Exercises
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- Exercise 1 Part a
- Exercise 1 Part b
- Exercise 1 Part c
- Exercise 1 Part d
- Exercise 2 Part a
- Exercise 2 Part b
- Exercise 3
- Exercise 4
- Exercise 5 Part a
- Exercise 5 Part b
- Exercise 6 Part a
- Exercise 6 Part b
- Exercise 7 Part a
- Exercise 7 Part b
- Exercise 7 Part c
- Exercise 7 Part d
- Exercise 8
- Exercise 9
- Exercise 10 Part a
- Exercise 10 Part b
- Exercise 11 Part a
- Exercise 11 Part b
- Exercise 12 Part a
- Exercise 12 Part b
- Exercise 13

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[{"Name":"Exercises","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 Part a","Duration":"4m 17s","ChapterTopicVideoID":8709,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8709.jpeg","UploadDate":"2017-02-13T05:29:00.2070000","DurationForVideoObject":"PT4M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.455","Text":"In this exercise, we have to compute this type 1 line integral along the curve C,"},{"Start":"00:07.455 ","End":"00:10.650","Text":"and C is given in parametric form."},{"Start":"00:10.650 ","End":"00:16.275","Text":"There are several formulas of variations of the formula we can use."},{"Start":"00:16.275 ","End":"00:25.630","Text":"The simplest is that ds is just the square root of dx squared plus dy squared."},{"Start":"00:25.630 ","End":"00:28.640","Text":"But that\u0027s a bit awkward to work with,"},{"Start":"00:28.640 ","End":"00:35.420","Text":"and so usually, we have dx over dt squared,"},{"Start":"00:35.420 ","End":"00:43.500","Text":"plus dy over dt squared, dt."},{"Start":"00:43.500 ","End":"00:48.400","Text":"It\u0027s like the dt squared square root cancels out with the dt."},{"Start":"00:48.920 ","End":"00:51.630","Text":"This is the Leibniz form."},{"Start":"00:51.630 ","End":"00:54.880","Text":"Those who don\u0027t like the dx, dt notation,"},{"Start":"00:54.880 ","End":"00:59.805","Text":"could always use the formula x-prime of t"},{"Start":"00:59.805 ","End":"01:10.450","Text":"squared plus y-prime of t squared dt."},{"Start":"01:11.300 ","End":"01:13.470","Text":"Not to be too confusing,"},{"Start":"01:13.470 ","End":"01:14.700","Text":"let me stick with 1 of them."},{"Start":"01:14.700 ","End":"01:19.170","Text":"Let me choose the last 1. I\u0027ll move it here."},{"Start":"01:19.170 ","End":"01:21.260","Text":"After we\u0027ve computed ds,"},{"Start":"01:21.260 ","End":"01:25.760","Text":"we also have to remember that this integral becomes an integral with"},{"Start":"01:25.760 ","End":"01:32.700","Text":"respect to t. We take it that t goes from 0 to 2Pi."},{"Start":"01:32.700 ","End":"01:37.190","Text":"We also substitute where we have x and y,"},{"Start":"01:37.190 ","End":"01:39.230","Text":"we don\u0027t have a y, we just have an x."},{"Start":"01:39.230 ","End":"01:43.805","Text":"But in this case, we have 1 minus x squared is cosine"},{"Start":"01:43.805 ","End":"01:49.905","Text":"squared t. Then I have to put ds in."},{"Start":"01:49.905 ","End":"01:52.590","Text":"Let\u0027s compute ds and then get back here."},{"Start":"01:52.590 ","End":"01:54.420","Text":"This has to be continued."},{"Start":"01:54.420 ","End":"01:57.270","Text":"This is the square root."},{"Start":"01:57.270 ","End":"02:07.680","Text":"Now, the derivative of x with respect to t is minus sine t, and that\u0027s squared."},{"Start":"02:07.680 ","End":"02:16.480","Text":"The derivative of y with respect to t is cosine t, also squared, dt."},{"Start":"02:16.480 ","End":"02:21.415","Text":"Now, sine squared plus cosine squared is 1."},{"Start":"02:21.415 ","End":"02:27.270","Text":"This comes out to be just 1dt, which is dt."},{"Start":"02:27.270 ","End":"02:31.720","Text":"Ds comes out to be dt."},{"Start":"02:32.290 ","End":"02:37.520","Text":"Now, we can use some trigonometrical identities to solve this."},{"Start":"02:37.520 ","End":"02:48.090","Text":"First of all, it\u0027s equal to the integral 1 minus cosine squared is sine squared t dt."},{"Start":"02:48.090 ","End":"02:51.770","Text":"Then as another trigonometrical identity,"},{"Start":"02:51.770 ","End":"02:58.800","Text":"that sine squared is 1/2 of 1 minus cosine twice the angle,"},{"Start":"02:58.800 ","End":"03:02.350","Text":"in this case, 2t dt."},{"Start":"03:02.360 ","End":"03:06.310","Text":"Now, we can actually do the integral."},{"Start":"03:06.620 ","End":"03:10.290","Text":"Let\u0027s leave the 1/2 outside."},{"Start":"03:10.290 ","End":"03:16.580","Text":"What we have is the integral of 1 is t. The"},{"Start":"03:16.580 ","End":"03:22.790","Text":"integral of cosine 2t is not quite sine 2t,"},{"Start":"03:22.790 ","End":"03:25.710","Text":"we have to divide by the 2."},{"Start":"03:26.120 ","End":"03:33.480","Text":"All this has to be taken from 0 to 2Pi."},{"Start":"03:33.480 ","End":"03:36.940","Text":"What does this come out to be?"},{"Start":"03:37.580 ","End":"03:40.410","Text":"I\u0027ll leave the 1/2 here."},{"Start":"03:40.410 ","End":"03:43.260","Text":"Now, if I put in 2Pi,"},{"Start":"03:43.260 ","End":"03:51.825","Text":"I get 2Pi minus sine of 4Pi,"},{"Start":"03:51.825 ","End":"03:54.030","Text":"it\u0027s a multiple of 2Pi,"},{"Start":"03:54.030 ","End":"03:56.190","Text":"so it\u0027s like 0."},{"Start":"03:56.190 ","End":"04:00.165","Text":"This would be 0. That\u0027s for the 2Pi,"},{"Start":"04:00.165 ","End":"04:07.910","Text":"and now for 0, I just get 0 minus sine of 0 is 0."},{"Start":"04:07.910 ","End":"04:11.575","Text":"Altogether, 1/2 of 2Pi,"},{"Start":"04:11.575 ","End":"04:15.030","Text":"and the answer is just Pi."},{"Start":"04:15.030 ","End":"04:17.980","Text":"We are done."}],"ID":8808},{"Watched":false,"Name":"Exercise 1 Part b","Duration":"13m 50s","ChapterTopicVideoID":8710,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8710.jpeg","UploadDate":"2017-02-13T05:32:00.6470000","DurationForVideoObject":"PT13M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"In this exercise, we have to compute the following line integral,"},{"Start":"00:05.490 ","End":"00:10.440","Text":"where the curve C is given in parametric form, x,"},{"Start":"00:10.440 ","End":"00:16.540","Text":"y in terms of t and the range where t goes from."},{"Start":"00:16.730 ","End":"00:22.350","Text":"There\u0027s only really one basic formula that we have to use."},{"Start":"00:22.350 ","End":"00:27.090","Text":"That is that ds,"},{"Start":"00:27.090 ","End":"00:29.760","Text":"one of the variations of the formula,"},{"Start":"00:29.760 ","End":"00:37.230","Text":"is the square root of x prime as a function of t squared,"},{"Start":"00:37.230 ","End":"00:44.940","Text":"plus y prime as a function of t squared, dt."},{"Start":"00:44.940 ","End":"00:49.130","Text":"Probably, best to put an extra set of brackets here to avoid confusion."},{"Start":"00:49.130 ","End":"00:52.260","Text":"The whole thing here is squared."},{"Start":"00:53.030 ","End":"00:57.295","Text":"Let\u0027s compute this, and then we\u0027ll get back here."},{"Start":"00:57.295 ","End":"01:06.105","Text":"This is equal to the derivative of x"},{"Start":"01:06.105 ","End":"01:11.310","Text":"is 1 minus cosine t squared."},{"Start":"01:11.310 ","End":"01:19.710","Text":"Then the derivative of y is just sine t. 1 disappears,"},{"Start":"01:19.710 ","End":"01:21.900","Text":"it\u0027s a constant and minus cosine."},{"Start":"01:21.900 ","End":"01:25.515","Text":"That\u0027s sine t squared."},{"Start":"01:25.515 ","End":"01:29.530","Text":"Square root of that, dt."},{"Start":"01:30.230 ","End":"01:33.005","Text":"If we simplify this,"},{"Start":"01:33.005 ","End":"01:36.545","Text":"we get the square root of,"},{"Start":"01:36.545 ","End":"01:39.725","Text":"it\u0027s just simple algebra here."},{"Start":"01:39.725 ","End":"01:46.650","Text":"I have 1 minus 2 cosine t plus cosine squared t. I will just write the side,"},{"Start":"01:46.650 ","End":"01:51.645","Text":"1 minus 2 cosine t plus cosine squared t,"},{"Start":"01:51.645 ","End":"01:56.600","Text":"plus sine squared t. The cosine squared and sine squared is 1,"},{"Start":"01:56.600 ","End":"01:58.730","Text":"together with the 1 gives me 2."},{"Start":"01:58.730 ","End":"02:03.320","Text":"I have twice, 2 minus 2 cosine t,"},{"Start":"02:03.320 ","End":"02:09.110","Text":"which I can write as 2 times 1 minus cosine t. That\u0027s ds."},{"Start":"02:09.110 ","End":"02:10.670","Text":"Now getting back to the integral,"},{"Start":"02:10.670 ","End":"02:12.590","Text":"we\u0027ll replace the curve,"},{"Start":"02:12.590 ","End":"02:19.665","Text":"with the range of t. T goes from 0 to Pi."},{"Start":"02:19.665 ","End":"02:22.425","Text":"Then I replace the x,"},{"Start":"02:22.425 ","End":"02:24.570","Text":"as the x from the curve."},{"Start":"02:24.570 ","End":"02:33.010","Text":"That t minus sine t. Then the ds,"},{"Start":"02:35.600 ","End":"02:38.430","Text":"sorry, I forgot the dt here,"},{"Start":"02:38.430 ","End":"02:42.780","Text":"which is the square root of twice,"},{"Start":"02:42.780 ","End":"02:45.490","Text":"1 minus cosine of t, dt."},{"Start":"02:45.490 ","End":"02:49.445","Text":"But I\u0027m going to use a trigonometrical formula here,"},{"Start":"02:49.445 ","End":"02:57.785","Text":"that 1 minus cosine t is 2 sine squared t over 2."},{"Start":"02:57.785 ","End":"02:59.480","Text":"It\u0027s just a variation of one of"},{"Start":"02:59.480 ","End":"03:02.960","Text":"the standard trigonometric formulas after switching sides a bit."},{"Start":"03:02.960 ","End":"03:06.930","Text":"I have here 2 times 2,"},{"Start":"03:06.930 ","End":"03:13.000","Text":"times sine squared t over 2."},{"Start":"03:13.000 ","End":"03:18.770","Text":"Now, what I have under the square root sign is 4 sine squared t over 2."},{"Start":"03:18.770 ","End":"03:24.620","Text":"The square root would be 2 sine t over 2."