","End":"15:52.915","Text":"whether you want dudv or dvdu."},{"Start":"15:52.915 ","End":"15:56.350","Text":"They\u0027re both about equally easy or difficult."},{"Start":"15:56.350 ","End":"16:00.850","Text":"I say, let\u0027s take vertical slices and go, first of all,"},{"Start":"16:00.850 ","End":"16:05.304","Text":"with u from minus 1/2-1 on the outside,"},{"Start":"16:05.304 ","End":"16:08.095","Text":"and for each such u,"},{"Start":"16:08.095 ","End":"16:11.215","Text":"v will go from 1-2."},{"Start":"16:11.215 ","End":"16:15.910","Text":"Actually, this will be a dvdu integral."},{"Start":"16:15.910 ","End":"16:18.415","Text":"I\u0027m going to continue over here."},{"Start":"16:18.415 ","End":"16:22.435","Text":"What we have now is the integral."},{"Start":"16:22.435 ","End":"16:27.505","Text":"More space. That\u0027s better. See the picture."},{"Start":"16:27.505 ","End":"16:29.275","Text":"I need the integral."},{"Start":"16:29.275 ","End":"16:35.350","Text":"We said that u is going to go from 1/2-1,"},{"Start":"16:35.350 ","End":"16:37.030","Text":"and for each u,"},{"Start":"16:37.030 ","End":"16:38.260","Text":"were going to have,"},{"Start":"16:38.260 ","End":"16:40.855","Text":"and I\u0027m deliberately leaving some space here,"},{"Start":"16:40.855 ","End":"16:46.495","Text":"integral of v from 1-2,"},{"Start":"16:46.495 ","End":"16:50.665","Text":"v equals 1-2 dv."},{"Start":"16:50.665 ","End":"16:52.915","Text":"The outer one was du."},{"Start":"16:52.915 ","End":"16:56.455","Text":"Now, what we\u0027re left with is,"},{"Start":"16:56.455 ","End":"17:01.070","Text":"this bit is e^v times 1 over 2u."},{"Start":"17:03.420 ","End":"17:11.575","Text":"First of all, write it as 1 of e^v, 1 over 2u."},{"Start":"17:11.575 ","End":"17:15.190","Text":"But what I\u0027m saying now is I can save a step"},{"Start":"17:15.190 ","End":"17:19.180","Text":"because the u is a constant as far as v is concerned,"},{"Start":"17:19.180 ","End":"17:22.190","Text":"I can bring it in here upfront."},{"Start":"17:22.190 ","End":"17:23.730","Text":"You know what, just so you\u0027ll see it,"},{"Start":"17:23.730 ","End":"17:28.440","Text":"I\u0027ll erase it from here and write it here, 1 over 2u."},{"Start":"17:28.440 ","End":"17:33.330","Text":"Just putting it in front of the integral because u"},{"Start":"17:33.330 ","End":"17:39.025","Text":"doesn\u0027t depend on v. Now it\u0027s all just technical."},{"Start":"17:39.025 ","End":"17:44.420","Text":"Let\u0027s see. Here we\u0027ll do the inner integral first."},{"Start":"17:44.550 ","End":"17:47.050","Text":"Let me do this 1 at the side,"},{"Start":"17:47.050 ","End":"17:49.345","Text":"the 1 I highlighted, the inner 1,"},{"Start":"17:49.345 ","End":"17:59.290","Text":"the integral from 1 to 2 of e^vdv is just e^v itself from 1 to"},{"Start":"17:59.290 ","End":"18:01.630","Text":"2 which means that it\u0027s e"},{"Start":"18:01.630 ","End":"18:09.535","Text":"squared minus e. This is a constant so I can take it out in front."},{"Start":"18:09.535 ","End":"18:13.525","Text":"In fact, I can also take the 1/2 in front, totally in front."},{"Start":"18:13.525 ","End":"18:20.575","Text":"So what I\u0027m left with now is 1/2 and then I have this e squared minus e,"},{"Start":"18:20.575 ","End":"18:25.765","Text":"and I have the integral from 1/2 to 1."},{"Start":"18:25.765 ","End":"18:31.075","Text":"All that\u0027s left here is the 1 over u du."},{"Start":"18:31.075 ","End":"18:32.740","Text":"This is an immediate 1."},{"Start":"18:32.740 ","End":"18:35.390","Text":"This is a natural logarithm."},{"Start":"18:35.760 ","End":"18:38.725","Text":"What we get is let\u0027s see,"},{"Start":"18:38.725 ","End":"18:42.220","Text":"e squared minus e over 2 for this."},{"Start":"18:42.220 ","End":"18:52.480","Text":"Then I have natural log of u from 1/2 to 1."},{"Start":"18:52.480 ","End":"18:54.860","Text":"Let\u0027s see what that is."},{"Start":"18:55.680 ","End":"18:59.095","Text":"Natural log of 1 is 0,"},{"Start":"18:59.095 ","End":"19:04.330","Text":"so I get e squared minus e over 2,"},{"Start":"19:04.330 ","End":"19:10.490","Text":"0 minus natural log of 1/2."},{"Start":"19:12.000 ","End":"19:15.040","Text":"I can stop here, but I want to continue just a little bit more."},{"Start":"19:15.040 ","End":"19:21.100","Text":"Natural log of 1/2 is natural log of 1 minus natural log of 2."},{"Start":"19:21.100 ","End":"19:22.570","Text":"I end up getting,"},{"Start":"19:22.570 ","End":"19:26.905","Text":"if I simplify it as plus natural log of 2,"},{"Start":"19:26.905 ","End":"19:30.400","Text":"so altogether, maybe I\u0027ll put the 1/2 in front."},{"Start":"19:30.400 ","End":"19:33.730","Text":"I have a natural log of 2 because I have a minus,"},{"Start":"19:33.730 ","End":"19:36.730","Text":"minus and then I also have an e"},{"Start":"19:36.730 ","End":"19:44.645","Text":"squared minus e. I\u0027ll just highlight it and that\u0027s the answer."},{"Start":"19:44.645 ","End":"19:47.320","Text":"But wait, don\u0027t go yet,"},{"Start":"19:47.320 ","End":"19:48.820","Text":"or at least you can go if you want to."},{"Start":"19:48.820 ","End":"19:55.225","Text":"But I want to show you a shortcut trick that we can use often in this kind of problem."},{"Start":"19:55.225 ","End":"20:01.130","Text":"Notice that a lot of the work went into computing the Jacobian."},{"Start":"20:02.100 ","End":"20:04.900","Text":"Let me write again what it is."},{"Start":"20:04.900 ","End":"20:09.850","Text":"We had that the Jacobian was the determinant of"},{"Start":"20:09.850 ","End":"20:16.165","Text":"the partial derivatives x with respect to u,"},{"Start":"20:16.165 ","End":"20:18.745","Text":"x with respect to v,"},{"Start":"20:18.745 ","End":"20:20.934","Text":"y with respect to u,"},{"Start":"20:20.934 ","End":"20:24.520","Text":"and y with respect to v. To do this,"},{"Start":"20:24.520 ","End":"20:26.215","Text":"we had to first of all,"},{"Start":"20:26.215 ","End":"20:30.820","Text":"extract x and y in terms of u and v. Then we have to"},{"Start":"20:30.820 ","End":"20:35.680","Text":"do compute 4 partial derivatives and then do all the multiplication."},{"Start":"20:35.680 ","End":"20:43.945","Text":"Finally, we substitute in the integral the formula as absolute value of J."},{"Start":"20:43.945 ","End":"20:47.800","Text":"Now it turns out, there\u0027s a shorter way that often works."},{"Start":"20:47.800 ","End":"20:51.204","Text":"Instead of using absolute value of J,"},{"Start":"20:51.204 ","End":"21:01.675","Text":"we replace this by 1 over the absolute value of something called J star, J asterisk."},{"Start":"21:01.675 ","End":"21:04.870","Text":"Now, what is this J star?"},{"Start":"21:04.870 ","End":"21:11.800","Text":"This J star is what you get if you don\u0027t extract x,"},{"Start":"21:11.800 ","End":"21:13.150","Text":"y in terms of u, v,"},{"Start":"21:13.150 ","End":"21:15.190","Text":"but you have u and v in terms of x,"},{"Start":"21:15.190 ","End":"21:16.510","Text":"y like we did."},{"Start":"21:16.510 ","End":"21:19.000","Text":"So this is equal to the other way around."},{"Start":"21:19.000 ","End":"21:20.965","Text":"It\u0027s u with respect to x,"},{"Start":"21:20.965 ","End":"21:23.109","Text":"u with respect to y,"},{"Start":"21:23.109 ","End":"21:25.060","Text":"v with respect to x,"},{"Start":"21:25.060 ","End":"21:27.550","Text":"v with respect to y."},{"Start":"21:27.550 ","End":"21:29.545","Text":"Let\u0027s see if it works."},{"Start":"21:29.545 ","End":"21:31.660","Text":"Now I remember that in our case,"},{"Start":"21:31.660 ","End":"21:36.325","Text":"we got, where was the Jacobian?"},{"Start":"21:36.325 ","End":"21:37.870","Text":"There was the answer,"},{"Start":"21:37.870 ","End":"21:40.015","Text":"it was minus 1 over 2u,"},{"Start":"21:40.015 ","End":"21:44.140","Text":"or an absolute value without the minus."},{"Start":"21:44.140 ","End":"21:51.490","Text":"Let\u0027s see if we get the same thing using this J star and taking 1 over."},{"Start":"21:51.490 ","End":"21:58.855","Text":"Now this we know immediately because and I better just copy the equations again."},{"Start":"21:58.855 ","End":"22:06.370","Text":"I believe we had u equals y over x, v equals xy."},{"Start":"22:06.370 ","End":"22:12.580","Text":"You can go back and check that I remember correctly."},{"Start":"22:12.580 ","End":"22:14.680","Text":"Now if I do this,"},{"Start":"22:14.680 ","End":"22:19.285","Text":"J star u with respect to x is,"},{"Start":"22:19.285 ","End":"22:22.045","Text":"since y is a constant,"},{"Start":"22:22.045 ","End":"22:27.160","Text":"it\u0027s just minus y over x squared,"},{"Start":"22:27.160 ","End":"22:29.319","Text":"u with respect to y."},{"Start":"22:29.319 ","End":"22:31.090","Text":"This time x is the constant,"},{"Start":"22:31.090 ","End":"22:39.055","Text":"so it\u0027s 1 over x. v with respect to x is just y,"},{"Start":"22:39.055 ","End":"22:43.299","Text":"and v with respect to y is just x."},{"Start":"22:43.299 ","End":"22:46.165","Text":"If I multiply out this times this,"},{"Start":"22:46.165 ","End":"22:49.300","Text":"while x cancels with 1 of the x\u0027s in the denominator,"},{"Start":"22:49.300 ","End":"22:55.975","Text":"and I get minus y over x."},{"Start":"22:55.975 ","End":"22:58.780","Text":"If I multiply this with this,"},{"Start":"22:58.780 ","End":"23:01.690","Text":"I get y over x,"},{"Start":"23:01.690 ","End":"23:06.070","Text":"but I have to subtract it so it\u0027s minus y over x."},{"Start":"23:06.070 ","End":"23:15.160","Text":"Altogether, what I get is minus 2 y over x."},{"Start":"23:15.160 ","End":"23:18.040","Text":"Yes, we do have to do a little bit of work because"},{"Start":"23:18.040 ","End":"23:21.340","Text":"the final answer has to be in terms of u and v. But here,"},{"Start":"23:21.340 ","End":"23:22.825","Text":"we are very lucky."},{"Start":"23:22.825 ","End":"23:25.540","Text":"We have y over x as u,"},{"Start":"23:25.540 ","End":"23:32.215","Text":"so this comes out to be just equal to minus 2u."},{"Start":"23:32.215 ","End":"23:39.910","Text":"When I compute 1 over the absolute value of J star,"},{"Start":"23:39.910 ","End":"23:48.130","Text":"this becomes 1 over the absolute value is just 2u because u is positive in our region."},{"Start":"23:48.130 ","End":"23:52.645","Text":"That is the same as what we got before."},{"Start":"23:52.645 ","End":"23:58.795","Text":"Earlier on, we got the absolute value of 1 over 2u with a minus 1 over 2u."},{"Start":"23:58.795 ","End":"24:01.600","Text":"It\u0027s the same, but a lot less work."},{"Start":"24:01.600 ","End":"24:05.810","Text":"There\u0027s a shortcut you should know. Now we\u0027re done."}],"ID":8669},{"Watched":false,"Name":"Exercise 3","Duration":"17m 46s","ChapterTopicVideoID":8454,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8454.jpeg","UploadDate":"2017-02-13T01:50:31.7200000","DurationForVideoObject":"PT17M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.645","Text":"In this exercise, we\u0027re asked to compute the double integral, this thing."},{"Start":"00:06.645 ","End":"00:13.770","Text":"The region R is described as a triangle with vertices this, this, and this."},{"Start":"00:14.090 ","End":"00:19.815","Text":"I already prepared myself 2 sketches."},{"Start":"00:19.815 ","End":"00:21.930","Text":"We\u0027re going to do a transformation,"},{"Start":"00:21.930 ","End":"00:25.410","Text":"a change of variables from xy to uv,"},{"Start":"00:25.410 ","End":"00:28.245","Text":"and I have them all handy."},{"Start":"00:28.245 ","End":"00:34.785","Text":"Let\u0027s get started with drawing the original region, this triangle."},{"Start":"00:34.785 ","End":"00:36.720","Text":"Lets say, well, 0,"},{"Start":"00:36.720 ","End":"00:42.640","Text":"0 is here and let\u0027s say this is 1."},{"Start":"00:42.640 ","End":"00:49.570","Text":"If this is 1, then this would be maybe 1 and this would be 2,"},{"Start":"00:50.060 ","End":"00:53.985","Text":"0, 1, 2, 1."},{"Start":"00:53.985 ","End":"00:55.695","Text":"Now we can do the points,"},{"Start":"00:55.695 ","End":"01:00.315","Text":"this one here the origin is A 2,"},{"Start":"01:00.315 ","End":"01:05.280","Text":"0 here is B and"},{"Start":"01:05.280 ","End":"01:10.710","Text":"C 1, 1 here."},{"Start":"01:10.710 ","End":"01:16.890","Text":"This, and this, and here\u0027s the third side of the triangle."},{"Start":"01:16.930 ","End":"01:21.505","Text":"A bullet R and now to shade it."},{"Start":"01:21.505 ","End":"01:25.880","Text":"The 2 main problems with this integral and in general,"},{"Start":"01:25.880 ","End":"01:30.060","Text":"problems that would maybe warrant a substitution,"},{"Start":"01:30.060 ","End":"01:35.435","Text":"change of variables, is either badly shaped region."},{"Start":"01:35.435 ","End":"01:36.965","Text":"This one is not too bad,"},{"Start":"01:36.965 ","End":"01:38.794","Text":"although in any event,"},{"Start":"01:38.794 ","End":"01:44.480","Text":"we might have to split it up to 2 regions if we slice it vertically."},{"Start":"01:44.480 ","End":"01:46.280","Text":"That\u0027s not such a problem."},{"Start":"01:46.280 ","End":"01:49.220","Text":"The bigger problem is that the integral itself,"},{"Start":"01:49.220 ","End":"01:52.570","Text":"the thing to be integrated in so messy."},{"Start":"01:52.570 ","End":"01:56.870","Text":"We can actually kill 2 birds with 1 stone with the right substitution."},{"Start":"01:56.870 ","End":"01:59.540","Text":"We can get this region to be even nicer and"},{"Start":"01:59.540 ","End":"02:03.330","Text":"certainly this integral to be a lot friendlier."