},{"Start":"03:24.620 ","End":"03:27.440","Text":"But that would normally be an absolute value,"},{"Start":"03:27.440 ","End":"03:29.825","Text":"and you take the square root of something squared."},{"Start":"03:29.825 ","End":"03:34.355","Text":"But when t goes from 0 to Pi,"},{"Start":"03:34.355 ","End":"03:40.680","Text":"t over 2 goes from 0 to Pi over 2."},{"Start":"03:40.680 ","End":"03:42.510","Text":"The sine is positive,"},{"Start":"03:42.510 ","End":"03:45.380","Text":"because that\u0027s the whole first quadrant,"},{"Start":"03:45.380 ","End":"03:46.910","Text":"or at least non-negative."},{"Start":"03:46.910 ","End":"03:48.995","Text":"I don\u0027t need the absolute value."},{"Start":"03:48.995 ","End":"03:53.630","Text":"What I get is the integral from 0 to Pi."},{"Start":"03:53.630 ","End":"03:55.685","Text":"This whole thing now is just,"},{"Start":"03:55.685 ","End":"03:59.429","Text":"this without the absolute value."},{"Start":"04:00.970 ","End":"04:04.970","Text":"If I multiply out this with this,"},{"Start":"04:04.970 ","End":"04:12.785","Text":"we just get twice t minus sine t,"},{"Start":"04:12.785 ","End":"04:17.330","Text":"sine of t over 2, dt."},{"Start":"04:17.330 ","End":"04:23.270","Text":"I\u0027m going to split it up into 2 integrals from the minus."},{"Start":"04:23.270 ","End":"04:25.160","Text":"We get, on the one hand,"},{"Start":"04:25.160 ","End":"04:32.735","Text":"the integral from 0 to Pi of 2t,"},{"Start":"04:32.735 ","End":"04:37.595","Text":"sine of t over 2, dt."},{"Start":"04:37.595 ","End":"04:48.680","Text":"Then we have minus the integral from 0 to Pi of 2 sine t,"},{"Start":"04:48.680 ","End":"04:53.490","Text":"sine t over 2, dt."},{"Start":"04:53.500 ","End":"04:55.970","Text":"I\u0027ll do each one separately,"},{"Start":"04:55.970 ","End":"04:57.685","Text":"and then we\u0027ll do a subtraction."},{"Start":"04:57.685 ","End":"05:00.485","Text":"I just give them names."},{"Start":"05:00.485 ","End":"05:04.950","Text":"I\u0027ll call this one asterisk,"},{"Start":"05:04.950 ","End":"05:11.235","Text":"and I\u0027ll call the second integral double asterisk."},{"Start":"05:11.235 ","End":"05:16.680","Text":"We just have an integration problem now."},{"Start":"05:17.090 ","End":"05:20.340","Text":"That\u0027s all I really need now."},{"Start":"05:20.340 ","End":"05:24.525","Text":"I think I\u0027ll work on a split page."},{"Start":"05:24.525 ","End":"05:34.085","Text":"Also, here I\u0027ll compute the integral of 2t sine t over 2,"},{"Start":"05:34.085 ","End":"05:42.090","Text":"dt, from 0 to Pi."},{"Start":"05:42.090 ","End":"05:52.140","Text":"Then, afterwards, I\u0027ll do later the 0 to Pi of 2 sine t,"},{"Start":"05:52.140 ","End":"05:55.540","Text":"sine t over 2, dt."},{"Start":"05:55.880 ","End":"06:00.815","Text":"At the end we we\u0027ll subtract this one minus this one."},{"Start":"06:00.815 ","End":"06:05.960","Text":"For this one, we\u0027re going to do integration by parts."},{"Start":"06:05.960 ","End":"06:11.645","Text":"To remind you, the integration by parts says the integral of udv,"},{"Start":"06:11.645 ","End":"06:17.930","Text":"is uv, minus the integral of vdu."},{"Start":"06:17.930 ","End":"06:26.880","Text":"I\u0027m going to let the 2t be u."},{"Start":"06:26.880 ","End":"06:36.615","Text":"This one will be the v. The 2 quantities I\u0027m missing are du and v. Well,"},{"Start":"06:36.615 ","End":"06:39.960","Text":"du is simply 2dt,"},{"Start":"06:39.960 ","End":"06:47.555","Text":"and v is the antiderivative of sine t over 2,"},{"Start":"06:47.555 ","End":"06:50.765","Text":"which if you think about it,"},{"Start":"06:50.765 ","End":"06:55.920","Text":"the antiderivative of sine is minus cosine,"},{"Start":"06:58.660 ","End":"07:03.050","Text":"of t over 2, but we have to divide by the 1/2."},{"Start":"07:03.050 ","End":"07:06.375","Text":"Dividing by a 1/2 is this."},{"Start":"07:06.375 ","End":"07:08.270","Text":"If you\u0027re not sure,"},{"Start":"07:08.270 ","End":"07:10.670","Text":"then just differentiate this,"},{"Start":"07:10.670 ","End":"07:13.670","Text":"and you\u0027ll see that you get that."},{"Start":"07:13.670 ","End":"07:17.955","Text":"Now, we\u0027ve got, from here,"},{"Start":"07:17.955 ","End":"07:22.260","Text":"uv, let\u0027s see, u times v,"},{"Start":"07:22.260 ","End":"07:26.190","Text":"that would be minus because of the minus,"},{"Start":"07:26.190 ","End":"07:34.560","Text":"and then 2 times 2 is 4t cosine t over 2."},{"Start":"07:34.560 ","End":"07:37.410","Text":"But we\u0027re doing a definite integral."},{"Start":"07:37.410 ","End":"07:41.030","Text":"We have to take this between the limits."},{"Start":"07:41.030 ","End":"07:43.415","Text":"Maybe I\u0027ll put brackets here,"},{"Start":"07:43.415 ","End":"07:49.030","Text":"between the limits 0 to Pi."},{"Start":"07:49.030 ","End":"07:52.915","Text":"Then we have a minus vdu."},{"Start":"07:52.915 ","End":"07:57.815","Text":"We have minus the integral from 0 to Pi."},{"Start":"07:57.815 ","End":"08:02.875","Text":"Let\u0027s see this with this."},{"Start":"08:02.875 ","End":"08:05.714","Text":"You know what? The minus,"},{"Start":"08:05.714 ","End":"08:07.845","Text":"I\u0027ll just make this a plus."},{"Start":"08:07.845 ","End":"08:12.390","Text":"Then I have again 2 times 2 is 4,"},{"Start":"08:12.390 ","End":"08:24.090","Text":"and cosine of t over 2, dt."},{"Start":"08:24.090 ","End":"08:27.680","Text":"Here I have a substitution to do."},{"Start":"08:27.680 ","End":"08:30.590","Text":"Now when we put in 0 for t,"},{"Start":"08:30.590 ","End":"08:33.305","Text":"we\u0027re going to get 0 because of this t here."},{"Start":"08:33.305 ","End":"08:36.660","Text":"We just have to substitute the Pi."},{"Start":"08:36.660 ","End":"08:43.145","Text":"A cosine of Pi over 2 is cosine of 90 degrees,"},{"Start":"08:43.145 ","End":"08:45.965","Text":"is 0. That\u0027s also 0."},{"Start":"08:45.965 ","End":"08:48.640","Text":"Everything here is 0."},{"Start":"08:48.640 ","End":"08:51.905","Text":"Then I just have this other integral."},{"Start":"08:51.905 ","End":"08:59.055","Text":"The integral of cosine is almost sine t over 2."},{"Start":"08:59.055 ","End":"09:02.415","Text":"But because of the 1/2 I have to divide by a 1/2,"},{"Start":"09:02.415 ","End":"09:05.040","Text":"which is like multiplying by 2."},{"Start":"09:05.040 ","End":"09:08.385","Text":"It\u0027s 8 sine t over 2."},{"Start":"09:08.385 ","End":"09:12.670","Text":"Again, differentiate this if you\u0027re not sure and you\u0027ll see that you get this."},{"Start":"09:12.670 ","End":"09:18.840","Text":"This I have to take from 0 to Pi."},{"Start":"09:20.440 ","End":"09:22.894","Text":"What we have here,"},{"Start":"09:22.894 ","End":"09:24.740","Text":"forgetting the 8 for the moment,"},{"Start":"09:24.740 ","End":"09:28.449","Text":"is we have sine Pi over 2 minus sine 0."},{"Start":"09:28.449 ","End":"09:30.950","Text":"Sine Pi over 2 is 1."},{"Start":"09:30.950 ","End":"09:32.570","Text":"Sine of 90 degrees,"},{"Start":"09:32.570 ","End":"09:34.565","Text":"so we just get 1."},{"Start":"09:34.565 ","End":"09:42.060","Text":"This whole thing comes out to be just the 8 from here,"},{"Start":"09:42.060 ","End":"09:45.690","Text":"because this is 0, and we said this minus this is 1."},{"Start":"09:45.690 ","End":"09:50.715","Text":"That\u0027s 8. That finishes the asterisk."},{"Start":"09:50.715 ","End":"09:54.400","Text":"Now let\u0027s go to do the double asterisk."},{"Start":"09:54.400 ","End":"09:59.110","Text":"Here, again, I\u0027m going to use a trigonometric formula."},{"Start":"09:59.110 ","End":"10:03.110","Text":"There\u0027s a trigonometric formula for,"},{"Start":"10:03.220 ","End":"10:09.200","Text":"I meant to say sine Alpha, sine Beta."},{"Start":"10:09.200 ","End":"10:13.475","Text":"This is equal to, it\u0027s 1/2 something."},{"Start":"10:13.475 ","End":"10:15.170","Text":"Instead of the 1/2,"},{"Start":"10:15.170 ","End":"10:17.500","Text":"I\u0027ll put the 2 here."},{"Start":"10:17.500 ","End":"10:19.785","Text":"What this is equal to,"},{"Start":"10:19.785 ","End":"10:25.345","Text":"is cosine of Alpha minus Beta."},{"Start":"10:25.345 ","End":"10:32.780","Text":"Take away the cosine of Alpha plus Beta."},{"Start":"10:32.780 ","End":"10:37.755","Text":"Here we\u0027ll use it with Alpha being t,"},{"Start":"10:37.755 ","End":"10:41.295","Text":"and Beta will be t over 2."},{"Start":"10:41.295 ","End":"10:43.980","Text":"What we get here,"},{"Start":"10:43.980 ","End":"10:51.660","Text":"is the integral from 0 to Pi of cosine."},{"Start":"10:51.660 ","End":"10:56.800","Text":"Now Alpha minus beta is just t over 2."},{"Start":"10:58.190 ","End":"11:01.470","Text":"Then we have a minus,"},{"Start":"11:01.470 ","End":"11:04.755","Text":"cosine Alpha plus Beta,"},{"Start":"11:04.755 ","End":"11:07.275","Text":"is t plus t over 2."},{"Start":"11:07.275 ","End":"11:11.500","Text":"It\u0027s 3t over 2."},{"Start":"11:11.810 ","End":"11:15.240","Text":"This dt."},{"Start":"11:15.240 ","End":"11:19.060","Text":"We split it up into 2 separate bits."},{"Start":"11:19.060 ","End":"11:24.270","Text":"Now the integral of cosine generally is sine."},{"Start":"11:26.420 ","End":"11:29.100","Text":"Because it\u0027s not t, it\u0027s t over 2,"},{"Start":"11:29.100 ","End":"11:30.675","Text":"I have to divide by a 1/2,"},{"Start":"11:30.675 ","End":"11:33.300","Text":"which is like multiplying by 2."},{"Start":"11:33.300 ","End":"11:35.285","Text":"Similarly for the other one,"},{"Start":"11:35.285 ","End":"11:36.830","Text":"I would normally say,"},{"Start":"11:36.830 ","End":"11:39.200","Text":"if it was just cosine of t,"},{"Start":"11:39.200 ","End":"11:42.560","Text":"it would be sine of whatever it is."},{"Start":"11:42.560 ","End":"11:46.700","Text":"But because it\u0027s not t it\u0027s, times 3 over 2,"},{"Start":"11:46.700 ","End":"11:48.995","Text":"I have to divide by 3 over 2,"},{"Start":"11:48.995 ","End":"11:52.475","Text":"which is like multiplying by 2/3."},{"Start":"11:52.475 ","End":"11:54.125","Text":"This is what I get,"},{"Start":"11:54.125 ","End":"11:58.575","Text":"and this has to be evaluated from 0 to Pi."},{"Start":"11:58.575 ","End":"12:00.585","Text":"Let\u0027s see what we get."},{"Start":"12:00.585 ","End":"12:03.135","Text":"If we put in Pi,"},{"Start":"12:03.135 ","End":"12:10.454","Text":"we get sine of Pi over 2 is 1, so this is 2."},{"Start":"12:10.454 ","End":"12:20.470","Text":"3Pi over 2 is 270 degrees."},{"Start":"12:21.710 ","End":"12:27.585","Text":"Sine of that is minus 1."},{"Start":"12:27.585 ","End":"12:32.790","Text":"It\u0027s going to be plus 2/3,"},{"Start":"12:32.790 ","End":"12:36.090","Text":"because it\u0027s minus 2/3 times minus 1."},{"Start":"12:36.090 ","End":"12:38.910","Text":"That\u0027s the part for Pi."},{"Start":"12:38.910 ","End":"12:41.930","Text":"Now I have to take away the part to 0."},{"Start":"12:41.930 ","End":"12:45.200","Text":"Well, when t is 0,"},{"Start":"12:45.200 ","End":"12:47.330","Text":"the sine of t is also 0."},{"Start":"12:47.330 ","End":"12:50.340","Text":"It\u0027s just 0 minus 0."},{"Start":"12:50.340 ","End":"12:56.745","Text":"In short, what we get is 2 and 2/3."},{"Start":"12:56.745 ","End":"13:03.350","Text":"That\u0027s the answer for the double asterisk part here."},{"Start":"13:03.350 ","End":"13:06.325","Text":"Now I just have to subtract."},{"Start":"13:06.325 ","End":"13:09.665","Text":"Finally our line integral,"},{"Start":"13:09.665 ","End":"13:12.305","Text":"which is this minus this,"},{"Start":"13:12.305 ","End":"13:17.205","Text":"becomes 8 minus 2 and 2/3,"},{"Start":"13:17.205 ","End":"13:19.995","Text":"which is 5 and a 1/3."},{"Start":"13:19.995 ","End":"13:21.500","Text":"You can leave it like that,"},{"Start":"13:21.500 ","End":"13:23.390","Text":"or if you like improper fractions,"},{"Start":"13:23.390 ","End":"13:26.695","Text":"that\u0027s 16 over 3."},{"Start":"13:26.695 ","End":"13:29.550","Text":"I\u0027ll just highlight it."},{"Start":"13:29.550 ","End":"13:33.030","Text":"We are done. I want to apologize that,"},{"Start":"13:33.030 ","End":"13:35.640","Text":"it was such a nuisance some,"},{"Start":"13:35.640 ","End":"13:37.665","Text":"such a length integral."},{"Start":"13:37.665 ","End":"13:46.714","Text":"We did more integration than we talk about the line integral concept."},{"Start":"13:46.714 ","End":"13:49.950","Text":"Anyway, it\u0027s solved."}],"ID":8809},{"Watched":false,"Name":"Exercise 1 Part c","Duration":"3m 53s","ChapterTopicVideoID":8711,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8711.jpeg","UploadDate":"2017-02-13T05:32:52.9870000","DurationForVideoObject":"PT3M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.329","Text":"In this exercise, we have to compute this line integral over the curve,"},{"Start":"00:06.329 ","End":"00:09.670","Text":"but we\u0027re not given the curve in parametric form,"},{"Start":"00:09.670 ","End":"00:11.370","Text":"we\u0027re just given a description,"},{"Start":"00:11.370 ","End":"00:15.570","Text":"and it\u0027s the line segment joining this point to this point."},{"Start":"00:15.570 ","End":"00:21.119","Text":"The first thing I want to do is to describe C in parametric form,"},{"Start":"00:21.119 ","End":"00:25.830","Text":"and there are several formulas that are all similar,"},{"Start":"00:25.830 ","End":"00:27.435","Text":"I\u0027ll use one of the forms."},{"Start":"00:27.435 ","End":"00:31.380","Text":"One of the forms says that the x and the y,"},{"Start":"00:31.380 ","End":"00:34.244","Text":"I use the angular brackets for vectors,"},{"Start":"00:34.244 ","End":"00:40.544","Text":"is equal to 1 minus t times the start point,"},{"Start":"00:40.544 ","End":"00:49.690","Text":"which is 0,0 plus t times the end point which is 1,2."},{"Start":"00:50.750 ","End":"00:55.620","Text":"If I expand this out, I\u0027ll just get, well,"},{"Start":"00:55.620 ","End":"01:03.740","Text":"these are both 0s, so I get just t,"},{"Start":"01:03.740 ","End":"01:08.015","Text":"2t, which if I write it out,"},{"Start":"01:08.015 ","End":"01:18.245","Text":"tells me basically that x is equal to t and y equals 2t,"},{"Start":"01:18.245 ","End":"01:24.350","Text":"oh and I forgot to say this always goes from 0-1, the parameter."},{"Start":"01:24.350 ","End":"01:29.745","Text":"Here also t goes from 0-1."},{"Start":"01:29.745 ","End":"01:32.855","Text":"That\u0027s the parametric form of the curve."},{"Start":"01:32.855 ","End":"01:35.810","Text":"Now what we do is we do the substitution,"},{"Start":"01:35.810 ","End":"01:38.360","Text":"there\u0027s a long formula and a short formula."},{"Start":"01:38.360 ","End":"01:42.860","Text":"I just use the short formula for just the ds part,"},{"Start":"01:42.860 ","End":"01:52.310","Text":"which is the square root of it\u0027s x prime with respect to t squared,"},{"Start":"01:52.310 ","End":"01:56.715","Text":"and y prime squared,"},{"Start":"01:56.715 ","End":"01:59.010","Text":"and all this is dt,"},{"Start":"01:59.010 ","End":"02:00.420","Text":"and as for the rest of it,"},{"Start":"02:00.420 ","End":"02:03.630","Text":"you just set the integral along the curve,"},{"Start":"02:03.630 ","End":"02:07.530","Text":"you put the integral from the limits of t,"},{"Start":"02:07.530 ","End":"02:14.015","Text":"so that would be 0-1, and then you replace x and y according to the formula."},{"Start":"02:14.015 ","End":"02:21.770","Text":"X plus y would be t plus 2t, and then ds."},{"Start":"02:21.770 ","End":"02:24.825","Text":"Well, let\u0027s compute what ds is in our case."},{"Start":"02:24.825 ","End":"02:34.200","Text":"In our case, x prime is 1 and y prime with respect to t is 2,"},{"Start":"02:34.200 ","End":"02:37.589","Text":"so we get 1 squared, plus 2 squared,"},{"Start":"02:37.589 ","End":"02:43.840","Text":"square root, and that comes out to be root 5."},{"Start":"02:44.060 ","End":"02:51.780","Text":"So ds is root 5 dt."},{"Start":"02:51.780 ","End":"02:53.740","Text":"Now I can take constants in front."},{"Start":"02:53.740 ","End":"02:56.485","Text":"First of all, t plus 2t is 3t,"},{"Start":"02:56.485 ","End":"03:01.925","Text":"so the 3 I can put in front and the square root of 5,"},{"Start":"03:01.925 ","End":"03:10.290","Text":"and all we\u0027re left with is the integral of t-dt from 0-1."},{"Start":"03:10.290 ","End":"03:16.555","Text":"Let\u0027s see The integral of t. I\u0027ll just do this bit at the side."},{"Start":"03:16.555 ","End":"03:21.100","Text":"The integral from 0-1 of t-dt is"},{"Start":"03:21.100 ","End":"03:27.300","Text":"1.5t squared, taken from 0-1."},{"Start":"03:27.300 ","End":"03:30.000","Text":"At 0 I get nothing, and I plug in 1,"},{"Start":"03:30.000 ","End":"03:35.250","Text":"I get 1.5, so what I get is 3 root 5,"},{"Start":"03:35.250 ","End":"03:41.830","Text":"and this comes out to be 1.5,"},{"Start":"03:42.050 ","End":"03:44.840","Text":"just rewrite it slightly."},{"Start":"03:44.840 ","End":"03:50.435","Text":"Looks nicer if I write it as 3 over 2 square root of 5,"},{"Start":"03:50.435 ","End":"03:53.580","Text":"and that\u0027s the answer."}],"ID":8810},{"Watched":false,"Name":"Exercise 1 Part d","Duration":"10m 57s","ChapterTopicVideoID":8712,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8712.jpeg","UploadDate":"2017-02-13T05:35:15.0770000","DurationForVideoObject":"PT10M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.625","Text":"In this exercise, we\u0027re given a line integral of type 1 to compute."},{"Start":"00:05.625 ","End":"00:11.925","Text":"But the curve C is described as the perimeter of a triangle."},{"Start":"00:11.925 ","End":"00:14.355","Text":"Let\u0027s sketch the triangle."},{"Start":"00:14.355 ","End":"00:16.110","Text":"O is the origin,"},{"Start":"00:16.110 ","End":"00:18.465","Text":"that\u0027s this point O."},{"Start":"00:18.465 ","End":"00:22.140","Text":"A is the point, 0, 1,"},{"Start":"00:22.140 ","End":"00:24.000","Text":"which means that y is 1,"},{"Start":"00:24.000 ","End":"00:26.985","Text":"so that would be somewhere, let\u0027s say here."},{"Start":"00:26.985 ","End":"00:29.985","Text":"B is 1, 0,"},{"Start":"00:29.985 ","End":"00:33.150","Text":"so that would be somewhere here."},{"Start":"00:33.150 ","End":"00:39.450","Text":"Then OAB, usually we take it in order."},{"Start":"00:39.450 ","End":"00:48.885","Text":"From O to A and then from A to B."},{"Start":"00:48.885 ","End":"00:53.310","Text":"Then finally from B back to O."},{"Start":"00:53.310 ","End":"00:57.525","Text":"It\u0027s a closed path."},{"Start":"00:57.525 ","End":"01:00.960","Text":"What we have to do is,"},{"Start":"01:00.960 ","End":"01:03.580","Text":"since this is defined piecewise,"},{"Start":"01:03.580 ","End":"01:08.880","Text":"is just take this and break it up into 3 separate integrals."},{"Start":"01:08.900 ","End":"01:14.215","Text":"This integral of x plus y squared,"},{"Start":"01:14.215 ","End":"01:16.565","Text":"ds over the curve C,"},{"Start":"01:16.565 ","End":"01:18.050","Text":"I\u0027ll just break it up."},{"Start":"01:18.050 ","End":"01:23.720","Text":"I\u0027ll first of all take the integral of the same thing along OA."},{"Start":"01:23.720 ","End":"01:30.215","Text":"Then I\u0027ll take the integral of the whatever it is along AB,"},{"Start":"01:30.215 ","End":"01:37.930","Text":"and then the integral of whatever it is along BO."},{"Start":"01:39.710 ","End":"01:43.860","Text":"I\u0027m going to need to parametrize each of these segments,"},{"Start":"01:43.860 ","End":"01:48.910","Text":"so I\u0027ll just remind you of the formula for a parametrized segment."},{"Start":"01:48.910 ","End":"01:50.530","Text":"It\u0027s actually more than 1."},{"Start":"01:50.530 ","End":"01:54.680","Text":"Previously, I used the formula that x,"},{"Start":"01:54.680 ","End":"02:00.580","Text":"y is equal to 1 minus t times the first point."