},{"Start":"02:04.700 ","End":"02:11.075","Text":"What I\u0027d like to do first is write down the equations of the 3 sides of the triangle."},{"Start":"02:11.075 ","End":"02:18.180","Text":"For example, the side AC goes through 0,"},{"Start":"02:18.180 ","End":"02:19.725","Text":"0, and 1, 1."},{"Start":"02:19.725 ","End":"02:30.065","Text":"You can easily see that that is the equation y equals x. AB is the easiest, of course."},{"Start":"02:30.065 ","End":"02:32.240","Text":"That\u0027s just the x-axis,"},{"Start":"02:32.240 ","End":"02:35.105","Text":"so y equals 0."},{"Start":"02:35.105 ","End":"02:42.950","Text":"The CB, this is a 45-degree slope 1,"},{"Start":"02:42.950 ","End":"02:45.140","Text":"this is minus 45 degrees slope,"},{"Start":"02:45.140 ","End":"02:50.110","Text":"is minus 1, so y is going to be minus x plus something."},{"Start":"02:50.110 ","End":"02:52.430","Text":"If you just substitute a value,"},{"Start":"02:52.430 ","End":"02:59.415","Text":"you\u0027ll see that the equation is y equals minus x plus 2."},{"Start":"02:59.415 ","End":"03:02.420","Text":"I\u0027m not going to go into too much detail because you"},{"Start":"03:02.420 ","End":"03:06.120","Text":"know how to find line through a pair of points."},{"Start":"03:06.840 ","End":"03:13.180","Text":"Sometimes the region tells us what substitution or change of variables to make."},{"Start":"03:13.180 ","End":"03:15.175","Text":"Sometimes it\u0027s the integral."},{"Start":"03:15.175 ","End":"03:19.045","Text":"Hopefully, we can have something that will satisfy both."},{"Start":"03:19.045 ","End":"03:21.370","Text":"But the thing that\u0027s most obvious to me,"},{"Start":"03:21.370 ","End":"03:27.485","Text":"is that I don\u0027t want to have this expression and this expression,"},{"Start":"03:27.485 ","End":"03:29.710","Text":"I\u0027d like to substitute these."},{"Start":"03:29.710 ","End":"03:33.310","Text":"One of them was u and the other one was v. That would be much easier to"},{"Start":"03:33.310 ","End":"03:37.315","Text":"integrate and hopefully this will also make the region come out nice."},{"Start":"03:37.315 ","End":"03:42.330","Text":"This question was cooked up so that it does work that way."},{"Start":"03:42.330 ","End":"03:52.070","Text":"Let\u0027s just change variables and say that u is 1/2 of x plus y,"},{"Start":"03:52.070 ","End":"03:59.980","Text":"and v will be 1/2 of x minus y."},{"Start":"04:00.350 ","End":"04:04.410","Text":"Doing this, let\u0027s see what we\u0027ll get for these 3 lines."},{"Start":"04:04.410 ","End":"04:06.145","Text":"What they will become."},{"Start":"04:06.145 ","End":"04:08.720","Text":"For the first one, y equals x,"},{"Start":"04:08.720 ","End":"04:16.765","Text":"I can just bring the y over to the other side and say x minus y equals 0."},{"Start":"04:16.765 ","End":"04:20.590","Text":"I can also multiply both sides by 1/2."},{"Start":"04:20.590 ","End":"04:25.015","Text":"I can then say that 1/2 of x minus y equals 0,"},{"Start":"04:25.015 ","End":"04:32.300","Text":"so this one just become v equals 0 because the half there it is,"},{"Start":"04:32.300 ","End":"04:35.900","Text":"x minus y is v. As for the next one,"},{"Start":"04:35.900 ","End":"04:40.760","Text":"y equals 0, I\u0027m going to try and extract y from these 2."},{"Start":"04:40.760 ","End":"04:42.545","Text":"In fact, if you look at this,"},{"Start":"04:42.545 ","End":"04:47.120","Text":"if I subtract the top point minus the bottom one,"},{"Start":"04:47.120 ","End":"04:52.805","Text":"then I\u0027ll get u minus v and the 1/2x minus 1/2 x will cancel,"},{"Start":"04:52.805 ","End":"04:59.665","Text":"and 1/2y minus minus 1/2y will just give me y."},{"Start":"04:59.665 ","End":"05:09.845","Text":"The equation that y equals 0 will give me that u minus v equals 0."},{"Start":"05:09.845 ","End":"05:13.680","Text":"In other words, v equals"},{"Start":"05:13.680 ","End":"05:20.310","Text":"u will be the second equation from here."},{"Start":"05:20.310 ","End":"05:21.845","Text":"Then the last one,"},{"Start":"05:21.845 ","End":"05:23.720","Text":"y equals x, sorry,"},{"Start":"05:23.720 ","End":"05:25.925","Text":"y equals minus x plus 2."},{"Start":"05:25.925 ","End":"05:31.980","Text":"This I could write as x plus y equals 2."},{"Start":"05:32.590 ","End":"05:38.120","Text":"Now, x plus y is very close to u."},{"Start":"05:38.120 ","End":"05:41.060","Text":"If I just divide by 2 on both sides,"},{"Start":"05:41.060 ","End":"05:45.680","Text":"I\u0027ll get 1/2 of x plus y equals 1."},{"Start":"05:45.680 ","End":"05:51.555","Text":"Now I can see that the 1/2x plus y is u,"},{"Start":"05:51.555 ","End":"05:55.090","Text":"so the last equation is u equals 1."},{"Start":"05:58.580 ","End":"06:03.915","Text":"Now I\u0027m going to plot these 3 lines in the uv plane."},{"Start":"06:03.915 ","End":"06:06.240","Text":"The first one, v equals 0,"},{"Start":"06:06.240 ","End":"06:10.170","Text":"is just the u-axis here."},{"Start":"06:10.170 ","End":"06:15.960","Text":"V equals u is the 45-degree line through the origin."},{"Start":"06:15.960 ","End":"06:21.425","Text":"This would be the v equals u and u equals 1."},{"Start":"06:21.425 ","End":"06:24.750","Text":"Let\u0027s say this is 1 here,"},{"Start":"06:24.750 ","End":"06:29.520","Text":"that would be a vertical line here."},{"Start":"06:29.520 ","End":"06:37.260","Text":"Actually it continues, 1 and this would be our new region,"},{"Start":"06:37.260 ","End":"06:43.040","Text":"I\u0027ll call it S. This is a fairly nice region"},{"Start":"06:43.040 ","End":"06:49.070","Text":"and it will be easy whether I slice it vertically or horizontally,"},{"Start":"06:49.070 ","End":"06:51.000","Text":"it\u0027s going to come out nicely."},{"Start":"06:51.000 ","End":"06:54.120","Text":"Let\u0027s see what the integral becomes."},{"Start":"06:54.430 ","End":"06:59.300","Text":"We get the double integral instead of the old region,"},{"Start":"06:59.300 ","End":"07:02.490","Text":"we put the new region."},{"Start":"07:02.930 ","End":"07:07.235","Text":"Then I substitute the variables sine."},{"Start":"07:07.235 ","End":"07:10.610","Text":"This part is just u,"},{"Start":"07:10.610 ","End":"07:19.925","Text":"and this part is cosine v. The brackets are just for emphasis."},{"Start":"07:19.925 ","End":"07:27.685","Text":"Instead of dA, what we do is we put the absolute value of the Jacobian."},{"Start":"07:27.685 ","End":"07:30.975","Text":"I\u0027ll remind you what this is in a moment."},{"Start":"07:30.975 ","End":"07:36.775","Text":"Also I\u0027ll write dudv,"},{"Start":"07:36.775 ","End":"07:39.575","Text":"but I\u0027m not committing myself."},{"Start":"07:39.575 ","End":"07:42.500","Text":"We can write this even if we want to do it dvdu,"},{"Start":"07:42.500 ","End":"07:48.900","Text":"we can still later decide whether to do it as type 1 or type 2 region."},{"Start":"07:49.150 ","End":"07:54.930","Text":"In any event, this is a pretty straightforward integral."},{"Start":"07:55.750 ","End":"08:01.295","Text":"It looks slightly easier to meet to do vertical slices,"},{"Start":"08:01.295 ","End":"08:07.845","Text":"which means that I will let u run from 0-1 and for each,"},{"Start":"08:07.845 ","End":"08:12.300","Text":"u will take v from here to here."},{"Start":"08:12.300 ","End":"08:15.585","Text":"This line has the equation,"},{"Start":"08:15.585 ","End":"08:17.700","Text":"I need v in terms of u,"},{"Start":"08:17.700 ","End":"08:20.295","Text":"I already have it v equals u."},{"Start":"08:20.295 ","End":"08:23.130","Text":"The bottom line is,"},{"Start":"08:23.130 ","End":"08:24.315","Text":"also I have it,"},{"Start":"08:24.315 ","End":"08:26.745","Text":"is v equals 0."},{"Start":"08:26.745 ","End":"08:33.930","Text":"You will go from 0-1 and v will go from 0-u."},{"Start":"08:33.930 ","End":"08:42.075","Text":"I\u0027ll write that as the integral space here."},{"Start":"08:42.075 ","End":"08:49.965","Text":"U goes from 0-1 that\u0027s here to here,"},{"Start":"08:49.965 ","End":"08:52.560","Text":"and that\u0027s the du."},{"Start":"08:52.560 ","End":"08:54.825","Text":"Actually it\u0027s going to be a dudv."},{"Start":"08:54.825 ","End":"09:02.355","Text":"Then for this, the integral of v,"},{"Start":"09:02.355 ","End":"09:07.060","Text":"v will go from 0-u."},{"Start":"09:08.360 ","End":"09:11.770","Text":"That will be dv."},{"Start":"09:13.940 ","End":"09:19.740","Text":"Then I need to transform the integral."},{"Start":"09:19.740 ","End":"09:24.855","Text":"Let\u0027s see, I think we can still see it."},{"Start":"09:24.855 ","End":"09:35.295","Text":"Sine of 1/2x plus y is u, I have it here,"},{"Start":"09:35.295 ","End":"09:40.305","Text":"cosine of v and"},{"Start":"09:40.305 ","End":"09:46.410","Text":"absolute value of j but dvdu."},{"Start":"09:46.410 ","End":"09:50.070","Text":"I said I\u0027d remind you what j is."},{"Start":"09:50.070 ","End":"09:52.780","Text":"It\u0027s the Jacobian."},{"Start":"09:52.780 ","End":"09:59.660","Text":"This Jacobian is the determinant of the 2 by 2 matrix,"},{"Start":"09:59.660 ","End":"10:04.515","Text":"which is partial derivative of x with respect to u,"},{"Start":"10:04.515 ","End":"10:06.765","Text":"x with respect to v,"},{"Start":"10:06.765 ","End":"10:08.895","Text":"y with respect to u,"},{"Start":"10:08.895 ","End":"10:17.865","Text":"y with respect to v. In case you forgot what a determinant is,"},{"Start":"10:17.865 ","End":"10:23.280","Text":"well, in general, the determinant of a 2 by 2 matrix a, b, c,"},{"Start":"10:23.280 ","End":"10:29.320","Text":"d, just means ad minus bc."},{"Start":"10:29.660 ","End":"10:35.970","Text":"In our case, it\u0027ll be xuyv"},{"Start":"10:35.970 ","End":"10:45.645","Text":"minus yuxv or xvyu doesn\u0027t matter."},{"Start":"10:45.645 ","End":"10:47.520","Text":"What we do need though,"},{"Start":"10:47.520 ","End":"10:52.830","Text":"is to get x and y in terms of u and v as opposed to u and v in terms of x"},{"Start":"10:52.830 ","End":"10:58.560","Text":"and y or to solve these 2 equations for x and y."},{"Start":"10:58.560 ","End":"11:00.150","Text":"Actually, we\u0027ve made a good start."},{"Start":"11:00.150 ","End":"11:01.965","Text":"If you look at what we wrote here,"},{"Start":"11:01.965 ","End":"11:07.830","Text":"we subtracted the 2 equations and we got the y is u minus v. I\u0027ll write"},{"Start":"11:07.830 ","End":"11:14.040","Text":"that y equals u minus v. Actually,"},{"Start":"11:14.040 ","End":"11:20.910","Text":"we could do a similar trick if I added these 2 equations and got u plus v,"},{"Start":"11:20.910 ","End":"11:28.425","Text":"u plus v would just be 1/2x plus 1/2x is x and the y\u0027s would cancel so x is"},{"Start":"11:28.425 ","End":"11:36.410","Text":"equal to u plus v. I should have written maybe x above and below it doesn\u0027t matter."},{"Start":"11:36.410 ","End":"11:40.115","Text":"Anyway, I have the reverse transformation."},{"Start":"11:40.115 ","End":"11:44.520","Text":"Now I can go ahead and compute the Jacobian."},{"Start":"11:44.870 ","End":"11:51.885","Text":"Continuing here, x with respect to u is,"},{"Start":"11:51.885 ","End":"11:58.200","Text":"you know what, why don\u0027t I just write down what this comes out to be?"},{"Start":"11:58.200 ","End":"12:02.565","Text":"I think it\u0027ll be just easier to organize it this way."},{"Start":"12:02.565 ","End":"12:07.725","Text":"X with respect to u and v are both 1, 1,"},{"Start":"12:07.725 ","End":"12:13.515","Text":"1, and y with respect to u and v,"},{"Start":"12:13.515 ","End":"12:16.185","Text":"that\u0027s 1 and minus 1."},{"Start":"12:16.185 ","End":"12:23.910","Text":"This is our Jacobian and like I said,"},{"Start":"12:23.910 ","End":"12:31.080","Text":"with the determinants, you just take this diagonal multiplied less this diagonal."},{"Start":"12:31.080 ","End":"12:33.090","Text":"We get this times,"},{"Start":"12:33.090 ","End":"12:36.915","Text":"this is 1 times minus 1 is minus 1."},{"Start":"12:36.915 ","End":"12:41.865","Text":"This times this is 1 that we subtract, it\u0027s minus 2."},{"Start":"12:41.865 ","End":"12:47.595","Text":"When we substituted this j,"},{"Start":"12:47.595 ","End":"12:51.870","Text":"absolute value will just be 2."},{"Start":"12:51.870 ","End":"12:54.960","Text":"Now I can simplify this integral."},{"Start":"12:54.960 ","End":"12:57.370","Text":"I forgot to put equals."},{"Start":"12:57.620 ","End":"13:01.605","Text":"We can take the 2 right in front."},{"Start":"13:01.605 ","End":"13:03.750","Text":"It\u0027s a constant."},{"Start":"13:03.750 ","End":"13:10.840","Text":"Now you\u0027ve got the integral from 0-1 for you."},{"Start":"13:11.660 ","End":"13:17.685","Text":"Since v is a constant as far as u goes,"},{"Start":"13:17.685 ","End":"13:22.960","Text":"sorry, I\u0027m doing the inner integral first."},{"Start":"13:24.080 ","End":"13:30.360","Text":"I\u0027m going to take the sine u, I\u0027m taking out,"},{"Start":"13:30.360 ","End":"13:37.140","Text":"I\u0027ll just write it sine of u integral from 0-u,"},{"Start":"13:37.140 ","End":"13:44.015","Text":"let me put the names of the variables too from 0-u of"},{"Start":"13:44.015 ","End":"13:54.090","Text":"cosine vdv and then another du."},{"Start":"13:54.220 ","End":"13:58.865","Text":"What I\u0027m going to do first is this integral,"},{"Start":"13:58.865 ","End":"14:07.139","Text":"the integral dv and we\u0027ll do it at the side somewhere."},{"Start":"14:07.139 ","End":"14:08.850","Text":"I have some room here."},{"Start":"14:08.850 ","End":"14:10.800","Text":"Let me just call this asterisk."},{"Start":"14:10.800 ","End":"14:16.590","Text":"I\u0027m going over here and we want to do the integral"},{"Start":"14:16.590 ","End":"14:23.990","Text":"from 0-u of cosine vdv."