},{"Start":"02:00.580 ","End":"02:04.864","Text":"Let\u0027s say, the first was x naught, y naught,"},{"Start":"02:04.864 ","End":"02:08.235","Text":"and then t times the second point,"},{"Start":"02:08.235 ","End":"02:12.165","Text":"whatever that is, let\u0027s call it x1, y1."},{"Start":"02:12.165 ","End":"02:16.230","Text":"We\u0027re going to use this 3 times once,"},{"Start":"02:16.340 ","End":"02:18.855","Text":"well, with each of the 3 sides."},{"Start":"02:18.855 ","End":"02:22.070","Text":"There is a variation of this which is equivalent,"},{"Start":"02:22.070 ","End":"02:24.140","Text":"which is sometimes I find more useful,"},{"Start":"02:24.140 ","End":"02:26.570","Text":"is just to say that this is x naught,"},{"Start":"02:26.570 ","End":"02:32.445","Text":"y naught plus t times the difference of the 2,"},{"Start":"02:32.445 ","End":"02:35.265","Text":"x1 minus x naught,"},{"Start":"02:35.265 ","End":"02:37.930","Text":"y1 minus y naught."},{"Start":"02:37.930 ","End":"02:39.440","Text":"Sometimes I use this form,"},{"Start":"02:39.440 ","End":"02:40.530","Text":"sometimes I use this form."},{"Start":"02:40.530 ","End":"02:43.085","Text":"I may even repeat the x, y."},{"Start":"02:43.085 ","End":"02:48.930","Text":"Let\u0027s work with the second form in this exercise."},{"Start":"02:49.310 ","End":"02:53.810","Text":"I can now parametrize each of them."},{"Start":"02:53.940 ","End":"02:57.460","Text":"I forgot to add that in each of these cases,"},{"Start":"02:57.460 ","End":"03:00.310","Text":"t goes from 0 to 1."},{"Start":"03:00.310 ","End":"03:05.280","Text":"Let\u0027s parametrize first of all, OA."},{"Start":"03:05.280 ","End":"03:12.630","Text":"For OA, what I have is that O is this point working with this formula."},{"Start":"03:12.630 ","End":"03:14.070","Text":"You know what, I\u0027ll erase this one,"},{"Start":"03:14.070 ","End":"03:16.275","Text":"so I don\u0027t get confused."},{"Start":"03:16.275 ","End":"03:20.130","Text":"Let\u0027s see. What I get here, is x naught,"},{"Start":"03:20.130 ","End":"03:22.920","Text":"y naught is the O,"},{"Start":"03:22.920 ","End":"03:27.420","Text":"and x1, y1 is A."},{"Start":"03:27.420 ","End":"03:32.280","Text":"Let me just rewrite these coordinates, 0,"},{"Start":"03:32.280 ","End":"03:38.820","Text":"0, 0, 1, and 1, 0."},{"Start":"03:38.820 ","End":"03:47.540","Text":"Here we get that x, y is 0,"},{"Start":"03:47.540 ","End":"03:52.325","Text":"0 plus t times the difference,"},{"Start":"03:52.325 ","End":"03:57.715","Text":"this minus this, which is 0, 1."},{"Start":"03:57.715 ","End":"04:01.680","Text":"That just tells us that,"},{"Start":"04:01.680 ","End":"04:12.165","Text":"I can write it that x equals 0 and y equals t,"},{"Start":"04:12.165 ","End":"04:16.170","Text":"t goes from 0 to 1."},{"Start":"04:16.170 ","End":"04:21.675","Text":"That\u0027s the parametrization for OA."},{"Start":"04:21.675 ","End":"04:26.160","Text":"Next. Next one is AB."},{"Start":"04:26.160 ","End":"04:32.340","Text":"AB, x, y is the first point which is A,"},{"Start":"04:32.340 ","End":"04:40.675","Text":"which is 0,1 plus t times the difference B minus A."},{"Start":"04:40.675 ","End":"04:47.190","Text":"B minus A is 1 minus 0,"},{"Start":"04:47.190 ","End":"04:49.364","Text":"and then 0 minus 1,"},{"Start":"04:49.364 ","End":"04:53.135","Text":"so it\u0027s 1 minus 1,"},{"Start":"04:53.135 ","End":"05:02.965","Text":"which means that x equals 0 plus 1t, x equals t,"},{"Start":"05:02.965 ","End":"05:13.605","Text":"and y equals 1 minus t. I take this minus t times this,"},{"Start":"05:13.605 ","End":"05:18.880","Text":"and again, t between 0 and 1."},{"Start":"05:18.950 ","End":"05:23.460","Text":"Lastly BO. Where x,"},{"Start":"05:23.460 ","End":"05:27.190","Text":"y is going to be the point for B"},{"Start":"05:27.610 ","End":"05:34.850","Text":"plus t times the difference of O minus B,"},{"Start":"05:34.850 ","End":"05:39.060","Text":"which is minus 1, 0."},{"Start":"05:39.060 ","End":"05:43.145","Text":"What we get is that x is equal to"},{"Start":"05:43.145 ","End":"05:51.990","Text":"1 minus t and y equals just 0,"},{"Start":"05:51.990 ","End":"05:57.465","Text":"and still t goes as always from 0 to 1."},{"Start":"05:57.465 ","End":"05:59.500","Text":"Before I write the integrals,"},{"Start":"05:59.500 ","End":"06:03.140","Text":"I want to write another formula for what ds is equal to."},{"Start":"06:03.140 ","End":"06:10.025","Text":"In general, ds is going to be the square root of x"},{"Start":"06:10.025 ","End":"06:18.955","Text":"prime with respect to t plus y prime squared dt."},{"Start":"06:18.955 ","End":"06:23.140","Text":"Let\u0027s do ds for each of these 3 cases."},{"Start":"06:23.140 ","End":"06:26.960","Text":"In this case, I will get that ds is"},{"Start":"06:26.960 ","End":"06:32.285","Text":"the square root of this derivative squared plus this derivative squared,"},{"Start":"06:32.285 ","End":"06:36.845","Text":"0 squared plus 1 squared is 1."},{"Start":"06:36.845 ","End":"06:41.870","Text":"It\u0027s just 1dt. In here,"},{"Start":"06:41.870 ","End":"06:43.670","Text":"if I differentiate this, I get 1,"},{"Start":"06:43.670 ","End":"06:47.360","Text":"I differentiate this, I get minus 1."},{"Start":"06:47.360 ","End":"06:51.185","Text":"The square root of this squared plus this squared will be square root of 2,"},{"Start":"06:51.185 ","End":"06:54.570","Text":"so ds is square root of 2dt."},{"Start":"06:54.620 ","End":"06:58.640","Text":"Here, the derivative of this is minus 1,"},{"Start":"06:58.640 ","End":"06:59.840","Text":"of this is 0."},{"Start":"06:59.840 ","End":"07:02.990","Text":"Taking the square root of the sum of squares, again,"},{"Start":"07:02.990 ","End":"07:09.820","Text":"I get 1, so here also ds is equal to dt."},{"Start":"07:11.570 ","End":"07:16.335","Text":"Now, I\u0027m going to write each of the 3 integrals."},{"Start":"07:16.335 ","End":"07:20.640","Text":"The first integral, the one that belongs to"},{"Start":"07:20.640 ","End":"07:26.180","Text":"OA is going to be the integral where t goes from 0 to."},{"Start":"07:26.180 ","End":"07:29.700","Text":"Well, they\u0027re all going to go from 0 to 1."},{"Start":"07:30.470 ","End":"07:34.744","Text":"Just have to substitute x plus y squared,"},{"Start":"07:34.744 ","End":"07:41.254","Text":"and here x plus y squared is 0 plus t squared."},{"Start":"07:41.254 ","End":"07:43.760","Text":"This is going to be t squared."},{"Start":"07:43.760 ","End":"07:49.770","Text":"Ds is 1dt, so this is just dt."},{"Start":"07:50.210 ","End":"07:52.590","Text":"We\u0027ll see what this equals in a moment."},{"Start":"07:52.590 ","End":"07:55.000","Text":"I\u0027ll work on them in parallel maybe."},{"Start":"07:55.000 ","End":"08:00.125","Text":"The AB integral is going to be the integral from 0 to 1."},{"Start":"08:00.125 ","End":"08:02.060","Text":"In fact, they\u0027re all, as I said,"},{"Start":"08:02.060 ","End":"08:05.640","Text":"going to be the integral from 0 to 1."},{"Start":"08:05.640 ","End":"08:07.910","Text":"Now, x plus y squared,"},{"Start":"08:07.910 ","End":"08:09.710","Text":"in the second case,"},{"Start":"08:09.710 ","End":"08:16.760","Text":"x plus y squared is t plus 1 minus t squared."},{"Start":"08:16.760 ","End":"08:21.030","Text":"This is 1 minus 2t plus t squared,"},{"Start":"08:21.030 ","End":"08:27.410","Text":"so altogether I\u0027ll get 1 minus t plus t squared."},{"Start":"08:27.410 ","End":"08:31.375","Text":"Then I need the ds, which is root 2."},{"Start":"08:31.375 ","End":"08:35.745","Text":"I can put the root 2 in front."},{"Start":"08:35.745 ","End":"08:41.505","Text":"Here I put the root 2 and here I put the dt from here."},{"Start":"08:41.505 ","End":"08:44.940","Text":"Let\u0027s see what the last integral is."},{"Start":"08:44.940 ","End":"08:48.510","Text":"I need x plus y squared,"},{"Start":"08:48.510 ","End":"08:50.730","Text":"x is this, y is 0,"},{"Start":"08:50.730 ","End":"08:59.035","Text":"so it\u0027s just the 1 minus t and ds is dt."},{"Start":"08:59.035 ","End":"09:03.140","Text":"That\u0027s 3 integrals to compute."},{"Start":"09:03.140 ","End":"09:11.670","Text":"Let\u0027s see, the first 1 would be 1/3t cubed from 0 to 1."},{"Start":"09:11.670 ","End":"09:13.710","Text":"Plug in 0 is nothing, plug in 1,"},{"Start":"09:13.710 ","End":"09:20.280","Text":"it\u0027s a 1/3, so this is just 1/3. Let\u0027s see."},{"Start":"09:20.280 ","End":"09:25.095","Text":"This one will be the square root of 2,"},{"Start":"09:25.095 ","End":"09:29.175","Text":"and then t minus 1/2t"},{"Start":"09:29.175 ","End":"09:36.285","Text":"squared plus 1/3t cubed from 0 to 1,"},{"Start":"09:36.285 ","End":"09:40.050","Text":"0 gives 0, so I just have to plug in 1."},{"Start":"09:40.050 ","End":"09:44.055","Text":"1 minus a1/2 is a 1/2,"},{"Start":"09:44.055 ","End":"09:47.025","Text":"plus a 1/3 is 5/6."},{"Start":"09:47.025 ","End":"09:54.840","Text":"I get 5/6 root 2. Let\u0027s see."},{"Start":"09:54.840 ","End":"10:03.455","Text":"This is t minus 1/2t squared from 0 to 1,"},{"Start":"10:03.455 ","End":"10:05.090","Text":"0 doesn\u0027t give anything,"},{"Start":"10:05.090 ","End":"10:06.725","Text":"could just plug in 1,"},{"Start":"10:06.725 ","End":"10:11.840","Text":"it\u0027s 1 minus 1/2."},{"Start":"10:11.840 ","End":"10:15.360","Text":"This comes out to be 1/2."},{"Start":"10:15.360 ","End":"10:20.190","Text":"Finally, I have to add the 3."},{"Start":"10:20.190 ","End":"10:26.400","Text":"The integral along C of whatever it was,"},{"Start":"10:26.400 ","End":"10:28.860","Text":"is just this plus this plus this,"},{"Start":"10:28.860 ","End":"10:36.090","Text":"I get 1/3 plus 5/6 root"},{"Start":"10:36.090 ","End":"10:39.760","Text":"2 plus 1/2,"},{"Start":"10:39.760 ","End":"10:43.695","Text":"and could simplify it a bit,"},{"Start":"10:43.695 ","End":"10:46.155","Text":"1/2 plus a 1/3 is 5/6."},{"Start":"10:46.155 ","End":"10:54.720","Text":"I can take the 5/6 then outside the brackets and write it as 1 plus root 2."},{"Start":"10:54.720 ","End":"10:58.870","Text":"I\u0027ll leave the answer in this form. We are done."}],"ID":8811},{"Watched":false,"Name":"Exercise 2 Part a","Duration":"3m 18s","ChapterTopicVideoID":8713,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8713.jpeg","UploadDate":"2017-02-13T05:35:59.4330000","DurationForVideoObject":"PT3M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.055","Text":"In this exercise, we have a type 1 line integral."},{"Start":"00:05.055 ","End":"00:07.560","Text":"This time it\u0027s in 3-dimensions."},{"Start":"00:07.560 ","End":"00:11.010","Text":"Previously, we had some in 2-dimensions."},{"Start":"00:11.010 ","End":"00:16.830","Text":"The main difference in the formula is that in 2-dimensions,"},{"Start":"00:16.830 ","End":"00:21.870","Text":"one of the formulas we had for ds was the square root of x"},{"Start":"00:21.870 ","End":"00:29.970","Text":"prime with respect to t squared plus y prime with respect to t squared."},{"Start":"00:29.970 ","End":"00:35.190","Text":"Then we had dt. The difference is in 3D that we just extended a"},{"Start":"00:35.190 ","End":"00:41.465","Text":"bit and add also a z prime squared before we put the dt."},{"Start":"00:41.465 ","End":"00:43.940","Text":"In 2-dimensions is just these 2,"},{"Start":"00:43.940 ","End":"00:45.515","Text":"in 3-dimensions is these 3,"},{"Start":"00:45.515 ","End":"00:48.845","Text":"other than that, it\u0027s the same idea."},{"Start":"00:48.845 ","End":"00:52.370","Text":"The integral along the curve,"},{"Start":"00:52.370 ","End":"00:55.235","Text":"we replace by the integral of the parameter,"},{"Start":"00:55.235 ","End":"00:57.485","Text":"which in this case is t,"},{"Start":"00:57.485 ","End":"01:01.630","Text":"and it goes from 0 to Pi."},{"Start":"01:01.630 ","End":"01:08.260","Text":"Then we substitute each of the xyz according to the parametric equation."},{"Start":"01:08.260 ","End":"01:14.485","Text":"We have x squared is cosine squared t,"},{"Start":"01:14.485 ","End":"01:19.195","Text":"y squared is sine squared t,"},{"Start":"01:19.195 ","End":"01:23.960","Text":"and z squared is just t squared."},{"Start":"01:23.960 ","End":"01:26.790","Text":"Then I need the ds."},{"Start":"01:26.790 ","End":"01:29.559","Text":"Let\u0027s just compute that over here."},{"Start":"01:29.559 ","End":"01:35.550","Text":"This is going to equal x prime"},{"Start":"01:35.550 ","End":"01:42.975","Text":"is minus sine t squared,"},{"Start":"01:42.975 ","End":"01:48.010","Text":"y prime is cosine t,"},{"Start":"01:48.380 ","End":"01:51.135","Text":"and I want that squared,"},{"Start":"01:51.135 ","End":"01:54.750","Text":"and z prime is 1,"},{"Start":"01:54.750 ","End":"01:56.670","Text":"so I need 1 squared,"},{"Start":"01:56.670 ","End":"01:58.905","Text":"and I need the square root of that,"},{"Start":"01:58.905 ","End":"02:01.150","Text":"and at the end, the dt."},{"Start":"02:01.150 ","End":"02:06.095","Text":"Now, sine squared plus cosine squared is 1,"},{"Start":"02:06.095 ","End":"02:07.970","Text":"1 plus 1 is 2."},{"Start":"02:07.970 ","End":"02:11.300","Text":"So we have the square root of 2dt,"},{"Start":"02:11.300 ","End":"02:14.290","Text":"which I put here."},{"Start":"02:14.290 ","End":"02:17.790","Text":"I can put the dt here,"},{"Start":"02:17.790 ","End":"02:26.330","Text":"and the square root of 2 I\u0027d rather put in front right here instead of here."},{"Start":"02:26.330 ","End":"02:31.025","Text":"Once again, we see cosine squared plus sine squared,"},{"Start":"02:31.025 ","End":"02:34.125","Text":"and this is equal to 1."},{"Start":"02:34.125 ","End":"02:38.780","Text":"When I do the integral I have the square root of 2,"},{"Start":"02:38.780 ","End":"02:48.755","Text":"the integral of 1 is just t. The integral of t squared is 1/3 t cubed."},{"Start":"02:48.755 ","End":"02:54.360","Text":"All this needs to go from 0 to Pi."},{"Start":"02:55.100 ","End":"02:58.380","Text":"When t is 0, we get nothing."},{"Start":"02:58.380 ","End":"03:00.930","Text":"We just have to plug in Pi."},{"Start":"03:00.930 ","End":"03:11.755","Text":"We get the square root of 2 of Pi plus 1/3 Pi cubed."},{"Start":"03:11.755 ","End":"03:14.390","Text":"There\u0027s not really much to simplify."},{"Start":"03:14.390 ","End":"03:18.390","Text":"I\u0027ll just leave the answer like this and we\u0027re done."}],"ID":8812},{"Watched":false,"Name":"Exercise 2 Part b","Duration":"6m 22s","ChapterTopicVideoID":8714,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8714.jpeg","UploadDate":"2017-02-13T05:37:16.9570000","DurationForVideoObject":"PT6M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"Here we have a type 1 lightning to grow."},{"Start":"00:03.285 ","End":"00:06.420","Text":"This time it\u0027s in 3D because we have an x,"},{"Start":"00:06.420 ","End":"00:08.235","Text":"a y, and a z."},{"Start":"00:08.235 ","End":"00:10.725","Text":"It\u0027s given in parametric form."},{"Start":"00:10.725 ","End":"00:13.575","Text":"When we have it in parametric form,"},{"Start":"00:13.575 ","End":"00:21.140","Text":"1 of the formulas I use for the ds is there several variations of"},{"Start":"00:21.140 ","End":"00:29.285","Text":"formulas I use this 1 that ds is the square root of x prime."},{"Start":"00:29.285 ","End":"00:31.580","Text":"Sometimes I write x prime of t,"},{"Start":"00:31.580 ","End":"00:35.330","Text":"usually I just abbreviate it to x prime squared."},{"Start":"00:35.330 ","End":"00:44.040","Text":"We assume x, y, and z are functions of t along the curve plus y prime squared."},{"Start":"00:44.040 ","End":"00:46.070","Text":"If it was 2D, we\u0027d stop here,"},{"Start":"00:46.070 ","End":"00:47.750","Text":"but we are in 3D,"},{"Start":"00:47.750 ","End":"00:50.375","Text":"so we also have z prime squared,"},{"Start":"00:50.375 ","End":"00:53.430","Text":"and then it\u0027s dt."},{"Start":"00:53.430 ","End":"00:55.320","Text":"I will compute this in a moment,"},{"Start":"00:55.320 ","End":"00:59.060","Text":"and we have to remember that the line"},{"Start":"00:59.060 ","End":"01:05.570","Text":"integral is replaced by a simple 1 variable integral,"},{"Start":"01:05.570 ","End":"01:07.744","Text":"which is the parameter t,"},{"Start":"01:07.744 ","End":"01:13.420","Text":"which goes from 0-3."},{"Start":"01:13.420 ","End":"01:19.895","Text":"Then I need to compute what is x cubed plus 3z that are along the curve."},{"Start":"01:19.895 ","End":"01:24.245","Text":"Let me also do that at the side."},{"Start":"01:24.245 ","End":"01:26.440","Text":"We can just do it in here,"},{"Start":"01:26.440 ","End":"01:32.595","Text":"x cubed is t cubed,"},{"Start":"01:32.595 ","End":"01:38.240","Text":"and 3z is 3 times 1 third,"},{"Start":"01:38.240 ","End":"01:40.070","Text":"3 times 1/3 is 1,"},{"Start":"01:40.070 ","End":"01:43.200","Text":"so plus t cubed."},{"Start":"01:43.200 ","End":"01:46.965","Text":"Then we need the ds,"},{"Start":"01:46.965 ","End":"01:51.380","Text":"so we need the square root of,"},{"Start":"01:51.380 ","End":"01:57.635","Text":"let\u0027s see, x prime is 1 squared."},{"Start":"01:57.635 ","End":"02:03.890","Text":"Y prime is derivative of t squared is 2t,"},{"Start":"02:03.890 ","End":"02:11.330","Text":"so it\u0027s 2 over root 2t squared,"},{"Start":"02:11.330 ","End":"02:18.890","Text":"and then z prime is 3 times t squared,"},{"Start":"02:18.890 ","End":"02:23.540","Text":"but times 1/3 just comes out t squared,"},{"Start":"02:23.540 ","End":"02:28.980","Text":"also squared, and then dt."},{"Start":"02:30.400 ","End":"02:36.965","Text":"Now, I\u0027ll do this square root thing at the side."},{"Start":"02:36.965 ","End":"02:44.655","Text":"What I get is the square root of 1 squared is 1."},{"Start":"02:44.655 ","End":"02:50.880","Text":"This thing squared, 2 over root 2 squared is 4 over 2, which is 2,"},{"Start":"02:50.880 ","End":"02:54.255","Text":"so that\u0027s 2t squared,"},{"Start":"02:54.255 ","End":"03:00.600","Text":"and t squared squared is t^4."},{"Start":"03:00.600 ","End":"03:08.360","Text":"This is exactly 1 plus t squared squared."},{"Start":"03:08.360 ","End":"03:10.040","Text":"You could have made a substitution,"},{"Start":"03:10.040 ","End":"03:13.400","Text":"u equals t squared or something."},{"Start":"03:13.400 ","End":"03:15.095","Text":"Anyway, if you square this,"},{"Start":"03:15.095 ","End":"03:22.025","Text":"you\u0027ll see this 1 squared is this twice this times this plus the last 1 squared."},{"Start":"03:22.025 ","End":"03:24.590","Text":"When I take the square root,"},{"Start":"03:24.590 ","End":"03:30.860","Text":"it\u0027s actually the absolute value"},{"Start":"03:30.860 ","End":"03:38.890","Text":"of just 1 plus t squared."},{"Start":"03:39.350 ","End":"03:44.910","Text":"But the absolute value this thing is always non-negative,"},{"Start":"03:44.910 ","End":"03:47.750","Text":"so I don\u0027t need the absolute value."},{"Start":"03:47.750 ","End":"03:51.995","Text":"It just comes out to be 1 plus t squared."},{"Start":"03:51.995 ","End":"03:55.880","Text":"We get the integral from 0-3."},{"Start":"03:55.880 ","End":"03:59.810","Text":"Now this plus this is just 2t cubed,"},{"Start":"03:59.810 ","End":"04:07.110","Text":"and from here I get 1 plus t squared dt."},{"Start":"04:09.230 ","End":"04:13.335","Text":"I\u0027ll break it up into 2 integrals,"},{"Start":"04:13.335 ","End":"04:15.060","Text":"or I can just do it as 1."},{"Start":"04:15.060 ","End":"04:16.545","Text":"Let\u0027s do it as 1 then."},{"Start":"04:16.545 ","End":"04:19.125","Text":"It\u0027s the integral from 0-3."},{"Start":"04:19.125 ","End":"04:23.055","Text":"This with this is 2t cubed,"},{"Start":"04:23.055 ","End":"04:30.210","Text":"and this with this is 2t^5."},{"Start":"04:30.210 ","End":"04:34.820","Text":"All this 0-3dt."