},{"Start":"14:23.990 ","End":"14:27.035","Text":"Well, the integral of cosine is sine."},{"Start":"14:27.035 ","End":"14:29.165","Text":"I\u0027ve got sine v,"},{"Start":"14:29.165 ","End":"14:38.685","Text":"which I need to take between 0 and u sine of u,"},{"Start":"14:38.685 ","End":"14:47.940","Text":"when I let v equals u, I get sine of u less sine of 0,"},{"Start":"14:47.940 ","End":"14:52.450","Text":"but sine of 0 is 0."},{"Start":"14:52.700 ","End":"14:55.605","Text":"I just have sine u."},{"Start":"14:55.605 ","End":"15:01.905","Text":"This whole thing boils down to sine of u."},{"Start":"15:01.905 ","End":"15:07.540","Text":"Now I\u0027ve got, let\u0027s see, space."},{"Start":"15:08.180 ","End":"15:15.570","Text":"This will now equal twice sine u with sine u will be sine squared u."},{"Start":"15:15.570 ","End":"15:16.740","Text":"The integral, of course,"},{"Start":"15:16.740 ","End":"15:22.470","Text":"from 0-1 sine squared udu."},{"Start":"15:22.470 ","End":"15:28.170","Text":"I\u0027m going to use a trigonometrical identity here that says that"},{"Start":"15:28.170 ","End":"15:34.470","Text":"sine squared of u or any other letter in the books that were written,"},{"Start":"15:34.470 ","End":"15:36.240","Text":"maybe as alpha or theta,"},{"Start":"15:36.240 ","End":"15:44.290","Text":"is equal to 1/2 of 1 minus cosine 2u."},{"Start":"15:44.390 ","End":"15:47.955","Text":"If I make this substitution here,"},{"Start":"15:47.955 ","End":"15:51.990","Text":"the 2 and the 1/2 will cancel and I\u0027ll"},{"Start":"15:51.990 ","End":"15:59.325","Text":"get just the integral from 0-1,"},{"Start":"15:59.325 ","End":"16:08.295","Text":"2 sine squared u is 1 minus cosine 2u du."},{"Start":"16:08.295 ","End":"16:15.940","Text":"We\u0027re almost done. Now, this is equal to,"},{"Start":"16:17.150 ","End":"16:20.640","Text":"let us continue over here."},{"Start":"16:20.640 ","End":"16:25.950","Text":"I want to just do it over here."},{"Start":"16:25.950 ","End":"16:33.990","Text":"I\u0027ve got the, let\u0027s say 1 its integral is"},{"Start":"16:33.990 ","End":"16:39.240","Text":"u cosine 2u will give"},{"Start":"16:39.240 ","End":"16:44.925","Text":"me not exactly sine 2u because of the inner derivative,"},{"Start":"16:44.925 ","End":"16:48.990","Text":"I need to divide by 2."},{"Start":"16:48.990 ","End":"16:55.305","Text":"Then to evaluate this from 0-1,"},{"Start":"16:55.305 ","End":"16:58.125","Text":"if I plug in 1,"},{"Start":"16:58.125 ","End":"16:59.970","Text":"I will get"},{"Start":"16:59.970 ","End":"17:10.870","Text":"1 minus 1/2 sine of 2."},{"Start":"17:11.570 ","End":"17:14.820","Text":"Now if I plug in the 0,"},{"Start":"17:14.820 ","End":"17:21.760","Text":"I\u0027ll get that\u0027s just 0 sine of twice 0 sine 0 is 0."},{"Start":"17:21.920 ","End":"17:26.715","Text":"What we have is simply,"},{"Start":"17:26.715 ","End":"17:30.660","Text":"well I\u0027ll write it here,"},{"Start":"17:30.660 ","End":"17:41.295","Text":"is 1 minus 1/2 sine 2."},{"Start":"17:41.295 ","End":"17:46.330","Text":"I\u0027ll highlight it and we\u0027re done."}],"ID":8670},{"Watched":false,"Name":"Exercise 4","Duration":"20m 5s","ChapterTopicVideoID":8455,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8455.jpeg","UploadDate":"2017-02-13T01:55:37.8030000","DurationForVideoObject":"PT20M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.855","Text":"In this exercise, we\u0027re given a double integral as follows over a region R,"},{"Start":"00:06.855 ","End":"00:09.750","Text":"and R is a parallelogram with these vertices."},{"Start":"00:09.750 ","End":"00:13.050","Text":"I\u0027ve already plotted the vertices here."},{"Start":"00:13.050 ","End":"00:17.010","Text":"Now let me join lines through them."},{"Start":"00:17.010 ","End":"00:22.245","Text":"Here are the lines and it\u0027s a parallelogram."},{"Start":"00:22.245 ","End":"00:25.455","Text":"My drawing is not precise, but that\u0027s okay."},{"Start":"00:25.455 ","End":"00:28.110","Text":"This is the region R,"},{"Start":"00:28.110 ","End":"00:30.450","Text":"and why don\u0027t I shade it?"},{"Start":"00:30.450 ","End":"00:36.150","Text":"I\u0027d like the equation of each of these 4 lines that surround the parallelogram."},{"Start":"00:36.150 ","End":"00:40.195","Text":"Maybe I\u0027ll put the coordinates from here, copy them here."},{"Start":"00:40.195 ","End":"00:42.980","Text":"There we are. Let\u0027s do a little bit of review of how"},{"Start":"00:42.980 ","End":"00:45.335","Text":"to find the equation of a line through 2 points."},{"Start":"00:45.335 ","End":"00:47.990","Text":"Let\u0027s take this one, for example."},{"Start":"00:47.990 ","End":"00:50.120","Text":"That one would be AD."},{"Start":"00:50.120 ","End":"00:52.055","Text":"How do I find its equation?"},{"Start":"00:52.055 ","End":"00:55.360","Text":"Well, I can use the slope."},{"Start":"00:55.360 ","End":"00:58.635","Text":"Slope was rise over run."},{"Start":"00:58.635 ","End":"01:00.935","Text":"Let\u0027s see from the rise,"},{"Start":"01:00.935 ","End":"01:03.800","Text":"I can see that it\u0027s 5 minus 3."},{"Start":"01:03.800 ","End":"01:05.210","Text":"I go up 2 units,"},{"Start":"01:05.210 ","End":"01:09.155","Text":"and the run from minus 1 to 1 is also 2 units."},{"Start":"01:09.155 ","End":"01:16.050","Text":"The slope is 2 over 2, which is 1."},{"Start":"01:16.060 ","End":"01:18.830","Text":"Now when we have the slope,"},{"Start":"01:18.830 ","End":"01:27.175","Text":"we know that Y equals in general mx plus m let\u0027s say,"},{"Start":"01:27.175 ","End":"01:29.580","Text":"and in our case m,"},{"Start":"01:29.580 ","End":"01:31.650","Text":"which is the slope is 1."},{"Start":"01:31.650 ","End":"01:37.980","Text":"I can just replace it with 1 and I\u0027ll remove that 1 later, I don\u0027t need it."},{"Start":"01:37.980 ","End":"01:42.860","Text":"Then to find n, we just make sure that it passes through these points,"},{"Start":"01:42.860 ","End":"01:45.335","Text":"just pick one of them, let\u0027s say 1, 5."},{"Start":"01:45.335 ","End":"01:47.300","Text":"1, 5 has to be on it."},{"Start":"01:47.300 ","End":"01:50.195","Text":"So when x is 1, y has to be 5."},{"Start":"01:50.195 ","End":"01:53.575","Text":"5 equals 1 plus something,"},{"Start":"01:53.575 ","End":"01:56.140","Text":"that something has to be 4."},{"Start":"01:56.140 ","End":"02:00.145","Text":"There\u0027s the 4 and I\u0027ll erase the 1."},{"Start":"02:00.145 ","End":"02:04.895","Text":"There\u0027s an extra check, we can see that this point is also on it."},{"Start":"02:04.895 ","End":"02:09.590","Text":"X plus 4 here is minus 1 plus 4 is 3,"},{"Start":"02:09.590 ","End":"02:12.020","Text":"which is y, so we\u0027re okay."},{"Start":"02:12.020 ","End":"02:15.090","Text":"Next, let me do the parallel one to it."},{"Start":"02:15.090 ","End":"02:19.070","Text":"Let\u0027s do BC. Very similar."},{"Start":"02:19.070 ","End":"02:23.600","Text":"Notice that the rise is minus 1, less minus 3."},{"Start":"02:23.600 ","End":"02:27.845","Text":"It\u0027s also a rise of 2 and the run from 1 to 3,"},{"Start":"02:27.845 ","End":"02:30.575","Text":"the axis is also to 2,"},{"Start":"02:30.575 ","End":"02:32.530","Text":"2 over 2 is 1."},{"Start":"02:32.530 ","End":"02:35.205","Text":"Again, we get y equals 1x,"},{"Start":"02:35.205 ","End":"02:38.340","Text":"which is x plus something."},{"Start":"02:38.340 ","End":"02:43.100","Text":"To get that something, we just have to make sure that these points are on it,"},{"Start":"02:43.100 ","End":"02:44.635","Text":"let\u0027s take this one."},{"Start":"02:44.635 ","End":"02:46.995","Text":"If I plug in x equals 3,"},{"Start":"02:46.995 ","End":"02:48.830","Text":"y equals minus 1,"},{"Start":"02:48.830 ","End":"02:54.480","Text":"then minus 1 has to be 3 plus something."},{"Start":"02:54.480 ","End":"02:56.480","Text":"It\u0027s actually not a plus,"},{"Start":"02:56.480 ","End":"03:00.190","Text":"it has to be minus 4 to make it work."},{"Start":"03:00.190 ","End":"03:04.985","Text":"Minus 4, I\u0027ll just check that the other one works also."},{"Start":"03:04.985 ","End":"03:08.760","Text":"X minus 4 here is minus 3,"},{"Start":"03:08.760 ","End":"03:11.655","Text":"it\u0027s y it\u0027s okay. That\u0027s 2 of them."},{"Start":"03:11.655 ","End":"03:13.530","Text":"Now let\u0027s get to,"},{"Start":"03:13.530 ","End":"03:18.160","Text":"say this one, AB."},{"Start":"03:18.820 ","End":"03:25.250","Text":"For AB, the rise over the run is going to be a negative rise."},{"Start":"03:25.250 ","End":"03:27.455","Text":"If I go from A to B,"},{"Start":"03:27.455 ","End":"03:32.185","Text":"I actually go from 3 down to minus 3."},{"Start":"03:32.185 ","End":"03:35.119","Text":"The rise is negative 6."},{"Start":"03:35.119 ","End":"03:40.250","Text":"The run from minus 1 to 1 is 2."},{"Start":"03:40.250 ","End":"03:43.520","Text":"I have minus 6 over 2,"},{"Start":"03:43.520 ","End":"03:49.835","Text":"which is minus 3 for the slope times x."},{"Start":"03:49.835 ","End":"03:52.850","Text":"Then it\u0027s got to be plus a constant,"},{"Start":"03:52.850 ","End":"03:56.435","Text":"so we just plug-in one of these points."},{"Start":"03:56.435 ","End":"03:58.555","Text":"Let\u0027s say this one."},{"Start":"03:58.555 ","End":"04:03.510","Text":"If I put in x equals minus 1 and y equals 3,"},{"Start":"04:03.510 ","End":"04:06.240","Text":"I get 3 equals 3 plus n,"},{"Start":"04:06.240 ","End":"04:10.275","Text":"n is 0, so I just erase it."},{"Start":"04:10.275 ","End":"04:14.995","Text":"Then finally, DC."},{"Start":"04:14.995 ","End":"04:17.390","Text":"This line here."},{"Start":"04:17.390 ","End":"04:20.795","Text":"Also, if you compute the rise,"},{"Start":"04:20.795 ","End":"04:24.775","Text":"we go from 5 down to minus 1."},{"Start":"04:24.775 ","End":"04:27.480","Text":"The rise is minus 6."},{"Start":"04:27.480 ","End":"04:30.425","Text":"The run from 1 to 3 is 2."},{"Start":"04:30.425 ","End":"04:34.925","Text":"Minus 6 over 2 is also minus 3."},{"Start":"04:34.925 ","End":"04:42.540","Text":"We have y equals minus 3x plus some n. Substitute one of them,"},{"Start":"04:42.540 ","End":"04:44.445","Text":"let\u0027s say this point,"},{"Start":"04:44.445 ","End":"04:49.695","Text":"minus 3x is minus 3."},{"Start":"04:49.695 ","End":"04:54.515","Text":"I have to add something to minus 3 to get 5."},{"Start":"04:54.515 ","End":"04:58.880","Text":"This comes out to be plus 8."},{"Start":"04:58.880 ","End":"05:03.205","Text":"You could always check that it satisfies this one."},{"Start":"05:03.205 ","End":"05:09.945","Text":"For example, minus 3x is minus 9,"},{"Start":"05:09.945 ","End":"05:12.450","Text":"plus 8 is minus 1,"},{"Start":"05:12.450 ","End":"05:14.310","Text":"which is y and we are okay."},{"Start":"05:14.310 ","End":"05:17.005","Text":"Now we have the 4 equations."},{"Start":"05:17.005 ","End":"05:21.230","Text":"Now, this region R is very awkward for doing an"},{"Start":"05:21.230 ","End":"05:26.090","Text":"integral on because we have to subdivide it."},{"Start":"05:26.090 ","End":"05:30.140","Text":"We might have to take the region and carve it up."},{"Start":"05:30.140 ","End":"05:35.770","Text":"Maybe we\u0027d have a line from here to here,"},{"Start":"05:35.770 ","End":"05:40.695","Text":"and possibly another line here."},{"Start":"05:40.695 ","End":"05:48.715","Text":"Then I\u0027d have to subdivide R into R1, R2, R3."},{"Start":"05:48.715 ","End":"05:50.540","Text":"Actually in this case,"},{"Start":"05:50.540 ","End":"05:53.645","Text":"it turns out that D is exactly above B,"},{"Start":"05:53.645 ","End":"05:56.555","Text":"because I notice that they both have an x coordinate of 1,"},{"Start":"05:56.555 ","End":"05:58.250","Text":"so there\u0027s really only 2 regions."},{"Start":"05:58.250 ","End":"06:01.470","Text":"But in general you could expect to get 3 pieces,"},{"Start":"06:01.470 ","End":"06:02.630","Text":"and then for each piece,"},{"Start":"06:02.630 ","End":"06:05.000","Text":"it have a different upper and lower function,"},{"Start":"06:05.000 ","End":"06:08.330","Text":"and you\u0027d have to figure out what these points are."},{"Start":"06:08.330 ","End":"06:11.185","Text":"It could be messy."},{"Start":"06:11.185 ","End":"06:14.780","Text":"Hopefully, if we do a change of variables,"},{"Start":"06:14.780 ","End":"06:18.860","Text":"we might get this region to be much better behaved,"},{"Start":"06:18.860 ","End":"06:24.720","Text":"hopefully even a rectangle parallel with the axis, that would be nice."},{"Start":"06:24.850 ","End":"06:34.050","Text":"The idea is to somehow change these equations and make a change of variables."},{"Start":"06:34.130 ","End":"06:37.950","Text":"Notice that the first 2 are similar,"},{"Start":"06:37.950 ","End":"06:40.950","Text":"y equals x plus something and the second 2 are similar,"},{"Start":"06:40.950 ","End":"06:43.355","Text":"y equals minus 3x plus something."},{"Start":"06:43.355 ","End":"06:46.045","Text":"I can slightly rewrite them."},{"Start":"06:46.045 ","End":"06:53.265","Text":"This one I can rewrite as y minus x equals 4."},{"Start":"06:53.265 ","End":"07:02.489","Text":"This one I can rewrite as y minus x equals minus 4."},{"Start":"07:02.489 ","End":"07:12.780","Text":"This one I can rewrite as y plus 3x equals 0."},{"Start":"07:12.780 ","End":"07:20.410","Text":"This one I can write as y plus 3x equals 8."},{"Start":"07:20.950 ","End":"07:23.060","Text":"This came out a bit messy."},{"Start":"07:23.060 ","End":"07:25.405","Text":"I\u0027ll just erase the old ones."},{"Start":"07:25.405 ","End":"07:27.690","Text":"Okay, that looks better."},{"Start":"07:27.690 ","End":"07:31.240","Text":"Now, the y minus x,"},{"Start":"07:31.240 ","End":"07:32.960","Text":"it appears here and here,"},{"Start":"07:32.960 ","End":"07:38.515","Text":"and the y plus 3x that appears here and here suggest a substitution."},{"Start":"07:38.515 ","End":"07:40.530","Text":"Let\u0027s call this one u,"},{"Start":"07:40.530 ","End":"07:45.220","Text":"so we have u equals y minus x."},{"Start":"07:45.350 ","End":"07:48.165","Text":"We also have from these 2,"},{"Start":"07:48.165 ","End":"07:50.115","Text":"I\u0027ll let this be v,"},{"Start":"07:50.115 ","End":"07:55.515","Text":"I\u0027ll let v equals y plus 3x."},{"Start":"07:55.515 ","End":"07:58.400","Text":"After making this substitution,"},{"Start":"07:58.400 ","End":"08:03.210","Text":"I can now rewrite these 4 equations as follows."},{"Start":"08:03.210 ","End":"08:06.180","Text":"The first one, y minus x is u."},{"Start":"08:06.180 ","End":"08:09.495","Text":"This one is u equals 4."},{"Start":"08:09.495 ","End":"08:13.365","Text":"The second one, u equals minus 4."},{"Start":"08:13.365 ","End":"08:17.895","Text":"This one, this left-hand side is v,"},{"Start":"08:17.895 ","End":"08:20.625","Text":"that\u0027s v equals 0,"},{"Start":"08:20.625 ","End":"08:25.730","Text":"and the last one, v equals 8."},{"Start":"08:25.730 ","End":"08:31.105","Text":"Now, I want to plot these on a new graph."},{"Start":"08:31.105 ","End":"08:36.475","Text":"Let\u0027s say this is the u-axis and this is the v-axis."},{"Start":"08:36.475 ","End":"08:39.025","Text":"Then let\u0027s see."},{"Start":"08:39.025 ","End":"08:47.560","Text":"Maybe 4 is here and then u equals 4 would be a vertical line through 4."},{"Start":"08:47.560 ","End":"08:50.755","Text":"Minus 4 would be on the other side,"},{"Start":"08:50.755 ","End":"08:55.810","Text":"minus 4, so u is minus 4 is a vertical line here."},{"Start":"08:55.810 ","End":"09:05.860","Text":"v equals 0 is just the u-axis here and v equals 8 is somewhere,"},{"Start":"09:05.860 ","End":"09:07.705","Text":"I don\u0027t know, maybe here,"},{"Start":"09:07.705 ","End":"09:10.960","Text":"this is 8, this is 0."},{"Start":"09:10.960 ","End":"09:20.020","Text":"Now, our region is this rectangle here and I\u0027ll call it what\u0027s after r,"},{"Start":"09:20.020 ","End":"09:23.620","Text":"letter s and I\u0027ll shade it."},{"Start":"09:23.620 ","End":"09:29.275","Text":"We\u0027re almost ready to rewrite this integral in terms of"},{"Start":"09:29.275 ","End":"09:34.540","Text":"u and v. The steps to make a decision with this rectangle,"},{"Start":"09:34.540 ","End":"09:37.990","Text":"whether you want to make vertical slices or horizontal slices,"},{"Start":"09:37.990 ","End":"09:39.865","Text":"type 1 or type 2 region,"},{"Start":"09:39.865 ","End":"09:42.610","Text":"I say let\u0027s make it a type 1 region."},{"Start":"09:42.610 ","End":"09:47.820","Text":"For a typical u here,"},{"Start":"09:47.820 ","End":"09:57.405","Text":"you will go from minus 4-4, it\u0027s a minus."},{"Start":"09:57.405 ","End":"10:00.769","Text":"For each particular u,"},{"Start":"10:00.769 ","End":"10:08.140","Text":"I\u0027ll make a vertical slice v will enter here and will exit here,"},{"Start":"10:08.140 ","End":"10:10.780","Text":"v will go from 0-8."},{"Start":"10:10.780 ","End":"10:17.690","Text":"I\u0027ll get the integral v goes from 0-8."},{"Start":"10:18.300 ","End":"10:21.280","Text":"I know that the outer loop is u,"},{"Start":"10:21.280 ","End":"10:23.710","Text":"so the answer is going to be a du,"},{"Start":"10:23.710 ","End":"10:27.740","Text":"and before that there\u0027s going to be a dv for this."},{"Start":"10:27.960 ","End":"10:31.495","Text":"Now hit upon a small snag,"},{"Start":"10:31.495 ","End":"10:33.625","Text":"another thing we have to do,"},{"Start":"10:33.625 ","End":"10:37.090","Text":"we have the integrnd in terms of x and y,"},{"Start":"10:37.090 ","End":"10:42.010","Text":"I wanted in u and v. I want to compute the reverse formulas,"},{"Start":"10:42.010 ","End":"10:47.350","Text":"from this one I want to extract what y and x are in terms of v and"},{"Start":"10:47.350 ","End":"10:53.800","Text":"u and let me do that somewhere at this side."},{"Start":"10:53.800 ","End":"10:55.975","Text":"Let\u0027s say here."},{"Start":"10:55.975 ","End":"10:59.770","Text":"If I want to extract x I need to get rid of y."},{"Start":"10:59.770 ","End":"11:02.620","Text":"How about I subtract these 2 equations?"},{"Start":"11:02.620 ","End":"11:05.830","Text":"Now what I\u0027ll do the first from the second,"},{"Start":"11:05.830 ","End":"11:10.825","Text":"in other words equation 2 minus equation 1 here would give me v"},{"Start":"11:10.825 ","End":"11:17.620","Text":"minus u equals y minus y cancels 3x minus,"},{"Start":"11:17.620 ","End":"11:20.180","Text":"minus x is 4x."},{"Start":"11:20.940 ","End":"11:25.015","Text":"I would get from this that x"},{"Start":"11:25.015 ","End":"11:34.480","Text":"equals 1/4v minus 1/4u."},{"Start":"11:34.480 ","End":"11:36.445","Text":"Now I want to extract y."},{"Start":"11:36.445 ","End":"11:38.320","Text":"How should I do that?"},{"Start":"11:38.320 ","End":"11:42.970","Text":"What I\u0027ll do is take this equation and add"},{"Start":"11:42.970 ","End":"11:47.545","Text":"3 times this equation and it\u0027ll be minus 3x and the x will cancel."},{"Start":"11:47.545 ","End":"11:51.205","Text":"I\u0027m going to take this plus 3 times this,"},{"Start":"11:51.205 ","End":"12:02.440","Text":"so v plus 3u will equal y plus 3y is 4y,"},{"Start":"12:02.440 ","End":"12:11.440","Text":"3x plus 3 times minus x is nothing as we planned and this gives us that y equals"},{"Start":"12:11.440 ","End":"12:18.010","Text":"1/4v plus 3/4u and"},{"Start":"12:18.010 ","End":"12:22.015","Text":"this is the inverse formula for this substitution."},{"Start":"12:22.015 ","End":"12:24.925","Text":"Now that I have what x and y are,"},{"Start":"12:24.925 ","End":"12:27.415","Text":"I can plug them in here."},{"Start":"12:27.415 ","End":"12:34.030","Text":"Let me just do a little computation somewhere maybe here."},{"Start":"12:34.030 ","End":"12:42.820","Text":"Let me just figure out what is 4x plus 8y is equal to 4x, multiply this by 4,"},{"Start":"12:42.820 ","End":"12:51.670","Text":"I\u0027ve just got v minus u and 8y if I multiply this by 8,"},{"Start":"12:51.670 ","End":"12:56.965","Text":"I\u0027ve got 2v plus"},{"Start":"12:56.965 ","End":"13:05.115","Text":"6u and so I end up with if I collect,"},{"Start":"13:05.115 ","End":"13:15.440","Text":"I\u0027ve got 5u and 3v."},{"Start":"13:15.440 ","End":"13:18.900","Text":"Here I can write"},{"Start":"13:18.900 ","End":"13:27.169","Text":"5u plus 3v but that\u0027s not all."},{"Start":"13:27.169 ","End":"13:33.010","Text":"The theorem, the formula says that we replace da not just by dv,"},{"Start":"13:33.010 ","End":"13:34.570","Text":"du or du, dv,"},{"Start":"13:34.570 ","End":"13:40.840","Text":"we have an extra thing here which is the absolute value of the Jacobian."},{"Start":"13:40.840 ","End":"13:44.930","Text":"I\u0027m going to remind you what the Jacobian is."},{"Start":"13:45.660 ","End":"13:55.119","Text":"The Jacobian is equal to the determinant of x with respect to u,"},{"Start":"13:55.119 ","End":"13:57.205","Text":"x with respect to v,"},{"Start":"13:57.205 ","End":"14:00.550","Text":"partial derivatives, y with respect to u,"},{"Start":"14:00.550 ","End":"14:05.890","Text":"y with respect to v. This is determinant not the same as absolute value,"},{"Start":"14:05.890 ","End":"14:07.810","Text":"they\u0027re both written with bars."},{"Start":"14:07.810 ","End":"14:09.310","Text":"Don\u0027t confuse the 2, Oh,"},{"Start":"14:09.310 ","End":"14:12.220","Text":"and if you have forgotten or you haven\u0027t learned what"},{"Start":"14:12.220 ","End":"14:15.595","Text":"a determinant is in the case of a 2 by 2 matrix."},{"Start":"14:15.595 ","End":"14:18.175","Text":"Let\u0027s see where should I write it."},{"Start":"14:18.175 ","End":"14:19.750","Text":"I\u0027ve bit of room here."},{"Start":"14:19.750 ","End":"14:23.830","Text":"The determinant of any 4 numbers, a, b, c,"},{"Start":"14:23.830 ","End":"14:27.580","Text":"d is just this diagonal,"},{"Start":"14:27.580 ","End":"14:31.330","Text":"ad minus product of the other diagonal."},{"Start":"14:31.330 ","End":"14:34.120","Text":"This times this minus this times this."},{"Start":"14:34.120 ","End":"14:37.040","Text":"Let\u0027s see if we can fill this in."},{"Start":"14:37.110 ","End":"14:41.410","Text":"To do this, we have to have x and y in terms of u and v,"},{"Start":"14:41.410 ","End":"14:42.805","Text":"but we\u0027ve already done that."},{"Start":"14:42.805 ","End":"14:47.155","Text":"If we hadn\u0027t, we\u0027d have to do it now. Now let\u0027s see."},{"Start":"14:47.155 ","End":"14:49.014","Text":"I\u0027ll put it in the bars."},{"Start":"14:49.014 ","End":"14:52.270","Text":"What is derivative of x with respect to u?"},{"Start":"14:52.270 ","End":"14:55.720","Text":"That\u0027s just minus 1/4,"},{"Start":"14:55.720 ","End":"14:58.630","Text":"with respect to v, it\u0027s 1/4."},{"Start":"14:58.630 ","End":"15:01.435","Text":"As for y with respect to u,"},{"Start":"15:01.435 ","End":"15:08.440","Text":"it\u0027s 3/4, and with respect to v, it\u0027s 1/4."},{"Start":"15:08.440 ","End":"15:13.375","Text":"Now, I have to multiply this diagonal minus this diagonal."},{"Start":"15:13.375 ","End":"15:19.090","Text":"I get minus 16th, minus 3/16,"},{"Start":"15:19.090 ","End":"15:26.590","Text":"that\u0027s minus 4/16, this is equal to minus 1/4."},{"Start":"15:26.590 ","End":"15:30.340","Text":"But here, absolute value of J,"},{"Start":"15:30.340 ","End":"15:32.020","Text":"I don\u0027t put minus 1/4,"},{"Start":"15:32.020 ","End":"15:35.870","Text":"I put plus 1/4 because I throw out the sign."},{"Start":"15:36.180 ","End":"15:40.550","Text":"Now we can rewrite this."},{"Start":"15:40.680 ","End":"15:43.300","Text":"Really rewrite, just a slight revision."},{"Start":"15:43.300 ","End":"15:45.040","Text":"I\u0027ll put 1/4 upfront,"},{"Start":"15:45.040 ","End":"15:50.020","Text":"so I have 1/4 times the integral,"},{"Start":"15:50.020 ","End":"15:54.190","Text":"u goes from minus 4-4."},{"Start":"15:54.190 ","End":"16:03.730","Text":"Integral v from 0-8 of 5u plus 3v,"},{"Start":"16:03.730 ","End":"16:10.060","Text":"dv, du, and we do these things from inside out."},{"Start":"16:10.060 ","End":"16:12.085","Text":"First of all, the inner integral,"},{"Start":"16:12.085 ","End":"16:16.720","Text":"which is with respect to v. Everything now is"},{"Start":"16:16.720 ","End":"16:21.520","Text":"just technical so I don\u0027t care if the diagrams go out of u."},{"Start":"16:21.520 ","End":"16:26.020","Text":"Continuing, I\u0027d like to do this integral at the side"},{"Start":"16:26.020 ","End":"16:30.595","Text":"and then return to it just the shaded part."},{"Start":"16:30.595 ","End":"16:40.119","Text":"I want the integral from 0-8 of 5u minus 3v,"},{"Start":"16:40.119 ","End":"16:46.225","Text":"dv and that is equal to if I take the integral now,"},{"Start":"16:46.225 ","End":"16:47.860","Text":"remember u is a constant,"},{"Start":"16:47.860 ","End":"16:54.580","Text":"so I get 5uv and the integral of v is a 1/2v squared,"},{"Start":"16:54.580 ","End":"16:59.620","Text":"so minus 3 over 2v squared."},{"Start":"16:59.620 ","End":"17:03.235","Text":"All this taken from 0-8,"},{"Start":"17:03.235 ","End":"17:07.730","Text":"and it\u0027s v that goes from 0-8."},{"Start":"17:11.010 ","End":"17:14.675","Text":"Let me start the other way. If I plug in v equals 0,"},{"Start":"17:14.675 ","End":"17:17.420","Text":"everything becomes 0, so I don\u0027t need that."},{"Start":"17:17.420 ","End":"17:24.400","Text":"All I have to do is substitute 8 for v. When v is 8,"},{"Start":"17:24.400 ","End":"17:31.030","Text":"here I get 8 times 5 is 40u,"},{"Start":"17:31.030 ","End":"17:34.025","Text":"and here when v is 8,"},{"Start":"17:34.025 ","End":"17:38.855","Text":"8 squared is 64."},{"Start":"17:38.855 ","End":"17:43.460","Text":"64 times 1.5 is 96."},{"Start":"17:43.460 ","End":"17:45.185","Text":"That\u0027s what I make it."},{"Start":"17:45.185 ","End":"17:51.540","Text":"Now we can take this expression and put it back up there."},{"Start":"17:51.540 ","End":"17:54.475","Text":"But wait, I see we have 1/4 here."},{"Start":"17:54.475 ","End":"18:00.200","Text":"Why don\u0027t I just take 4 out of here and then we can maybe cancel something."},{"Start":"18:00.200 ","End":"18:07.300","Text":"This is 4 times 10u minus 24."},{"Start":"18:07.300 ","End":"18:10.165","Text":"If I put this here,"},{"Start":"18:10.165 ","End":"18:17.960","Text":"4 can cancel with 1/4 and this just becomes 10u minus 24."},{"Start":"18:18.060 ","End":"18:23.035","Text":"The integral from minus 4-4,"},{"Start":"18:23.035 ","End":"18:33.075","Text":"the 1/4 has already canceled with the 4 and this whole thing is 10u minus 24, du."},{"Start":"18:33.075 ","End":"18:35.010","Text":"Very straightforward."},{"Start":"18:35.010 ","End":"18:42.075","Text":"Now, the integral of 10u is 5u squared,"},{"Start":"18:42.075 ","End":"18:45.430","Text":"here we get minus 24u."},{"Start":"18:45.430 ","End":"18:51.570","Text":"This evaluated between minus 4 and 4."},{"Start":"18:51.570 ","End":"18:53.115","Text":"Oh, hang on a moment,"},{"Start":"18:53.115 ","End":"18:55.565","Text":"I just noticed that I missed copied something."},{"Start":"18:55.565 ","End":"18:58.235","Text":"This is a plus and somehow I made it a minus."},{"Start":"18:58.235 ","End":"19:00.455","Text":"This is plus, this is plus,"},{"Start":"19:00.455 ","End":"19:02.535","Text":"and this is plus."