},{"Start":"04:34.820 ","End":"04:40.380","Text":"Now to the integral raised the power of 1 is 4, divide by 4."},{"Start":"04:41.400 ","End":"04:46.335","Text":"1.5t^4, here, I raise it to 6 and divide by 6,"},{"Start":"04:46.335 ","End":"04:56.135","Text":"so it\u0027s 1/3t^6, and all this has to be evaluated from 0-3."},{"Start":"04:56.135 ","End":"04:59.720","Text":"Now, 0 when we substitute gives nothing,"},{"Start":"04:59.720 ","End":"05:03.230","Text":"so we just have to substitute the 3. What do we get?"},{"Start":"05:03.230 ","End":"05:08.030","Text":"1.5, 3^4,"},{"Start":"05:08.030 ","End":"05:17.900","Text":"plus 1 third times 3^6."},{"Start":"05:17.900 ","End":"05:23.965","Text":"Well, I could cancel the 1/3with 1 of the threes and make it just 5,"},{"Start":"05:23.965 ","End":"05:32.670","Text":"and then I could take 3^4 outside the brackets, 3^4."},{"Start":"05:32.670 ","End":"05:34.050","Text":"What am I left with?"},{"Start":"05:34.050 ","End":"05:40.120","Text":"1.5 plus 3. That\u0027s 3.5."},{"Start":"05:41.960 ","End":"05:49.125","Text":"So it\u0027s 81, and this is 7 over 2"},{"Start":"05:49.125 ","End":"05:54.100","Text":"times 7 over 2,"},{"Start":"05:54.980 ","End":"06:02.770","Text":"567 over 2."},{"Start":"06:02.770 ","End":"06:06.620","Text":"I could leave it like that or well, yeah,"},{"Start":"06:06.620 ","End":"06:15.080","Text":"I could do the division and you get something like 283.5 and write that down 283.5,"},{"Start":"06:15.080 ","End":"06:17.059","Text":"whichever one you prefer."},{"Start":"06:17.059 ","End":"06:21.690","Text":"I\u0027ll stick with this and we are done."}],"ID":8813},{"Watched":false,"Name":"Exercise 3","Duration":"11m 3s","ChapterTopicVideoID":8715,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8715.jpeg","UploadDate":"2017-02-13T05:40:02.4870000","DurationForVideoObject":"PT11M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this exercise, we need to compute the length of the following curve,"},{"Start":"00:05.145 ","End":"00:11.400","Text":"which is given an implicit form with x and y so we\u0027re working in 2 dimensions."},{"Start":"00:11.400 ","End":"00:15.570","Text":"Now, suppose I restrict x and y to the first quadrant,"},{"Start":"00:15.570 ","End":"00:19.935","Text":"you could try plugging in a few points for example, if x is 0,"},{"Start":"00:19.935 ","End":"00:26.490","Text":"we get that y is 1 and if we let y is 0,"},{"Start":"00:26.490 ","End":"00:31.285","Text":"then we get that x is 1 and if you try a few more points,"},{"Start":"00:31.285 ","End":"00:38.810","Text":"basically what we get is some curve like this and that\u0027s just for the first quadrant."},{"Start":"00:38.810 ","End":"00:42.560","Text":"Now notice if I replace x by minus x,"},{"Start":"00:42.560 ","End":"00:45.200","Text":"because it\u0027s to the power of 2/3,"},{"Start":"00:45.200 ","End":"00:47.315","Text":"it\u0027s squared and then the cube root."},{"Start":"00:47.315 ","End":"00:52.880","Text":"But I could replace x by minus x and I could also replace y by minus y,"},{"Start":"00:52.880 ","End":"00:58.445","Text":"so really I get a symmetry with respect to both the x- and the y-axis."},{"Start":"00:58.445 ","End":"01:01.130","Text":"If I draw in the points, minus 1,"},{"Start":"01:01.130 ","End":"01:04.615","Text":"0, and 0, minus 1,"},{"Start":"01:04.615 ","End":"01:07.140","Text":"we get a star shape,"},{"Start":"01:07.140 ","End":"01:14.145","Text":"something like this and this is the curve."},{"Start":"01:14.145 ","End":"01:18.420","Text":"Because of the total symmetry,"},{"Start":"01:18.420 ","End":"01:22.700","Text":"what we can do is just compute this part in"},{"Start":"01:22.700 ","End":"01:27.490","Text":"the first quadrant and then multiply the final answer by 4."},{"Start":"01:27.490 ","End":"01:31.970","Text":"I\u0027m just making a note to myself to remember to take this length"},{"Start":"01:31.970 ","End":"01:37.175","Text":"and then multiply it by 4."},{"Start":"01:37.175 ","End":"01:39.775","Text":"Now, length of curve,"},{"Start":"01:39.775 ","End":"01:41.530","Text":"if this was the curve,"},{"Start":"01:41.530 ","End":"01:48.035","Text":"let\u0027s say this is the curve C. Suppose I had it in parametric form or any other form,"},{"Start":"01:48.035 ","End":"01:58.330","Text":"the length is the integral along the curve of just 1 ds."},{"Start":"01:58.330 ","End":"02:02.770","Text":"What I\u0027d like to do first is try and write C in parametric form."},{"Start":"02:02.770 ","End":"02:04.135","Text":"I\u0027m talking about this,"},{"Start":"02:04.135 ","End":"02:12.690","Text":"x to the 2/3 plus y to the 2/3 equals 1."},{"Start":"02:12.690 ","End":"02:15.530","Text":"From experience,"},{"Start":"02:15.530 ","End":"02:21.295","Text":"what I know is that when I have something squared plus something squared equals 1,"},{"Start":"02:21.295 ","End":"02:23.965","Text":"often a trigonometric substitution works."},{"Start":"02:23.965 ","End":"02:28.074","Text":"I could write this as the cube root of x"},{"Start":"02:28.074 ","End":"02:35.430","Text":"squared plus the cube root of y squared equals 1."},{"Start":"02:35.430 ","End":"02:38.535","Text":"What I\u0027d like to do is let"},{"Start":"02:38.535 ","End":"02:44.745","Text":"the cube root of x equal cosine t"},{"Start":"02:44.745 ","End":"02:50.335","Text":"and the cube root of y to be sine t,"},{"Start":"02:50.335 ","End":"02:53.900","Text":"because cosine squared plus sine squared is 1."},{"Start":"02:53.900 ","End":"02:57.810","Text":"From experience, this substitution works."},{"Start":"03:01.280 ","End":"03:05.105","Text":"First of all, I want to write this differently,"},{"Start":"03:05.105 ","End":"03:13.140","Text":"this is the same as saying that x equals cosine cube t and"},{"Start":"03:13.140 ","End":"03:22.860","Text":"y equals sine cube t. Then it satisfies this equation,"},{"Start":"03:22.860 ","End":"03:27.680","Text":"to the power of 3/2 gives me cosine squared plus sine squared is 1."},{"Start":"03:27.680 ","End":"03:31.625","Text":"I also have to figure out where t goes from in 2."},{"Start":"03:31.625 ","End":"03:34.960","Text":"Notice that when t is 0,"},{"Start":"03:34.960 ","End":"03:38.550","Text":"I get cosine 0 is 1,"},{"Start":"03:38.550 ","End":"03:42.150","Text":"so x is 1 cubed and y is 0 cubed,"},{"Start":"03:42.150 ","End":"03:44.075","Text":"that gives me this point."},{"Start":"03:44.075 ","End":"03:46.985","Text":"That\u0027s where t equals 0."},{"Start":"03:46.985 ","End":"03:52.830","Text":"If I let t equals 90 degrees or Pi/2,"},{"Start":"03:52.830 ","End":"03:58.050","Text":"cosine of Pi/2 is 0 and sine Pi/2 is 1,"},{"Start":"03:58.050 ","End":"04:03.640","Text":"so this corresponds to t equals Pi/2 and we\u0027re"},{"Start":"04:03.640 ","End":"04:09.810","Text":"traveling along the curve from 0-Pi/2."},{"Start":"04:09.810 ","End":"04:14.570","Text":"What we get is the integral"},{"Start":"04:14.570 ","End":"04:21.560","Text":"where t goes from 0-Pi/2."},{"Start":"04:21.560 ","End":"04:23.645","Text":"We\u0027re still missing the ds."},{"Start":"04:23.645 ","End":"04:26.240","Text":"When we have a parametric form,"},{"Start":"04:26.240 ","End":"04:31.770","Text":"we can use the formula that ds is the square root"},{"Start":"04:31.770 ","End":"04:39.815","Text":"of x prime squared plus y prime squared dt,"},{"Start":"04:39.815 ","End":"04:45.875","Text":"where the prime means derivative with respect to t. Now,"},{"Start":"04:45.875 ","End":"04:47.585","Text":"if I look at this,"},{"Start":"04:47.585 ","End":"04:50.900","Text":"I\u0027ll make the computation."},{"Start":"04:50.900 ","End":"04:53.480","Text":"In our case, x prime"},{"Start":"04:53.480 ","End":"05:02.365","Text":"is 3 cosine squared t,"},{"Start":"05:02.365 ","End":"05:07.680","Text":"it\u0027s like something cubed so it\u0027s 3 times that something squared but times the inner"},{"Start":"05:07.680 ","End":"05:15.765","Text":"derivative which is minus sine t. All this is x prime."},{"Start":"05:15.765 ","End":"05:18.390","Text":"Let me just write that x prime is this,"},{"Start":"05:18.390 ","End":"05:22.445","Text":"y prime is very similar."},{"Start":"05:22.445 ","End":"05:25.385","Text":"Just here, I have sine squared t,"},{"Start":"05:25.385 ","End":"05:26.810","Text":"and here I don\u0027t have the minus."},{"Start":"05:26.810 ","End":"05:31.630","Text":"Derivative of sine is just cosine t. Now,"},{"Start":"05:31.630 ","End":"05:39.260","Text":"I can plug it in here and get that ds equals the square root of, now,"},{"Start":"05:39.260 ","End":"05:44.270","Text":"x prime squared is 3"},{"Start":"05:44.270 ","End":"05:53.080","Text":"squared times cosine to the 4th t,"},{"Start":"05:53.080 ","End":"05:54.725","Text":"and the minus doesn\u0027t matter,"},{"Start":"05:54.725 ","End":"06:00.560","Text":"sine squared t plus,"},{"Start":"06:00.560 ","End":"06:03.600","Text":"let\u0027s make this a bit longer."},{"Start":"06:04.040 ","End":"06:12.329","Text":"Similar thing here, 3 squared sine to the 4th t,"},{"Start":"06:12.329 ","End":"06:22.820","Text":"cosine squared t. What we can do is we can take the common stuff out."},{"Start":"06:22.820 ","End":"06:25.310","Text":"We have 3 squared, cosine squared,"},{"Start":"06:25.310 ","End":"06:28.675","Text":"sine squared in common."},{"Start":"06:28.675 ","End":"06:34.850","Text":"I can write this as 3 squared cosine squared sine"},{"Start":"06:34.850 ","End":"06:41.140","Text":"squared t. What we\u0027re left with is,"},{"Start":"06:41.140 ","End":"06:44.735","Text":"here we\u0027re just left with a cosine squared."},{"Start":"06:44.735 ","End":"06:49.020","Text":"Here we\u0027re left with a sine squared."},{"Start":"06:50.210 ","End":"06:52.710","Text":"This here is 1,"},{"Start":"06:52.710 ","End":"06:55.510","Text":"square root of 1 is 1."