},{"Start":"19:02.535 ","End":"19:04.970","Text":"Even over here, this is plus,"},{"Start":"19:04.970 ","End":"19:07.565","Text":"this is plus, this is plus, this is plus."},{"Start":"19:07.565 ","End":"19:11.640","Text":"Apologies, but no harms done yet."},{"Start":"19:12.330 ","End":"19:14.465","Text":"Let\u0027s see what we get."},{"Start":"19:14.465 ","End":"19:17.695","Text":"If we plug in 4, 4 squared is 16,"},{"Start":"19:17.695 ","End":"19:23.875","Text":"16 times 5 is 80."},{"Start":"19:23.875 ","End":"19:27.055","Text":"For this part, I get 80,"},{"Start":"19:27.055 ","End":"19:30.430","Text":"and if I plug in 4 here,"},{"Start":"19:30.430 ","End":"19:35.020","Text":"24 times 4 is 96."},{"Start":"19:35.020 ","End":"19:37.180","Text":"I have plug in minus 4,"},{"Start":"19:37.180 ","End":"19:40.930","Text":"I here it\u0027ll come out the same, 80 also."},{"Start":"19:40.930 ","End":"19:46.450","Text":"But here, I\u0027ll get minus 96,"},{"Start":"19:46.450 ","End":"19:51.580","Text":"80 minus 80 cancels,"},{"Start":"19:51.580 ","End":"19:57.295","Text":"and 96 minus, minus 96 is twice 96 which is"},{"Start":"19:57.295 ","End":"20:05.460","Text":"192 and this is our final answer. We\u0027re done."}],"ID":8671},{"Watched":false,"Name":"Exercise 5","Duration":"11m 57s","ChapterTopicVideoID":8456,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8456.jpeg","UploadDate":"2017-02-13T01:58:26.5170000","DurationForVideoObject":"PT11M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this exercise, we have to compute the double integral as follows."},{"Start":"00:04.680 ","End":"00:10.410","Text":"Over the region, R. R is the region enclosed by the ellipse."},{"Start":"00:10.410 ","End":"00:12.824","Text":"We know that this is an ellipse and in general,"},{"Start":"00:12.824 ","End":"00:16.440","Text":"x squared over a squared plus y squared"},{"Start":"00:16.440 ","End":"00:20.605","Text":"over b squared equals 1 is the general form of an ellipse,"},{"Start":"00:20.605 ","End":"00:25.010","Text":"where a is where it cuts the x-axis."},{"Start":"00:25.010 ","End":"00:28.145","Text":"In this case, a is 3 and b is 4,"},{"Start":"00:28.145 ","End":"00:30.470","Text":"because this is 3 squared and this is 4 squared."},{"Start":"00:30.470 ","End":"00:33.770","Text":"You didn\u0027t need to have the sketch of the ellipse to do this."},{"Start":"00:33.770 ","End":"00:36.870","Text":"I just thought it would be nice to include it."},{"Start":"00:36.880 ","End":"00:44.000","Text":"Now, my idea here is to use polar coordinates."},{"Start":"00:44.000 ","End":"00:45.860","Text":"But for polar coordinates,"},{"Start":"00:45.860 ","End":"00:50.915","Text":"I need a disk or circle, the insides."},{"Start":"00:50.915 ","End":"00:55.280","Text":"I have an ellipse, not a circle."},{"Start":"00:55.280 ","End":"01:02.345","Text":"I\u0027d like to do a transformation that will get this ellipse into a circle or"},{"Start":"01:02.345 ","End":"01:05.810","Text":"disc is the correct word because you include the"},{"Start":"01:05.810 ","End":"01:09.590","Text":"interior and maybe even do it as the unit circle,"},{"Start":"01:09.590 ","End":"01:11.735","Text":"that would be really nice."},{"Start":"01:11.735 ","End":"01:16.955","Text":"Now if I just rewrite this, inspired by this,"},{"Start":"01:16.955 ","End":"01:24.200","Text":"I could just write the original ellipse in the form x over 3,"},{"Start":"01:24.200 ","End":"01:30.410","Text":"all squared because that would give me x squared over 9 plus y over 4 squared."},{"Start":"01:30.410 ","End":"01:33.800","Text":"That\u0027s y squared over 16 equals 1."},{"Start":"01:33.800 ","End":"01:39.810","Text":"Now the obvious substitution is to let 1 of these be"},{"Start":"01:39.810 ","End":"01:46.139","Text":"U and the other 1 be v. Let\u0027s say u is x over 3,"},{"Start":"01:46.139 ","End":"01:50.025","Text":"v is y over 4."},{"Start":"01:50.025 ","End":"01:55.640","Text":"Then this ellipse becomes quite"},{"Start":"01:55.640 ","End":"02:02.795","Text":"simply u squared plus v squared equals 1."},{"Start":"02:02.795 ","End":"02:06.050","Text":"That\u0027s u squared v squared 1."},{"Start":"02:06.050 ","End":"02:09.680","Text":"We know this is an equation of a circle,"},{"Start":"02:09.680 ","End":"02:13.010","Text":"but of course, we take the insides also."},{"Start":"02:13.010 ","End":"02:17.990","Text":"Here\u0027s the picture of the unit circle in the uv plane."},{"Start":"02:17.990 ","End":"02:25.610","Text":"Let\u0027s give it a name instead of R we\u0027ll call it S. We want to write the double"},{"Start":"02:25.610 ","End":"02:28.820","Text":"integral of something over this unit circle"},{"Start":"02:28.820 ","End":"02:33.710","Text":"S. We\u0027ll start off by writing the double integral."},{"Start":"02:33.710 ","End":"02:40.385","Text":"That\u0027s the conversion of this over S of the square root."},{"Start":"02:40.385 ","End":"02:47.120","Text":"Now already we\u0027re stuck because we have u and v in terms of x and y,"},{"Start":"02:47.120 ","End":"02:53.120","Text":"but we don\u0027t have x and y in terms of u and v. Let me remedy that."},{"Start":"02:53.120 ","End":"02:56.000","Text":"I have some space over here."},{"Start":"02:56.000 ","End":"02:58.250","Text":"We can do this in our heads."},{"Start":"02:58.250 ","End":"03:00.425","Text":"From u equals x over 3,"},{"Start":"03:00.425 ","End":"03:03.890","Text":"we get that x equals 3u,"},{"Start":"03:03.890 ","End":"03:06.275","Text":"and from v equals y over 4,"},{"Start":"03:06.275 ","End":"03:09.050","Text":"we get that y equals 4v."},{"Start":"03:09.050 ","End":"03:14.295","Text":"This is the inverse transformation or change of variables."},{"Start":"03:14.295 ","End":"03:17.495","Text":"Now we can continue here."},{"Start":"03:17.495 ","End":"03:19.850","Text":"I want to figure out what\u0027s under the square root sign."},{"Start":"03:19.850 ","End":"03:23.885","Text":"Let me also do this at the side 16x squared,"},{"Start":"03:23.885 ","End":"03:26.570","Text":"x squared is 9u squared."},{"Start":"03:26.570 ","End":"03:31.795","Text":"9 times 16 is 144, is 144u squared."},{"Start":"03:31.795 ","End":"03:34.189","Text":"They also need 9y squared."},{"Start":"03:34.189 ","End":"03:38.504","Text":"9y squared is 9 times 16,"},{"Start":"03:38.504 ","End":"03:42.465","Text":"also a 144v squared."},{"Start":"03:42.465 ","End":"03:51.330","Text":"Here I get 144u"},{"Start":"03:51.330 ","End":"03:56.250","Text":"squared plus 144v squared."},{"Start":"03:56.250 ","End":"03:58.890","Text":"Now, what does dA convert to?"},{"Start":"03:58.890 ","End":"04:04.660","Text":"Well, it\u0027s not simply dudv or dvdu."},{"Start":"04:04.660 ","End":"04:07.445","Text":"There\u0027s also an extra bit in the formula,"},{"Start":"04:07.445 ","End":"04:16.590","Text":"which is the absolute value of the Jacobian J. I\u0027ll remind you what the Jacobian is."},{"Start":"04:16.660 ","End":"04:21.770","Text":"The Jacobian of a change of variables with"},{"Start":"04:21.770 ","End":"04:28.700","Text":"2 variables is just the determinant of 4 partial derivatives,"},{"Start":"04:28.700 ","End":"04:30.350","Text":"x with respect to u,"},{"Start":"04:30.350 ","End":"04:32.375","Text":"x with respect to v,"},{"Start":"04:32.375 ","End":"04:34.415","Text":"y with respect to u,"},{"Start":"04:34.415 ","End":"04:35.900","Text":"y with respect to"},{"Start":"04:35.900 ","End":"04:42.935","Text":"v. If you tell me that you\u0027ve forgotten or you don\u0027t know what the determinant is."},{"Start":"04:42.935 ","End":"04:47.195","Text":"Well, a 2 by 2 determinant is defined as follows."},{"Start":"04:47.195 ","End":"04:52.530","Text":"Determinant of 4 numbers, a, b, c,"},{"Start":"04:52.530 ","End":"04:57.320","Text":"d. A 2 by 2 matrix is product of this diagonal minus this diagonal."},{"Start":"04:57.320 ","End":"05:01.595","Text":"In other words, ad minus bc."},{"Start":"05:01.595 ","End":"05:07.430","Text":"Back here, we get absolute, not absolute value."},{"Start":"05:07.430 ","End":"05:08.915","Text":"It\u0027s the determinant."},{"Start":"05:08.915 ","End":"05:11.510","Text":"It actually does look a bit like an absolute value."},{"Start":"05:11.510 ","End":"05:13.130","Text":"They both have bars."},{"Start":"05:13.130 ","End":"05:18.785","Text":"X with respect to u and v from x with respect to u is 3,"},{"Start":"05:18.785 ","End":"05:20.430","Text":"but with respect to v,"},{"Start":"05:20.430 ","End":"05:25.895","Text":"it\u0027s 0 because there is no v. As for y with respect to u it\u0027s 0,"},{"Start":"05:25.895 ","End":"05:29.405","Text":"and with respect to v, it\u0027s 4."},{"Start":"05:29.405 ","End":"05:31.085","Text":"If we compute this,"},{"Start":"05:31.085 ","End":"05:35.405","Text":"it\u0027s this diagonals product minus this diagonals product."},{"Start":"05:35.405 ","End":"05:37.864","Text":"It\u0027s 12 minus 0."},{"Start":"05:37.864 ","End":"05:41.570","Text":"This is just equal to 12."},{"Start":"05:41.570 ","End":"05:43.460","Text":"That\u0027s already is positive,"},{"Start":"05:43.460 ","End":"05:46.280","Text":"so the absolute value is still 12."},{"Start":"05:46.280 ","End":"05:48.859","Text":"So this thing we computed as 12."},{"Start":"05:48.859 ","End":"05:52.400","Text":"This can be simplified a bit because I could"},{"Start":"05:52.400 ","End":"05:55.835","Text":"take a 144 outside the brackets under the root."},{"Start":"05:55.835 ","End":"05:57.320","Text":"But when I take it out,"},{"Start":"05:57.320 ","End":"06:01.945","Text":"the square root of 144 is 12."},{"Start":"06:01.945 ","End":"06:05.630","Text":"If I take 12 out here together with this 12,"},{"Start":"06:05.630 ","End":"06:08.585","Text":"that will give me 144."},{"Start":"06:08.585 ","End":"06:14.480","Text":"In short, after taking this 12 and this 12 and pulling the 144 in front,"},{"Start":"06:14.480 ","End":"06:19.460","Text":"I get a 144 double integral over S,"},{"Start":"06:19.460 ","End":"06:25.460","Text":"which is the unit circle of the square root of u"},{"Start":"06:25.460 ","End":"06:32.780","Text":"squared plus v squared dudv or dvdu doesn\u0027t matter."},{"Start":"06:32.780 ","End":"06:36.860","Text":"I\u0027d like to do this integral with polar coordinates."},{"Start":"06:36.860 ","End":"06:39.970","Text":"Let me go and get my polar formula."},{"Start":"06:39.970 ","End":"06:42.950","Text":"Well, I copy-pasted the formula,"},{"Start":"06:42.950 ","End":"06:45.170","Text":"but it involves x and y,"},{"Start":"06:45.170 ","End":"06:48.680","Text":"and we have u and v. Let\u0027s just change every x to a u and"},{"Start":"06:48.680 ","End":"06:53.100","Text":"every y to a v. Here I had an x, I put a u."},{"Start":"06:53.100 ","End":"06:57.330","Text":"Here I had a y, I put a v. Here I had x squared plus y squared,"},{"Start":"06:57.330 ","End":"06:59.445","Text":"I put u squared plus v squared."},{"Start":"06:59.445 ","End":"07:03.695","Text":"Now, these are our conversion formulas for our Theta."},{"Start":"07:03.695 ","End":"07:06.335","Text":"Now for the unit circle,"},{"Start":"07:06.335 ","End":"07:15.715","Text":"we know the bounds for Theta and r. I\u0027ll just remind you in polar,"},{"Start":"07:15.715 ","End":"07:19.855","Text":"the circle we get by going all the way around,"},{"Start":"07:19.855 ","End":"07:21.780","Text":"all the way around,"},{"Start":"07:21.780 ","End":"07:28.060","Text":"and so on until we get back here."},{"Start":"07:28.730 ","End":"07:35.475","Text":"We start with Theta equals 0."},{"Start":"07:35.475 ","End":"07:40.760","Text":"That\u0027s Theta is the angle and we go"},{"Start":"07:40.760 ","End":"07:46.040","Text":"all the way around counterclockwise until we get to the same point."},{"Start":"07:46.040 ","End":"07:49.975","Text":"But at this point, Theta is equal to 2Pi."},{"Start":"07:49.975 ","End":"07:59.185","Text":"As for r, r always goes from the center where it\u0027s 0 to the circumference where it\u0027s 1."},{"Start":"07:59.185 ","End":"08:04.140","Text":"If I write it, it\u0027s 0 less than or"},{"Start":"08:04.140 ","End":"08:12.210","Text":"equal to Theta less than or equal to 360 degrees or 2Pi."},{"Start":"08:12.950 ","End":"08:17.410","Text":"R is from 0-1."},{"Start":"08:17.410 ","End":"08:24.440","Text":"This is the polar description of this disc circle with its interior."},{"Start":"08:24.440 ","End":"08:31.020","Text":"Now we use the polar conversion on this 1 using the formulas"},{"Start":"08:31.020 ","End":"08:41.065","Text":"and r. We would write normally as,"},{"Start":"08:41.065 ","End":"08:44.235","Text":"I\u0027m trying to avoid a name clash."},{"Start":"08:44.235 ","End":"08:48.260","Text":"Normally we have a dA here,"},{"Start":"08:48.260 ","End":"08:52.500","Text":"but it could have just as been dxdy."},{"Start":"08:53.860 ","End":"08:58.540","Text":"Then when I switched all the x and y to u and v,"},{"Start":"08:58.540 ","End":"09:03.450","Text":"I would have made that dxdy as dudv or dvdu."},{"Start":"09:03.450 ","End":"09:05.770","Text":"It doesn\u0027t really matter."},{"Start":"09:07.160 ","End":"09:14.840","Text":"What we have here is that dudv converts to rdrd Theta."},{"Start":"09:14.840 ","End":"09:19.325","Text":"If I could just write what happens after we convert it."},{"Start":"09:19.325 ","End":"09:27.895","Text":"We\u0027ll get, just continue over here, 144."},{"Start":"09:27.895 ","End":"09:31.795","Text":"Now, the outer integral is always the Theta,"},{"Start":"09:31.795 ","End":"09:37.075","Text":"that\u0027s from 0-2Pi, and that\u0027s d Theta."},{"Start":"09:37.075 ","End":"09:42.725","Text":"Then inside that, we have the integral of r going from 0-1."},{"Start":"09:42.725 ","End":"09:46.600","Text":"Then we need the square root of u squared plus v squared."},{"Start":"09:46.600 ","End":"09:52.965","Text":"This is the square root of u squared plus v squared is r squared"},{"Start":"09:52.965 ","End":"10:01.545","Text":"and dudv is replaced by rdrd Theta."},{"Start":"10:01.545 ","End":"10:05.015","Text":"Yes, Theta on the outside, r on the inside."},{"Start":"10:05.015 ","End":"10:07.700","Text":"We can slightly simplify this."},{"Start":"10:07.700 ","End":"10:12.155","Text":"The square root of r squared is just r and r times r is r squared."},{"Start":"10:12.155 ","End":"10:18.454","Text":"Instead of this, imagine that I\u0027ve written r squared and I\u0027ll save a line."},{"Start":"10:18.454 ","End":"10:24.270","Text":"Let\u0027s compute the inner integral first."},{"Start":"10:24.770 ","End":"10:28.745","Text":"I\u0027ll do this format at the side over here."},{"Start":"10:28.745 ","End":"10:34.640","Text":"The integral from 0-1 of r squared dr"},{"Start":"10:34.640 ","End":"10:41.760","Text":"is equal to 1/3r cubed from 0-1."},{"Start":"10:43.100 ","End":"10:45.575","Text":"When r is 0, it\u0027s nothing."},{"Start":"10:45.575 ","End":"10:47.525","Text":"When r is 1, this is 1,"},{"Start":"10:47.525 ","End":"10:52.465","Text":"so this is just equal to 1/3."},{"Start":"10:52.465 ","End":"10:54.770","Text":"Now I put that back in here."},{"Start":"10:54.770 ","End":"10:58.100","Text":"All this shaded bit is 1/3,"},{"Start":"10:58.100 ","End":"11:00.970","Text":"which I can pull in front."},{"Start":"11:00.970 ","End":"11:06.330","Text":"This thing becomes 1/3 of a 144 is"},{"Start":"11:06.330 ","End":"11:12.345","Text":"48 times the integral"},{"Start":"11:12.345 ","End":"11:18.720","Text":"from 0-2Pi of just all, there\u0027s nothing left."},{"Start":"11:18.720 ","End":"11:22.750","Text":"I\u0027ll write just 1d Theta."},{"Start":"11:22.870 ","End":"11:25.640","Text":"Now when we have the integral of 1,"},{"Start":"11:25.640 ","End":"11:28.760","Text":"we know that it\u0027s just the upper limit minus the lower limit."},{"Start":"11:28.760 ","End":"11:32.840","Text":"I mean, integral of this with respect to Theta is Theta."},{"Start":"11:32.840 ","End":"11:34.880","Text":"When Theta\u0027s 2Pi, then Theta\u0027s 2Pi,"},{"Start":"11:34.880 ","End":"11:36.860","Text":"and we subtract when Theta is 0."},{"Start":"11:36.860 ","End":"11:41.265","Text":"It\u0027s 48 times 2Pi minus 0 and then what?"},{"Start":"11:41.265 ","End":"11:43.969","Text":"I\u0027ll even write 2Pi minus 0."},{"Start":"11:43.969 ","End":"11:45.890","Text":"But the final answer,"},{"Start":"11:45.890 ","End":"11:48.110","Text":"48 times 2 is 96."},{"Start":"11:48.110 ","End":"11:58.500","Text":"I get 96Pi and this is the answer. we are done."}],"ID":8672},{"Watched":false,"Name":"Exercise 6","Duration":"11m 41s","ChapterTopicVideoID":8457,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8457.jpeg","UploadDate":"2017-02-13T02:01:21.9400000","DurationForVideoObject":"PT11M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this exercise, we have to compute this double integral,"},{"Start":"00:04.290 ","End":"00:09.910","Text":"and R is the region bounded by the following 4 curves."},{"Start":"00:10.340 ","End":"00:15.880","Text":"Let\u0027s assume we\u0027re working in the first quadrant where everything\u0027s positive."},{"Start":"00:15.890 ","End":"00:23.385","Text":"In this exercise I\u0027m not going to use a graph or a sketch for a change."},{"Start":"00:23.385 ","End":"00:25.485","Text":"If I look at these, well,"},{"Start":"00:25.485 ","End":"00:26.820","Text":"the last 2 look very similar."},{"Start":"00:26.820 ","End":"00:28.875","Text":"I have xy squared in both of them."},{"Start":"00:28.875 ","End":"00:31.275","Text":"If I slightly rewrite the first 2,"},{"Start":"00:31.275 ","End":"00:34.015","Text":"what I get from the first 1,"},{"Start":"00:34.015 ","End":"00:36.720","Text":"I can get xy equals 1,"},{"Start":"00:36.720 ","End":"00:40.525","Text":"then I\u0027ve got xy equals 2."},{"Start":"00:40.525 ","End":"00:44.015","Text":"Then let me just copy these as is."},{"Start":"00:44.015 ","End":"00:50.110","Text":"Then xy squared equals 1 and xy squared equals 2."},{"Start":"00:50.110 ","End":"00:54.060","Text":"Now, here\u0027s xy and here\u0027s xy."},{"Start":"00:54.060 ","End":"00:56.535","Text":"I\u0027ll highlight them."},{"Start":"00:56.535 ","End":"00:58.230","Text":"Here\u0027s xy squared,"},{"Start":"00:58.230 ","End":"01:00.225","Text":"and here\u0027s xy squared."},{"Start":"01:00.225 ","End":"01:02.940","Text":"I\u0027ll highlight those."},{"Start":"01:02.940 ","End":"01:10.170","Text":"What this clearly calls for is a substitution where I let xy be 1 thing."},{"Start":"01:10.170 ","End":"01:14.370","Text":"Let\u0027s say u is xy, and the other 1,"},{"Start":"01:14.370 ","End":"01:20.275","Text":"let it be v. v is xy squared."},{"Start":"01:20.275 ","End":"01:24.170","Text":"Now, let me say bounded by the curves,"},{"Start":"01:24.170 ","End":"01:28.760","Text":"looks like xy goes from 1 up to 2 and including the middle bits,"},{"Start":"01:28.760 ","End":"01:34.190","Text":"so the limits of the region in terms of u and"},{"Start":"01:34.190 ","End":"01:40.470","Text":"v are that u goes between 1 and 2."},{"Start":"01:42.950 ","End":"01:48.015","Text":"Similarly for v, xy squared also starts from 1 and ends at 2,"},{"Start":"01:48.015 ","End":"01:51.465","Text":"so v also goes from 1-2."},{"Start":"01:51.465 ","End":"01:57.545","Text":"This is in fact a square if you draw it on the uv plane,"},{"Start":"01:57.545 ","End":"02:00.320","Text":"but we\u0027re not going to draw it in the uv plane."},{"Start":"02:00.320 ","End":"02:03.665","Text":"Now when we transform an integral,"},{"Start":"02:03.665 ","End":"02:06.350","Text":"there are several things we need to do."},{"Start":"02:06.350 ","End":"02:08.990","Text":"First of all, we need to transform the region."},{"Start":"02:08.990 ","End":"02:10.520","Text":"Well, this is the region."},{"Start":"02:10.520 ","End":"02:14.285","Text":"But instead of naming it something like ROS or something,"},{"Start":"02:14.285 ","End":"02:18.530","Text":"I\u0027ll just say straight away that we\u0027ll do the integral from 1-2."},{"Start":"02:18.530 ","End":"02:20.555","Text":"It doesn\u0027t really matter which order."},{"Start":"02:20.555 ","End":"02:28.725","Text":"Let\u0027s say we\u0027ll take the outer loop as u from 1-2 and the inner loop v from 1-2."},{"Start":"02:28.725 ","End":"02:33.660","Text":"At the end I know then there\u0027ll be a dv du."},{"Start":"02:34.670 ","End":"02:41.465","Text":"In fact, the dA is equal to something called"},{"Start":"02:41.465 ","End":"02:48.720","Text":"the Jacobian times du dv or dv du either way."},{"Start":"02:48.720 ","End":"02:52.565","Text":"All that we\u0027re left with is this bit here,"},{"Start":"02:52.565 ","End":"02:54.080","Text":"which I have to put,"},{"Start":"02:54.080 ","End":"02:58.640","Text":"I\u0027ll just describe it as y squared,"},{"Start":"02:58.640 ","End":"03:08.090","Text":"but in terms of u and v. We\u0027ve got several unknowns."},{"Start":"03:08.090 ","End":"03:11.105","Text":"We want to know how to convert y-squared to u and v,"},{"Start":"03:11.105 ","End":"03:12.950","Text":"and we want to know what is this Jacobian,"},{"Start":"03:12.950 ","End":"03:14.785","Text":"and how to compute it."},{"Start":"03:14.785 ","End":"03:17.190","Text":"I\u0027ll start with the Jacobian."},{"Start":"03:17.190 ","End":"03:20.780","Text":"The Jacobian, and this is the absolute value,"},{"Start":"03:20.780 ","End":"03:22.700","Text":"but the Jacobian is the determinant."},{"Start":"03:22.700 ","End":"03:24.020","Text":"It\u0027s also written with 2 bars,"},{"Start":"03:24.020 ","End":"03:25.760","Text":"but it\u0027s something else."},{"Start":"03:25.760 ","End":"03:29.030","Text":"The determinant of the partial derivatives"},{"Start":"03:29.030 ","End":"03:31.760","Text":"of x and y in terms of u and v. In other words,"},{"Start":"03:31.760 ","End":"03:33.410","Text":"x with respect to u,"},{"Start":"03:33.410 ","End":"03:35.360","Text":"x with respect to v,"},{"Start":"03:35.360 ","End":"03:37.520","Text":"y with respect to u,"},{"Start":"03:37.520 ","End":"03:40.700","Text":"y with respect to v. That\u0027s the determinant,"},{"Start":"03:40.700 ","End":"03:43.745","Text":"and in case you forgotten what a determinant is,"},{"Start":"03:43.745 ","End":"03:46.160","Text":"or at least a 2 by 2 determinant,"},{"Start":"03:46.160 ","End":"03:49.100","Text":"which has the general form of a, b, c,"},{"Start":"03:49.100 ","End":"03:55.580","Text":"d, you can just take it as a definition as ad minus bc."},{"Start":"03:55.580 ","End":"03:59.070","Text":"This diagonal minus this diagonal\u0027s product."},{"Start":"03:59.290 ","End":"04:04.400","Text":"For this, we need to differentiate x and y with respect to u and v. It"},{"Start":"04:04.400 ","End":"04:08.705","Text":"wouldn\u0027t it be nice to have x and y in terms of u and v. Also here,"},{"Start":"04:08.705 ","End":"04:12.350","Text":"we had y in terms of u and v. All things considered,"},{"Start":"04:12.350 ","End":"04:17.990","Text":"what we want to get now is an inverse transformation to this change of variables,"},{"Start":"04:17.990 ","End":"04:23.075","Text":"where we want to get what x equals in terms of u and v,"},{"Start":"04:23.075 ","End":"04:26.660","Text":"and y equals in terms of u and v.. Let\u0027s see if we can"},{"Start":"04:26.660 ","End":"04:31.310","Text":"do that by playing around with these 2."},{"Start":"04:31.700 ","End":"04:38.210","Text":"1 thing I can see straight away is that if I divide the"},{"Start":"04:38.210 ","End":"04:45.440","Text":"second by the first xy squared over xy will give me just y."},{"Start":"04:45.440 ","End":"04:47.180","Text":"Here I will get v over u,"},{"Start":"04:47.180 ","End":"04:51.000","Text":"so this is immediately v over u."},{"Start":"04:51.040 ","End":"04:59.065","Text":"The other thing I can do is if I want to get rid of y,"},{"Start":"04:59.065 ","End":"05:06.410","Text":"to get x, so how about if I first square this?"},{"Start":"05:06.410 ","End":"05:13.295","Text":"I\u0027ll just write it in small u squared equals x squared y squared."},{"Start":"05:13.295 ","End":"05:16.520","Text":"Now if I divide the top by the bottom,"},{"Start":"05:16.520 ","End":"05:18.260","Text":"the y squareds will disappear."},{"Start":"05:18.260 ","End":"05:23.900","Text":"x squared over x will give me exactly x and it will equal u squared"},{"Start":"05:23.900 ","End":"05:31.730","Text":"over v. We\u0027re progressing here with our integral."},{"Start":"05:31.730 ","End":"05:34.715","Text":"We can develop it further and say,"},{"Start":"05:34.715 ","End":"05:38.475","Text":"it\u0027s the integral u goes from 1-2,"},{"Start":"05:38.475 ","End":"05:40.785","Text":"v goes from 1-2."},{"Start":"05:40.785 ","End":"05:42.630","Text":"This part, I can already do,"},{"Start":"05:42.630 ","End":"05:46.160","Text":"the y squared in terms of u and v. I look at this."},{"Start":"05:46.160 ","End":"05:49.765","Text":"y squared is v squared over u squared."},{"Start":"05:49.765 ","End":"05:53.000","Text":"I have here v squared over u squared."},{"Start":"05:53.000 ","End":"05:56.615","Text":"Here I have still the dv du,"},{"Start":"05:56.615 ","End":"05:59.929","Text":"and the only gap I have is here,"},{"Start":"05:59.929 ","End":"06:02.240","Text":"where I need to compute the Jacobian."},{"Start":"06:02.240 ","End":"06:04.530","Text":"Let\u0027s do that."},{"Start":"06:05.180 ","End":"06:08.310","Text":"Let\u0027s see what this equals."},{"Start":"06:08.310 ","End":"06:12.135","Text":"Leave some space here to do 4 things."},{"Start":"06:12.135 ","End":"06:14.600","Text":"x with respect to u,"},{"Start":"06:14.600 ","End":"06:16.040","Text":"v is a constant,"},{"Start":"06:16.040 ","End":"06:23.025","Text":"so it comes out to be 2u over v. Then x with respect to v,"},{"Start":"06:23.025 ","End":"06:24.994","Text":"du squared is a constant."},{"Start":"06:24.994 ","End":"06:28.565","Text":"The derivative of 1 over v is minus 1 over v squared."},{"Start":"06:28.565 ","End":"06:32.840","Text":"So I get minus u squared over v squared."},{"Start":"06:32.840 ","End":"06:35.465","Text":"Then y with respect to u."},{"Start":"06:35.465 ","End":"06:40.190","Text":"Again, the 1 over u gives minus 1 over u squared and the v is a constant,"},{"Start":"06:40.190 ","End":"06:43.625","Text":"so I get minus v over u squared."},{"Start":"06:43.625 ","End":"06:45.770","Text":"Then this with respect to v,"},{"Start":"06:45.770 ","End":"06:48.415","Text":"just get 1 over u."},{"Start":"06:48.415 ","End":"06:53.505","Text":"Now if I multiply this out, let\u0027s see."},{"Start":"06:53.505 ","End":"07:01.220","Text":"This times this is 2u over v times 1 over u."},{"Start":"07:01.220 ","End":"07:02.720","Text":"The u cancels."},{"Start":"07:02.720 ","End":"07:07.355","Text":"This comes out to be 2 over v. That\u0027s for this diagonal."},{"Start":"07:07.355 ","End":"07:09.695","Text":"As for the other diagonal,"},{"Start":"07:09.695 ","End":"07:11.450","Text":"which I want to subtract,"},{"Start":"07:11.450 ","End":"07:16.160","Text":"this times this, is a minus times a minus is a plus."