},{"Start":"06:55.640 ","End":"07:00.310","Text":"This bit, the square root is just"},{"Start":"07:00.310 ","End":"07:07.260","Text":"3 cosine t sine t. Normally,"},{"Start":"07:07.260 ","End":"07:11.520","Text":"I\u0027d say absolute value but we\u0027re working in the first quadrant,"},{"Start":"07:11.520 ","End":"07:14.010","Text":"so all these things are positive and this,"},{"Start":"07:14.010 ","End":"07:16.230","Text":"as I said, is times 1."},{"Start":"07:16.230 ","End":"07:19.169","Text":"Getting back to here,"},{"Start":"07:19.169 ","End":"07:28.120","Text":"we get the integral from 0-Pi/2 of 3."},{"Start":"07:28.120 ","End":"07:29.560","Text":"I\u0027ll just change the order."},{"Start":"07:29.560 ","End":"07:32.615","Text":"I\u0027d like to write the sine before the cosine,"},{"Start":"07:32.615 ","End":"07:38.185","Text":"dt because ds, yeah."},{"Start":"07:38.185 ","End":"07:45.210","Text":"I should have written dt, dt, dt, yeah."},{"Start":"07:45.210 ","End":"07:47.320","Text":"Now, the trigonometric formula,"},{"Start":"07:47.320 ","End":"07:53.709","Text":"that if I had a 2 sine t,"},{"Start":"07:53.709 ","End":"07:57.055","Text":"usually written as Alpha in the books, but doesn\u0027t matter,"},{"Start":"07:57.055 ","End":"08:03.635","Text":"2 sine t, cosine t is sine of 2t."},{"Start":"08:03.635 ","End":"08:07.225","Text":"You usually see it. This is on the right, this is on the left."},{"Start":"08:07.225 ","End":"08:10.250","Text":"I don\u0027t have a 2, I have a 3."},{"Start":"08:10.250 ","End":"08:14.755","Text":"What I can do is I can force it to be a 2."},{"Start":"08:14.755 ","End":"08:21.500","Text":"If I make this a 2, then I have to compensate and put a 3/2 here,"},{"Start":"08:21.500 ","End":"08:24.095","Text":"and then everything will be okay."},{"Start":"08:24.095 ","End":"08:30.055","Text":"I have 3/2 times the integral from 0-Pi/2,"},{"Start":"08:30.055 ","End":"08:39.900","Text":"2 sine t cosine t is sine of 2t dt."},{"Start":"08:39.900 ","End":"08:45.095","Text":"The integral of sine is roughly minus cosine."},{"Start":"08:45.095 ","End":"08:50.015","Text":"What I get is 3/2 and"},{"Start":"08:50.015 ","End":"08:58.200","Text":"then I start off with minus cosine of 2t."},{"Start":"08:58.200 ","End":"09:01.975","Text":"But because it\u0027s not t, it\u0027s 2t,"},{"Start":"09:01.975 ","End":"09:06.050","Text":"I have to also put in a 1/2 because if I was to differentiate,"},{"Start":"09:06.050 ","End":"09:07.910","Text":"I\u0027d get times 2."},{"Start":"09:07.910 ","End":"09:12.150","Text":"This has to be taken from 0-Pi/2."},{"Start":"09:15.080 ","End":"09:17.900","Text":"Let\u0027s see what we get."},{"Start":"09:17.900 ","End":"09:21.095","Text":"One thing I like to do is instead of a minus,"},{"Start":"09:21.095 ","End":"09:30.890","Text":"I like to get rid of the minus and then switch these 2 around and what I have"},{"Start":"09:30.890 ","End":"09:35.480","Text":"pulling the 1/2 in front is I have 3/4 and then I"},{"Start":"09:35.480 ","End":"09:42.170","Text":"have cosine of 2t."},{"Start":"09:42.170 ","End":"09:45.150","Text":"But, in reverse order,"},{"Start":"09:45.850 ","End":"09:49.130","Text":"Pi/2-0, that\u0027s what got rid of the minus,"},{"Start":"09:49.130 ","End":"09:51.920","Text":"and the 1/2, I threw in with the 3/2."},{"Start":"09:51.920 ","End":"09:55.830","Text":"Let\u0027s see. If I have 3/4."},{"Start":"09:55.830 ","End":"09:58.720","Text":"Now. If I plug in 0,"},{"Start":"09:58.720 ","End":"10:01.930","Text":"I get cosine of,"},{"Start":"10:01.930 ","End":"10:06.085","Text":"twice 0 with cosine of 0 is 1."},{"Start":"10:06.085 ","End":"10:10.015","Text":"If I plug in Pi/2,"},{"Start":"10:10.015 ","End":"10:12.960","Text":"twice Pi/2 is Pi,"},{"Start":"10:12.960 ","End":"10:20.415","Text":"and cosine of Pi is minus 1."},{"Start":"10:20.415 ","End":"10:22.260","Text":"Let\u0027s see what do I get altogether?"},{"Start":"10:22.260 ","End":"10:24.440","Text":"1 minus minus 1 is 2,"},{"Start":"10:24.440 ","End":"10:28.490","Text":"2 cancels with the 4 partially,"},{"Start":"10:28.490 ","End":"10:32.135","Text":"I make this 3/2."},{"Start":"10:32.135 ","End":"10:35.540","Text":"But that\u0027s not the answer because if you remember,"},{"Start":"10:35.540 ","End":"10:39.610","Text":"we had to multiply this answer by 4."},{"Start":"10:39.610 ","End":"10:42.740","Text":"See, we had this is only a 1/4 of it,"},{"Start":"10:42.740 ","End":"10:45.820","Text":"so what I need for"},{"Start":"10:45.820 ","End":"10:53.320","Text":"the answer is 3/2 times 4."},{"Start":"10:53.320 ","End":"10:56.840","Text":"In other words, let\u0027s see what this comes out to,"},{"Start":"10:56.840 ","End":"11:04.320","Text":"6 and that\u0027s the answer I\u0027m going to highlight. Now we\u0027re done."}],"ID":8814},{"Watched":false,"Name":"Exercise 4","Duration":"4m 39s","ChapterTopicVideoID":8716,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8716.jpeg","UploadDate":"2017-02-13T05:41:04.5400000","DurationForVideoObject":"PT4M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this exercise, you have a question from physics or engineering,"},{"Start":"00:04.380 ","End":"00:10.810","Text":"but don\u0027t worry, I\u0027ll give you all the formulas you need. It\u0027s about a coil."},{"Start":"00:11.180 ","End":"00:15.629","Text":"I can recognize that this is in the shape of a helix."},{"Start":"00:15.629 ","End":"00:21.690","Text":"It\u0027s made of wire and we neglect the thickness of the wire."},{"Start":"00:21.690 ","End":"00:24.465","Text":"It\u0027s just like a line,"},{"Start":"00:24.465 ","End":"00:28.020","Text":"a curve given parametric form as x, y,"},{"Start":"00:28.020 ","End":"00:33.000","Text":"and z in terms of t and we\u0027re given the range t goes from zero to Pi."},{"Start":"00:33.000 ","End":"00:36.195","Text":"We\u0027re also given a density function."},{"Start":"00:36.195 ","End":"00:39.075","Text":"This is actually something called linear density,"},{"Start":"00:39.075 ","End":"00:47.085","Text":"not the usual density of mass per unit of volume but mass per unit length."},{"Start":"00:47.085 ","End":"00:48.420","Text":"Anyway, never mind that."},{"Start":"00:48.420 ","End":"00:53.280","Text":"This is a density function and it\u0027s proportional to z,"},{"Start":"00:53.280 ","End":"00:54.720","Text":"it\u0027s some constant times z,"},{"Start":"00:54.720 ","End":"00:59.825","Text":"the positive constant, and we have to compute the mass of the coil."},{"Start":"00:59.825 ","End":"01:04.940","Text":"In general, we want the formula for the mass given the density."},{"Start":"01:04.940 ","End":"01:14.060","Text":"In general, the mass is the integral along the parametrized curve,"},{"Start":"01:14.060 ","End":"01:19.245","Text":"c of Delta of x,"},{"Start":"01:19.245 ","End":"01:25.685","Text":"y, and z, ds like type 1 line integral."},{"Start":"01:25.685 ","End":"01:28.100","Text":"But in our case,"},{"Start":"01:28.100 ","End":"01:35.970","Text":"we know that delta is just kz."},{"Start":"01:35.970 ","End":"01:40.905","Text":"What we\u0027re missing still is ds and in 3 dimensions,"},{"Start":"01:40.905 ","End":"01:43.590","Text":"the formula for ds,"},{"Start":"01:43.590 ","End":"01:48.680","Text":"1 of the variations of the several ways of writing the formula."},{"Start":"01:48.680 ","End":"01:54.640","Text":"One way is just to write it as x prime squared,"},{"Start":"01:54.640 ","End":"01:58.110","Text":"plus y prime squared,"},{"Start":"01:58.110 ","End":"02:02.150","Text":"plus z prime squared dt,"},{"Start":"02:02.150 ","End":"02:05.760","Text":"where the prime means derivative with respect to t. Sometimes,"},{"Start":"02:05.760 ","End":"02:07.790","Text":"you write it as parentheses t,"},{"Start":"02:07.790 ","End":"02:09.920","Text":"I was just saving space."},{"Start":"02:09.920 ","End":"02:17.945","Text":"We could actually compute this here is quite easy to do because x prime"},{"Start":"02:17.945 ","End":"02:26.540","Text":"is minus sine t. Here I have minus sine t. Let me just write the derivatives."},{"Start":"02:26.540 ","End":"02:33.480","Text":"Y prime is cosine t and z prime is just 2."},{"Start":"02:33.480 ","End":"02:43.915","Text":"I need to put each of these squared and add them and then take the square root dt."},{"Start":"02:43.915 ","End":"02:48.430","Text":"Sine squared plus cosine squared is 1,"},{"Start":"02:48.430 ","End":"02:51.090","Text":"and 2 squared is 4."},{"Start":"02:51.090 ","End":"02:58.180","Text":"All together, I have the square root of 1 plus 4 is 5, square roots of 5dt."},{"Start":"03:01.370 ","End":"03:07.290","Text":"If we plug everything in and we plug in the Delta is kz,"},{"Start":"03:08.440 ","End":"03:11.314","Text":"we get the integral,"},{"Start":"03:11.314 ","End":"03:15.245","Text":"becomes the integral of 1 variable which is the parameter,"},{"Start":"03:15.245 ","End":"03:18.545","Text":"and it goes from 0 to Pi."},{"Start":"03:18.545 ","End":"03:22.620","Text":"Delta is just kz."},{"Start":"03:23.320 ","End":"03:33.170","Text":"But instead of z, we write what z is, which is 2t."},{"Start":"03:33.170 ","End":"03:36.730","Text":"It\u0027s k times 2t,"},{"Start":"03:36.730 ","End":"03:39.585","Text":"and then we need the ds,"},{"Start":"03:39.585 ","End":"03:43.180","Text":"which is root 5dt."},{"Start":"03:46.390 ","End":"03:49.940","Text":"I\u0027ll take some of the constants in front."},{"Start":"03:49.940 ","End":"03:53.270","Text":"I\u0027ll take out root 5,"},{"Start":"03:53.270 ","End":"03:56.360","Text":"I\u0027ll take out k, I was going to take the 2"},{"Start":"03:56.360 ","End":"04:00.190","Text":"out and then I realized that\u0027s useful for me to leave the 2 in."},{"Start":"04:00.190 ","End":"04:04.100","Text":"The reason I\u0027m leaving the 2 in is because the integral"},{"Start":"04:04.100 ","End":"04:08.585","Text":"of 2t comes out to be an t squared."},{"Start":"04:08.585 ","End":"04:13.140","Text":"What we get here is root 5k."},{"Start":"04:13.140 ","End":"04:19.080","Text":"Then we have t squared from zero to Pi."