},{"Start":"07:16.160 ","End":"07:21.140","Text":"u squared cancels with u squared and v over v"},{"Start":"07:21.140 ","End":"07:26.600","Text":"squared is just 1 over v. The product is a plus,"},{"Start":"07:26.600 ","End":"07:30.950","Text":"but I\u0027m subtracting, so I\u0027m subtracting 1 over v. Altogether I get"},{"Start":"07:30.950 ","End":"07:37.540","Text":"that the Jacobian is 1 over v. Now I can plug that in here,"},{"Start":"07:37.540 ","End":"07:45.169","Text":"so 1 over v. Remember I said we\u0027re assuming that x and y are both positive,"},{"Start":"07:45.169 ","End":"07:48.150","Text":"so the absolute value doesn\u0027t matter."},{"Start":"07:48.760 ","End":"07:51.980","Text":"But if x and y are positive,"},{"Start":"07:51.980 ","End":"07:55.595","Text":"then so are u and v and so anyway,"},{"Start":"07:55.595 ","End":"07:57.530","Text":"we can drop the absolute value."},{"Start":"07:57.530 ","End":"08:01.885","Text":"We just write 1 over v. Now let\u0027s see what cancels."},{"Start":"08:01.885 ","End":"08:04.560","Text":"v goes into v squared v times,"},{"Start":"08:04.560 ","End":"08:07.125","Text":"so I\u0027ll just do it like that."},{"Start":"08:07.125 ","End":"08:10.375","Text":"What am I left with? v over u squared."},{"Start":"08:10.375 ","End":"08:12.680","Text":"Now if I\u0027m doing the integral,"},{"Start":"08:12.680 ","End":"08:14.865","Text":"the first integral is dv,"},{"Start":"08:14.865 ","End":"08:16.970","Text":"u is a constant."},{"Start":"08:16.970 ","End":"08:23.450","Text":"So I can actually write pulling the 1 over u squared out."},{"Start":"08:23.450 ","End":"08:26.225","Text":"I\u0027ve got 1 over u squared,"},{"Start":"08:26.225 ","End":"08:32.700","Text":"integral of just v dv du the limits,"},{"Start":"08:32.700 ","End":"08:35.535","Text":"u goes from 1-2,"},{"Start":"08:35.535 ","End":"08:37.120","Text":"v goes from 1-2."},{"Start":"08:37.120 ","End":"08:38.510","Text":"You don\u0027t have to do this,"},{"Start":"08:38.510 ","End":"08:40.400","Text":"but I like to simplify."},{"Start":"08:40.400 ","End":"08:47.755","Text":"Forgot an expression multiplied and it\u0027s all in terms of u without v, I\u0027ll pull it out."},{"Start":"08:47.755 ","End":"08:51.530","Text":"As usual, we\u0027ll do the inside integral first,"},{"Start":"08:51.530 ","End":"08:53.800","Text":"the dv integral this time."},{"Start":"08:53.800 ","End":"08:56.839","Text":"I prefer to do this as a side exercise."},{"Start":"08:56.839 ","End":"08:59.015","Text":"You want to have some room over here."},{"Start":"08:59.015 ","End":"09:04.580","Text":"Integral from 1-2 of v dv is equal"},{"Start":"09:04.580 ","End":"09:11.330","Text":"to the integral of v is a half v squared from 1-2."},{"Start":"09:11.330 ","End":"09:16.129","Text":"If I plug in 2, I get 2 squared is 4 over 2 is 2."},{"Start":"09:16.129 ","End":"09:18.890","Text":"If I plug in 1, I get a half,"},{"Start":"09:18.890 ","End":"09:24.930","Text":"2 minus half is 1 and half or 3 over 2, whichever."},{"Start":"09:24.930 ","End":"09:29.720","Text":"This bit here that is highlighted is 3 over 2."},{"Start":"09:29.720 ","End":"09:33.720","Text":"It\u0027s a constant, so I can pull that in front,"},{"Start":"09:33.770 ","End":"09:42.540","Text":"and so we get that this equals 3 over 2 in front."},{"Start":"09:42.540 ","End":"09:50.620","Text":"The integral from 1-2 of 1 over u squared du."},{"Start":"09:53.090 ","End":"09:58.420","Text":"What is the integral of 1 over u squared du?"},{"Start":"09:58.910 ","End":"10:01.310","Text":"The derivative of 1 over u,"},{"Start":"10:01.310 ","End":"10:03.200","Text":"we know is minus 1 over u squared."},{"Start":"10:03.200 ","End":"10:08.570","Text":"So this is just going to be minus 1 over u. I say this is"},{"Start":"10:08.570 ","End":"10:14.975","Text":"3 over 2 times minus 1 over u from 1-2."},{"Start":"10:14.975 ","End":"10:19.310","Text":"If you\u0027re not sure, just do it with negative exponents, u^minus 2."},{"Start":"10:19.310 ","End":"10:21.005","Text":"Raise the power by 1,"},{"Start":"10:21.005 ","End":"10:23.430","Text":"u^minus 1 over minus 1,"},{"Start":"10:23.430 ","End":"10:25.440","Text":"you get the same thing."},{"Start":"10:25.440 ","End":"10:29.050","Text":"What does this come out to?"},{"Start":"10:31.130 ","End":"10:33.915","Text":"You got 3 over 2,"},{"Start":"10:33.915 ","End":"10:36.585","Text":"and I can keep that outside."},{"Start":"10:36.585 ","End":"10:41.765","Text":"If I plug in 2, I\u0027ve got minus a half."},{"Start":"10:41.765 ","End":"10:44.510","Text":"If I plug in 1,"},{"Start":"10:44.510 ","End":"10:50.854","Text":"I\u0027ve got minus 1 over 1 is minus 1."},{"Start":"10:50.854 ","End":"10:53.730","Text":"But there\u0027s a subtraction here."},{"Start":"10:54.160 ","End":"10:59.150","Text":"Minus a half, plus 1 is a half."},{"Start":"10:59.150 ","End":"11:10.215","Text":"So this thing is equal to 3 over 2."},{"Start":"11:10.215 ","End":"11:12.675","Text":"I might as well continue over here."},{"Start":"11:12.675 ","End":"11:23.830","Text":"As I said, 3 over 2 times the integral."},{"Start":"11:23.840 ","End":"11:26.580","Text":"Ignore that bit."},{"Start":"11:26.580 ","End":"11:29.130","Text":"Yeah, 3 over 2 times a half,"},{"Start":"11:29.130 ","End":"11:31.690","Text":"we figured this out as."},{"Start":"11:31.820 ","End":"11:36.270","Text":"We can just write 3 quarters."},{"Start":"11:36.270 ","End":"11:41.740","Text":"This is the answer. We\u0027re done."}],"ID":8673},{"Watched":false,"Name":"Exercise 7","Duration":"15m 1s","ChapterTopicVideoID":8458,"CourseChapterTopicPlaylistID":82673,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8458.jpeg","UploadDate":"2017-02-13T02:05:04.7500000","DurationForVideoObject":"PT15M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this exercise, we need to compute the double integral of"},{"Start":"00:03.870 ","End":"00:09.390","Text":"this function over region R. What I\u0027m interested in,"},{"Start":"00:09.390 ","End":"00:12.735","Text":"especially is the region R. What does this even mean?"},{"Start":"00:12.735 ","End":"00:15.240","Text":"It means the set of pairs x and y"},{"Start":"00:15.240 ","End":"00:20.880","Text":"such that absolute value of x plus absolute value of y less than or equal to 1."},{"Start":"00:20.880 ","End":"00:23.590","Text":"Now, what does this look like?"},{"Start":"00:23.600 ","End":"00:26.625","Text":"I prefer to do it with a sketch."},{"Start":"00:26.625 ","End":"00:28.800","Text":"It\u0027s possible to do it without a sketch."},{"Start":"00:28.800 ","End":"00:32.370","Text":"You could mechanically just say,"},{"Start":"00:32.370 ","End":"00:37.125","Text":"absolute value of x is either plus or minus x and this is plus or minus y."},{"Start":"00:37.125 ","End":"00:41.670","Text":"Then you could get 4 equations for the boundary,"},{"Start":"00:41.670 ","End":"00:44.170","Text":"x plus y equals 1,"},{"Start":"00:44.900 ","End":"00:49.014","Text":"x minus y equals 1,"},{"Start":"00:49.014 ","End":"00:52.070","Text":"minus x plus y equals 1,"},{"Start":"00:52.070 ","End":"00:55.745","Text":"and minus x minus y equals 1."},{"Start":"00:55.745 ","End":"01:02.325","Text":"Then figure out which side of the line you want."},{"Start":"01:02.325 ","End":"01:05.405","Text":"Anyway, it\u0027s possible to do it without a sketch."},{"Start":"01:05.405 ","End":"01:07.355","Text":"I would like to do it with a sketch."},{"Start":"01:07.355 ","End":"01:09.875","Text":"I\u0027ll also get these 4 equations,"},{"Start":"01:09.875 ","End":"01:12.440","Text":"but I\u0027d like to be a bit methodical so we see we\u0027re taking"},{"Start":"01:12.440 ","End":"01:15.770","Text":"the right side of each line in the inequality."},{"Start":"01:15.770 ","End":"01:19.460","Text":"Now, I\u0027m going to separate the 4 quadrants."},{"Start":"01:19.460 ","End":"01:21.005","Text":"In the first quadrant,"},{"Start":"01:21.005 ","End":"01:23.150","Text":"x and y are both positive."},{"Start":"01:23.150 ","End":"01:25.100","Text":"In the first quadrant,"},{"Start":"01:25.100 ","End":"01:27.125","Text":"we\u0027re going to take the first equation,"},{"Start":"01:27.125 ","End":"01:30.830","Text":"x plus y equals 1."},{"Start":"01:30.830 ","End":"01:33.470","Text":"In the fourth quadrant here,"},{"Start":"01:33.470 ","End":"01:35.180","Text":"x is still positive,"},{"Start":"01:35.180 ","End":"01:37.820","Text":"but y is negative and when y is negative,"},{"Start":"01:37.820 ","End":"01:39.500","Text":"this is minus y."},{"Start":"01:39.500 ","End":"01:44.020","Text":"Here we get x minus y equals 1."},{"Start":"01:44.020 ","End":"01:46.715","Text":"I\u0027ll sketch the lines in a moment."},{"Start":"01:46.715 ","End":"01:50.255","Text":"In the third quadrant,"},{"Start":"01:50.255 ","End":"01:52.735","Text":"no, lets take the second quadrant."},{"Start":"01:52.735 ","End":"01:57.830","Text":"Here, x is negative but y is positive."},{"Start":"01:57.830 ","End":"02:04.405","Text":"If x is negative, we take minus x as the absolute value, but plus y."},{"Start":"02:04.405 ","End":"02:07.999","Text":"In this quadrant both x and y are negative."},{"Start":"02:07.999 ","End":"02:13.765","Text":"I need to take minus x minus y is equal to 1."},{"Start":"02:13.765 ","End":"02:15.260","Text":"Now, if you plot these,"},{"Start":"02:15.260 ","End":"02:17.600","Text":"x plus y equals 1, in general,"},{"Start":"02:17.600 ","End":"02:24.525","Text":"x plus y equals A goes from A on the x-axis to A on the y-axis,"},{"Start":"02:24.525 ","End":"02:29.080","Text":"45 degree line actually continues."},{"Start":"02:29.660 ","End":"02:34.040","Text":"This one. X minus y equals 1."},{"Start":"02:34.040 ","End":"02:41.060","Text":"If you check, you will find it\u0027s a 45 degree line because y is equal to x minus 1,"},{"Start":"02:41.060 ","End":"02:44.570","Text":"it\u0027s the 45 degree line subtract lower 1."},{"Start":"02:44.570 ","End":"02:47.540","Text":"I\u0027m not going to do too many computations,"},{"Start":"02:47.540 ","End":"02:49.640","Text":"but you get this line here."},{"Start":"02:49.640 ","End":"02:54.890","Text":"The minus x minus y equals 1 is like x plus y is minus 1."},{"Start":"02:54.890 ","End":"02:59.140","Text":"That one comes out something like this."},{"Start":"02:59.140 ","End":"03:04.450","Text":"Minus x plus y equals 1 comes out like this."},{"Start":"03:04.450 ","End":"03:09.070","Text":"It\u0027s a bit crooked, but this is supposed to be 1 minus 1,"},{"Start":"03:09.070 ","End":"03:13.030","Text":"1 and minus 1."},{"Start":"03:13.040 ","End":"03:17.715","Text":"This one is, I didn\u0027t label which is which,"},{"Start":"03:17.715 ","End":"03:22.370","Text":"but well, each equation goes with the one in it\u0027s quadrant."},{"Start":"03:22.370 ","End":"03:28.730","Text":"This one goes with this because this bit is in this quadrant and so on."},{"Start":"03:28.730 ","End":"03:31.010","Text":"This is equation goes with this line."},{"Start":"03:31.010 ","End":"03:33.410","Text":"This equation with this line, this one this line."},{"Start":"03:33.410 ","End":"03:36.565","Text":"If we get the same ones I\u0027m going to erase this."},{"Start":"03:36.565 ","End":"03:39.080","Text":"Now, we didn\u0027t have equals 1,"},{"Start":"03:39.080 ","End":"03:41.885","Text":"we had less than or equals to 1."},{"Start":"03:41.885 ","End":"03:44.825","Text":"Really I should be having less than or equal to,"},{"Start":"03:44.825 ","End":"03:46.370","Text":"less than or equal to,"},{"Start":"03:46.370 ","End":"03:48.035","Text":"less than or equal to."},{"Start":"03:48.035 ","End":"03:51.245","Text":"Whenever you have one of these less than or equal to is you find that where it\u0027s"},{"Start":"03:51.245 ","End":"03:54.440","Text":"equal and then it\u0027s 1/2 plane of the other."},{"Start":"03:54.440 ","End":"03:56.990","Text":"It divides the plane into 2 parts."},{"Start":"03:56.990 ","End":"04:00.035","Text":"I\u0027m claiming it\u0027s always towards the center because the"},{"Start":"04:00.035 ","End":"04:06.735","Text":"0.00 certainly satisfies this equation,"},{"Start":"04:06.735 ","End":"04:09.760","Text":"if I put in 00,"},{"Start":"04:09.760 ","End":"04:13.700","Text":"I\u0027ll get 0, which is less than or equal to 1."},{"Start":"04:13.700 ","End":"04:16.550","Text":"In each case, whatever the expression is on the left,"},{"Start":"04:16.550 ","End":"04:19.760","Text":"I get 0 and it will be less than or equal to."},{"Start":"04:19.760 ","End":"04:21.965","Text":"It\u0027s always on this side of the line,"},{"Start":"04:21.965 ","End":"04:23.375","Text":"this side of the line,"},{"Start":"04:23.375 ","End":"04:24.650","Text":"this side of the line,"},{"Start":"04:24.650 ","End":"04:26.150","Text":"this side of the line."},{"Start":"04:26.150 ","End":"04:33.360","Text":"I\u0027m just justifying to say that it is this square which you would have figured anyway."},{"Start":"04:38.080 ","End":"04:41.525","Text":"I know this is a terrible sketch,"},{"Start":"04:41.525 ","End":"04:43.340","Text":"but we\u0027re studying math,"},{"Start":"04:43.340 ","End":"04:46.355","Text":"not art, and I think this will do."},{"Start":"04:46.355 ","End":"04:48.650","Text":"Let\u0027s give it a name."},{"Start":"04:48.650 ","End":"04:52.920","Text":"Well, that name will be R. That\u0027s this."},{"Start":"04:53.630 ","End":"04:56.945","Text":"Now, what is the substitution?"},{"Start":"04:56.945 ","End":"05:05.345","Text":"Notice that I can rewrite each of these to have either x plus y or x minus y."},{"Start":"05:05.345 ","End":"05:07.385","Text":"For example, the first one,"},{"Start":"05:07.385 ","End":"05:10.430","Text":"I\u0027ll write it as x plus y equals 1."},{"Start":"05:10.430 ","End":"05:14.495","Text":"I\u0027m talking about these lines. Let\u0027s see."},{"Start":"05:14.495 ","End":"05:19.025","Text":"If I go to this one and I multiply by minus 1,"},{"Start":"05:19.025 ","End":"05:24.305","Text":"I\u0027ll get x plus y equals minus 1."},{"Start":"05:24.305 ","End":"05:28.080","Text":"Again, I have an x plus y both here and here."},{"Start":"05:29.870 ","End":"05:33.035","Text":"For these 2, I\u0027ll get,"},{"Start":"05:33.035 ","End":"05:39.290","Text":"if I take this one as is I have x minus y equals 1."},{"Start":"05:39.290 ","End":"05:42.295","Text":"If I take this one and multiply by minus 1,"},{"Start":"05:42.295 ","End":"05:46.875","Text":"I get x minus y equals minus 1."},{"Start":"05:46.875 ","End":"05:53.945","Text":"All this really is begging us to substitute x plus y for something,"},{"Start":"05:53.945 ","End":"05:56.215","Text":"and let\u0027s call that u."},{"Start":"05:56.215 ","End":"06:02.285","Text":"For the other one, we want another letter v to be x minus y."},{"Start":"06:02.285 ","End":"06:05.075","Text":"This will be our substitution."},{"Start":"06:05.075 ","End":"06:07.670","Text":"If we do this,"},{"Start":"06:07.670 ","End":"06:12.635","Text":"this one will become u equals 1,"},{"Start":"06:12.635 ","End":"06:15.640","Text":"u equals minus 1."},{"Start":"06:15.640 ","End":"06:19.130","Text":"From here, v equals 1,"},{"Start":"06:19.130 ","End":"06:21.605","Text":"v equals minus 1."},{"Start":"06:21.605 ","End":"06:25.490","Text":"In fact, if you kept track of the inequalities,"},{"Start":"06:25.490 ","End":"06:29.455","Text":"these 2 we kept as is."},{"Start":"06:29.455 ","End":"06:33.080","Text":"If you were worrying about inequalities,"},{"Start":"06:33.080 ","End":"06:36.485","Text":"this one was a less than or equal to,"},{"Start":"06:36.485 ","End":"06:40.115","Text":"and this one was a less than or equal,"},{"Start":"06:40.115 ","End":"06:44.870","Text":"where was it, x minus y equals 1."},{"Start":"06:44.870 ","End":"06:47.225","Text":"This one was the less than or equal to."},{"Start":"06:47.225 ","End":"06:50.735","Text":"These 2 we multiplied by minus 1."},{"Start":"06:50.735 ","End":"06:54.515","Text":"We need to reverse the direction of the inequality."},{"Start":"06:54.515 ","End":"06:57.290","Text":"This is getting technical normally wouldn\u0027t do this."},{"Start":"06:57.290 ","End":"07:00.275","Text":"You would just see. I have a 1 and a minus 1."},{"Start":"07:00.275 ","End":"07:05.300","Text":"I\u0027m just going to assume that the domain is minus 1 less than or equal to u,"},{"Start":"07:05.300 ","End":"07:07.580","Text":"less than or equal to 1, and you\u0027d get away with it."},{"Start":"07:07.580 ","End":"07:10.865","Text":"But I was just trying to show you that it really does work the right way."},{"Start":"07:10.865 ","End":"07:12.380","Text":"It\u0027s less than or equal to 1."},{"Start":"07:12.380 ","End":"07:14.720","Text":"It is bigger or equal to minus 1."},{"Start":"07:14.720 ","End":"07:17.725","Text":"Because when you multiply an inequality by minus,"},{"Start":"07:17.725 ","End":"07:20.940","Text":"any negative number, you reverse it."},{"Start":"07:20.940 ","End":"07:26.445","Text":"Similarly, v is also between minus 1 and 1."},{"Start":"07:26.445 ","End":"07:31.160","Text":"This becomes a nice little square in the plane."},{"Start":"07:31.160 ","End":"07:33.490","Text":"We don\u0027t have to sketch it,"},{"Start":"07:33.490 ","End":"07:36.120","Text":"but I feel like doing it anyway."},{"Start":"07:36.120 ","End":"07:38.205","Text":"I think it\u0027s time for a good sketch."},{"Start":"07:38.205 ","End":"07:49.050","Text":"u, v, c_1 minus 1,1 minus 1 Even label them minus 1,1,"},{"Start":"07:49.050 ","End":"07:54.240","Text":"minus 1, 1. u is between minus 1 and 1,"},{"Start":"07:54.240 ","End":"07:58.055","Text":"so it\u0027s between these bars."},{"Start":"07:58.055 ","End":"08:01.900","Text":"V is between minus 1 and 1,"},{"Start":"08:01.900 ","End":"08:05.125","Text":"so it\u0027s between here and here."},{"Start":"08:05.125 ","End":"08:08.920","Text":"I\u0027ll just shade it a bit."},{"Start":"08:08.920 ","End":"08:14.270","Text":"This is going to be our new region, nice and rectangular."},{"Start":"08:14.640 ","End":"08:17.320","Text":"Now, we\u0027ve done several of these before,"},{"Start":"08:17.320 ","End":"08:20.410","Text":"and I know that we always come to a point where we need to"},{"Start":"08:20.410 ","End":"08:24.010","Text":"have not u and v in terms of x and y,"},{"Start":"08:24.010 ","End":"08:27.340","Text":"but x and y in terms of u and v. Might as well"},{"Start":"08:27.340 ","End":"08:31.210","Text":"do that technical part already and then start with the substituting."},{"Start":"08:31.210 ","End":"08:33.520","Text":"In brief, we need it in 2 places."},{"Start":"08:33.520 ","End":"08:38.695","Text":"First of all, we need to figure out what is x and what is y here."},{"Start":"08:38.695 ","End":"08:44.890","Text":"Although we could also get by with saying it straight away that this is u,"},{"Start":"08:44.890 ","End":"08:46.690","Text":"but normally you would need to."},{"Start":"08:46.690 ","End":"08:49.824","Text":"Also, we\u0027re going to have a Jacobian, if you remember,"},{"Start":"08:49.824 ","End":"08:56.395","Text":"and for that we will need x and y in terms of u and v. From these 2,"},{"Start":"08:56.395 ","End":"08:58.900","Text":"let me do it over here."},{"Start":"08:58.900 ","End":"09:01.840","Text":"If I add these 2 equations,"},{"Start":"09:01.840 ","End":"09:10.420","Text":"I will get that u plus v is equal to x plus y plus x minus y."},{"Start":"09:10.420 ","End":"09:14.005","Text":"That is just 2x."},{"Start":"09:14.005 ","End":"09:23.560","Text":"That gives us that x equals 1.5 u plus 1.5 v. Also,"},{"Start":"09:23.560 ","End":"09:25.630","Text":"if we subtract the equations,"},{"Start":"09:25.630 ","End":"09:27.820","Text":"doing u minus v,"},{"Start":"09:27.820 ","End":"09:29.934","Text":"we\u0027ll get x minus x cancels."},{"Start":"09:29.934 ","End":"09:33.550","Text":"Y minus minus y is 2y."},{"Start":"09:33.550 ","End":"09:43.015","Text":"This gives us that y equals 1.5 u minus 1.5 v. Now,"},{"Start":"09:43.015 ","End":"09:49.435","Text":"we\u0027re just about ready almost to do the change of variables,"},{"Start":"09:49.435 ","End":"09:51.595","Text":"but that\u0027s at least start."},{"Start":"09:51.595 ","End":"09:54.490","Text":"We get the double integral."},{"Start":"09:54.490 ","End":"09:56.590","Text":"Now we don\u0027t have this R anymore."},{"Start":"09:56.590 ","End":"09:58.075","Text":"We have the new region."},{"Start":"09:58.075 ","End":"09:59.395","Text":"I don\u0027t know what it\u0027s called,"},{"Start":"09:59.395 ","End":"10:02.065","Text":"S, but it\u0027s rectangular."},{"Start":"10:02.065 ","End":"10:04.405","Text":"I\u0027m going to do it as an iterated integral."},{"Start":"10:04.405 ","End":"10:06.370","Text":"It doesn\u0027t really matter which order,"},{"Start":"10:06.370 ","End":"10:14.575","Text":"I\u0027ll take the outer loop as u from minus 1 to 1 and so it\u0027ll end in du."},{"Start":"10:14.575 ","End":"10:18.700","Text":"The inner loop for each u we\u0027ll take a vertical slice from"},{"Start":"10:18.700 ","End":"10:23.395","Text":"minus 1 to 1 of v minus 1 to 1,"},{"Start":"10:23.395 ","End":"10:26.845","Text":"and so there\u0027ll be a dv for the inner loop."},{"Start":"10:26.845 ","End":"10:30.610","Text":"Then we need the e to the x plus y."},{"Start":"10:30.610 ","End":"10:34.450","Text":"We don\u0027t have to substitute x and y from here though you could,"},{"Start":"10:34.450 ","End":"10:35.875","Text":"if you added x plus y,"},{"Start":"10:35.875 ","End":"10:37.750","Text":"you would get this plus this is u,"},{"Start":"10:37.750 ","End":"10:39.730","Text":"or you could have just copied it straight"},{"Start":"10:39.730 ","End":"10:42.250","Text":"away from here if you\u0027d notice that x plus y is u,"},{"Start":"10:42.250 ","End":"10:44.350","Text":"either way it\u0027s e_ u."},{"Start":"10:44.350 ","End":"10:50.125","Text":"The missing bit is the absolute value of the Jacobian."},{"Start":"10:50.125 ","End":"10:53.125","Text":"That\u0027s the missing piece of the puzzle."},{"Start":"10:53.125 ","End":"10:57.160","Text":"Now the Jacobian, J it\u0027s called,"},{"Start":"10:57.160 ","End":"11:01.570","Text":"I will remind you, is the determinant of a 2-by-2 matrix,"},{"Start":"11:01.570 ","End":"11:06.880","Text":"which is the partial derivatives of x and y in terms of u and v. In short,"},{"Start":"11:06.880 ","End":"11:08.770","Text":"it\u0027s x with respect to u,"},{"Start":"11:08.770 ","End":"11:10.375","Text":"x with respect to v,"},{"Start":"11:10.375 ","End":"11:19.810","Text":"y by u, y by v. Some people tell me they forgot what a determinant is."},{"Start":"11:19.810 ","End":"11:22.240","Text":"I\u0027ll remind you of that also."},{"Start":"11:22.240 ","End":"11:27.010","Text":"Please the 2-by-2 be determinant where you have 4 numbers here, a, b, c,"},{"Start":"11:27.010 ","End":"11:28.540","Text":"and d. In this case,"},{"Start":"11:28.540 ","End":"11:32.590","Text":"the determinant comes out to be and you can take it as a definition,"},{"Start":"11:32.590 ","End":"11:35.260","Text":"as a d minus b,"},{"Start":"11:35.260 ","End":"11:39.340","Text":"c. Back here,"},{"Start":"11:39.340 ","End":"11:43.690","Text":"let\u0027s compute this Jacobian so we can put it in here and finally get this integral done."},{"Start":"11:43.690 ","End":"11:48.505","Text":"X with respect to u. I\u0027m looking over here now all the time."},{"Start":"11:48.505 ","End":"11:54.350","Text":"Let me just scroll down a bit and let\u0027s make place for it."},{"Start":"11:55.110 ","End":"11:59.200","Text":"X with respect to u is a /2,"},{"Start":"11:59.200 ","End":"12:02.530","Text":"and with respect to v, it\u0027s also a 1/2."},{"Start":"12:02.530 ","End":"12:05.410","Text":"Y with respect to u is a 1/2,"},{"Start":"12:05.410 ","End":"12:09.805","Text":"y with respect to v minus a /2."},{"Start":"12:09.805 ","End":"12:11.365","Text":"Remember what we said,"},{"Start":"12:11.365 ","End":"12:14.485","Text":"this diagonal minus this diagonal."},{"Start":"12:14.485 ","End":"12:24.970","Text":"So it\u0027s a 1/2 times minus a 11/2 is minus a 1/4 less this times this is a 1/4."},{"Start":"12:24.970 ","End":"12:30.370","Text":"We end up with being just minus a 1/2,"},{"Start":"12:30.370 ","End":"12:32.035","Text":"and that\u0027s my J."},{"Start":"12:32.035 ","End":"12:34.690","Text":"But here I want the absolute value of J."},{"Start":"12:34.690 ","End":"12:36.205","Text":"So instead of this,"},{"Start":"12:36.205 ","End":"12:39.040","Text":"I\u0027ll put in plus 1/2."},{"Start":"12:39.040 ","End":"12:42.655","Text":"Now we have everything we need to compute this integral."},{"Start":"12:42.655 ","End":"12:44.740","Text":"It\u0027s just technical."},{"Start":"12:44.740 ","End":"12:47.785","Text":"The 1/2 I can put in front."},{"Start":"12:47.785 ","End":"12:50.695","Text":"Why don\u0027t I continue where I have room over here."},{"Start":"12:50.695 ","End":"12:53.845","Text":"Let\u0027s go down there, here,"},{"Start":"12:53.845 ","End":"13:00.760","Text":"we get the integral u goes from minus 1 to 1."},{"Start":"13:00.760 ","End":"13:05.485","Text":"Then look the integral of v,"},{"Start":"13:05.485 ","End":"13:06.819","Text":"u is a constant,"},{"Start":"13:06.819 ","End":"13:10.105","Text":"so I can take this bit in front,"},{"Start":"13:10.105 ","End":"13:12.865","Text":"so I can write the e_u here,"},{"Start":"13:12.865 ","End":"13:18.985","Text":"and I\u0027ve got the integral of just J, which is a half."},{"Start":"13:18.985 ","End":"13:23.245","Text":"But I\u0027ll pull the half in front of the whole thing because it\u0027s a constant."},{"Start":"13:23.245 ","End":"13:31.730","Text":"All we\u0027re left with is dv/du."},{"Start":"13:33.480 ","End":"13:39.355","Text":"This is really the limits,"},{"Start":"13:39.355 ","End":"13:44.125","Text":"v equals minus 1 to 1."},{"Start":"13:44.125 ","End":"13:48.595","Text":"We start on the innermost integral, which is this."},{"Start":"13:48.595 ","End":"13:50.845","Text":"We can do this one in our heads."},{"Start":"13:50.845 ","End":"13:54.880","Text":"The integral of the function 1 from any 2 limits,"},{"Start":"13:54.880 ","End":"13:56.800","Text":"is just the upper minus the lower."},{"Start":"13:56.800 ","End":"14:01.210","Text":"So it\u0027s 1 minus, minus 1."},{"Start":"14:01.210 ","End":"14:05.920","Text":"I want 1 minus minus 1, which is 2."},{"Start":"14:05.920 ","End":"14:09.010","Text":"This thing comes out to be 2,"},{"Start":"14:09.010 ","End":"14:11.845","Text":"the bit that\u0027s highlighted."},{"Start":"14:11.845 ","End":"14:16.870","Text":"Now this 2 can be put in front with the half and it will cancel,"},{"Start":"14:16.870 ","End":"14:21.115","Text":"or I can just put a line through this and a line through this."},{"Start":"14:21.115 ","End":"14:24.565","Text":"In fact, this is the whole middle bit."},{"Start":"14:24.565 ","End":"14:30.340","Text":"All I\u0027m left with is the integral"},{"Start":"14:30.340 ","End":"14:36.295","Text":"from minus 1 to 1 of e_u du."},{"Start":"14:36.295 ","End":"14:40.045","Text":"The integral of e_u is e_u itself."},{"Start":"14:40.045 ","End":"14:45.430","Text":"So I\u0027ll just have to evaluate e_u minus 1 to 1."},{"Start":"14:45.430 ","End":"14:49.990","Text":"Upper minus lower, e_1 is e,"},{"Start":"14:49.990 ","End":"14:57.310","Text":"e_minus 1 is 1 over e. This looks like the final answer,"},{"Start":"14:57.310 ","End":"15:01.490","Text":"and so I highlight it and declare we are done."}],"ID":8674}],"Thumbnail":null,"ID":82673}]

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