},{"Start":"04:19.080 ","End":"04:21.870","Text":"If I plugin Pi, I get Pi squared,"},{"Start":"04:21.870 ","End":"04:24.300","Text":"I plugin 0, I get just 0."},{"Start":"04:24.300 ","End":"04:26.700","Text":"Altogether, this is Pi squared."},{"Start":"04:26.700 ","End":"04:37.920","Text":"The answer would be root 5k times Pi squared and that\u0027s the answer. That\u0027s all there is."}],"ID":8815},{"Watched":false,"Name":"Exercise 5 Part a","Duration":"7m ","ChapterTopicVideoID":8717,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8717.jpeg","UploadDate":"2017-02-13T05:42:39.0700000","DurationForVideoObject":"PT7M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.300","Text":"In this exercise, we have a line integral of type 2 over a parameterized curve C,"},{"Start":"00:09.300 ","End":"00:11.835","Text":"given it\u0027s in 2 dimensions,"},{"Start":"00:11.835 ","End":"00:15.030","Text":"x and y in terms of t and here\u0027s the range of"},{"Start":"00:15.030 ","End":"00:18.705","Text":"t so we just have to basically substitute everything into"},{"Start":"00:18.705 ","End":"00:25.855","Text":"language of t. The integral becomes the integral t goes from 0 to Pi over 2."},{"Start":"00:25.855 ","End":"00:29.765","Text":"I\u0027ll emphasize and write t equals 0 up to"},{"Start":"00:29.765 ","End":"00:34.070","Text":"Pi over 2 and then everything has to be substituted."},{"Start":"00:34.070 ","End":"00:42.285","Text":"We have here 2x we take along the curve so x is cosine t,"},{"Start":"00:42.285 ","End":"00:47.610","Text":"y is sine t and then I need dx."},{"Start":"00:47.610 ","End":"00:51.745","Text":"Let me just do dx and dy on this side."},{"Start":"00:51.745 ","End":"00:54.500","Text":"If I take the d of both sides,"},{"Start":"00:54.500 ","End":"00:59.585","Text":"we have 1dx which is dx and the other side,"},{"Start":"00:59.585 ","End":"01:06.200","Text":"I get minus sine t dt and"},{"Start":"01:06.200 ","End":"01:14.500","Text":"the dy becomes 1dy equals cosine t dt."},{"Start":"01:18.650 ","End":"01:21.880","Text":"Yeah, put the brackets there."},{"Start":"01:22.640 ","End":"01:26.285","Text":"What I get here,"},{"Start":"01:26.285 ","End":"01:33.270","Text":"dx is minus sine t and then dt."},{"Start":"01:38.210 ","End":"01:40.410","Text":"Later I\u0027m going to take the dt,"},{"Start":"01:40.410 ","End":"01:43.580","Text":"I\u0027m going to get a dt here and later I\u0027ll take it outside the brackets."},{"Start":"01:43.580 ","End":"01:49.445","Text":"Meanwhile, x squared is cosine squared"},{"Start":"01:49.445 ","End":"01:56.000","Text":"t and y squared"},{"Start":"01:56.000 ","End":"02:01.190","Text":"is sine squared t and here we have dy,"},{"Start":"02:01.190 ","End":"02:10.770","Text":"which is cosine t dt and we have a dt here and a dt here,"},{"Start":"02:10.770 ","End":"02:15.365","Text":"let\u0027s combine everything and put a single dt at the end."},{"Start":"02:15.365 ","End":"02:18.305","Text":"From 0 to Pi over 2,"},{"Start":"02:18.305 ","End":"02:21.270","Text":"let me start opening the brackets."},{"Start":"02:21.280 ","End":"02:23.435","Text":"What do we have here?"},{"Start":"02:23.435 ","End":"02:29.585","Text":"We have minus 2 cosine t,"},{"Start":"02:29.585 ","End":"02:36.135","Text":"sine squared t and from the second 1,"},{"Start":"02:36.135 ","End":"02:44.090","Text":"I have cosine squared cosine that\u0027s cosine cube t and then sine squared"},{"Start":"02:44.090 ","End":"02:52.920","Text":"times cosine is just sine squared t cosine t and all this dt."},{"Start":"02:53.560 ","End":"02:59.510","Text":"Notice that cosine times sine squared is the same as sine squared times"},{"Start":"02:59.510 ","End":"03:06.730","Text":"cosine so this could be combined and we just have minus 1 of these."},{"Start":"03:06.730 ","End":"03:14.415","Text":"We can also take the cosine outside the brackets so let\u0027s see,"},{"Start":"03:14.415 ","End":"03:17.080","Text":"from 0 to Pi over 2."},{"Start":"03:17.080 ","End":"03:21.160","Text":"I\u0027m taking cosine t outside the brackets."},{"Start":"03:21.160 ","End":"03:25.505","Text":"Now, then we said these 2 combine,"},{"Start":"03:25.505 ","End":"03:33.590","Text":"we get minus 2 sine squared plus sine squared is just minus sine squared t and from here,"},{"Start":"03:33.590 ","End":"03:43.500","Text":"I get just cosine squared t dt."},{"Start":"03:43.500 ","End":"03:45.390","Text":"There\u0027s more than 1 way to do this."},{"Start":"03:45.390 ","End":"03:47.625","Text":"I\u0027m going to suggest using"},{"Start":"03:47.625 ","End":"03:53.510","Text":"a trigonometric identity to convert sine squared into cosine squared because"},{"Start":"03:53.510 ","End":"04:04.040","Text":"sine squared t is 1 minus cosine squared t. If I do the computation,"},{"Start":"04:04.040 ","End":"04:10.370","Text":"we\u0027ll get cosine squared minus 1 plus cosine squared."},{"Start":"04:10.370 ","End":"04:20.855","Text":"In other words, I\u0027ll get the integral same limits of and you know what?"},{"Start":"04:20.855 ","End":"04:21.995","Text":"I changed my mind."},{"Start":"04:21.995 ","End":"04:25.820","Text":"I\u0027m going to convert the cosine into sine so I\u0027m"},{"Start":"04:25.820 ","End":"04:30.859","Text":"erasing this and I\u0027m going to write the cosine squared"},{"Start":"04:30.859 ","End":"04:37.370","Text":"as 1 minus sine squared t. The reason I prefer to stay with sine and not with"},{"Start":"04:37.370 ","End":"04:44.900","Text":"cosine is because I have the derivative of sine here and then I can use a substitution."},{"Start":"04:47.780 ","End":"04:50.070","Text":"I\u0027ll do this part first,"},{"Start":"04:50.070 ","End":"04:59.875","Text":"this is 1 minus twice sine squared t and then I\u0027ll take the cosine t dt"},{"Start":"04:59.875 ","End":"05:09.790","Text":"and now I\u0027ll do the substitution I mentioned by letting u"},{"Start":"05:09.790 ","End":"05:19.585","Text":"equal sine t and then 1du"},{"Start":"05:19.585 ","End":"05:27.470","Text":"equals cosine t dt and so this integral"},{"Start":"05:27.490 ","End":"05:38.390","Text":"becomes the integral of 1 minus 2u squared."},{"Start":"05:38.390 ","End":"05:43.250","Text":"The cosine t dt is du but that\u0027s not all because I also have to"},{"Start":"05:43.250 ","End":"05:49.475","Text":"substitute the limits unless I want to do it with an indefinite and return to t,"},{"Start":"05:49.475 ","End":"05:55.880","Text":"I\u0027ll rather stay in the land of u and so I say that when t is 0,"},{"Start":"05:55.880 ","End":"06:00.090","Text":"u equals and when t equals Pi over 2,"},{"Start":"06:00.090 ","End":"06:02.160","Text":"we\u0027ll see what u equals."},{"Start":"06:02.160 ","End":"06:06.419","Text":"Sine t, so sine of 0 is 0,"},{"Start":"06:06.419 ","End":"06:14.210","Text":"sine of Pi over 2 is 1 and so this is the integral from 0 to 1."},{"Start":"06:14.210 ","End":"06:19.435","Text":"This is very straightforward."},{"Start":"06:19.435 ","End":"06:21.755","Text":"This integral of 1 is u."},{"Start":"06:21.755 ","End":"06:23.960","Text":"The integral of this,"},{"Start":"06:23.960 ","End":"06:33.635","Text":"raise it to u cubed and divide by the 3 so 2/3 u cubed."},{"Start":"06:33.635 ","End":"06:37.370","Text":"This I want to take from 0 to"},{"Start":"06:37.370 ","End":"06:45.610","Text":"1 and if I plug in 0 for u,"},{"Start":"06:45.610 ","End":"06:48.725","Text":"I don\u0027t get anything, so I just need to plug in the 1,"},{"Start":"06:48.725 ","End":"06:53.030","Text":"so I get 1 minus 2/3,"},{"Start":"06:53.030 ","End":"06:59.760","Text":"which is 1/3 and that\u0027s the answer."}],"ID":8816},{"Watched":false,"Name":"Exercise 5 Part b","Duration":"2m 43s","ChapterTopicVideoID":8718,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8718.jpeg","UploadDate":"2017-02-13T05:43:16.8870000","DurationForVideoObject":"PT2M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"Here we have this type 2 line integral."},{"Start":"00:03.720 ","End":"00:07.110","Text":"It\u0027s in 2-dimensions, just x, and y."},{"Start":"00:07.110 ","End":"00:11.970","Text":"The line integral over a parametrized curve C,"},{"Start":"00:11.970 ","End":"00:15.720","Text":"where we have x and y in terms of t and we have the range for t."},{"Start":"00:15.720 ","End":"00:19.530","Text":"We just translate everything into t."},{"Start":"00:19.530 ","End":"00:26.710","Text":"So this integral becomes the integral where t goes from 0-1."},{"Start":"00:27.290 ","End":"00:32.489","Text":"Just write the t for emphasis and then I just substitute everything."},{"Start":"00:32.489 ","End":"00:35.610","Text":"So 2x is 2t,"},{"Start":"00:35.610 ","End":"00:38.940","Text":"y is t squared."},{"Start":"00:38.940 ","End":"00:41.545","Text":"Now I need dx."},{"Start":"00:41.545 ","End":"00:44.760","Text":"Let\u0027s do those at the side."},{"Start":"00:44.950 ","End":"00:48.170","Text":"From this x equals t,"},{"Start":"00:48.170 ","End":"00:49.970","Text":"I take the d of both sides."},{"Start":"00:49.970 ","End":"00:55.220","Text":"So I have dx equals dt and for y,"},{"Start":"00:55.220 ","End":"00:58.640","Text":"derivative of y is 1 with respect to y."},{"Start":"00:58.640 ","End":"01:03.900","Text":"So it\u0027s 1dy or just dy and here derivative is 2tdt."},{"Start":"01:05.170 ","End":"01:08.265","Text":"Now I\u0027ve got those as well."},{"Start":"01:08.265 ","End":"01:14.780","Text":"Dx is dt and I\u0027m going to have another one with a dt"},{"Start":"01:14.780 ","End":"01:18.385","Text":"and later I\u0027ll just take the dt outside the brackets."},{"Start":"01:18.385 ","End":"01:27.960","Text":"Let\u0027s see, x squared is t squared minus y is t squared."},{"Start":"01:27.960 ","End":"01:30.460","Text":"That\u0027s lucky."},{"Start":"01:30.770 ","End":"01:33.130","Text":"I\u0027ll write dy anyway,"},{"Start":"01:33.130 ","End":"01:37.880","Text":"even though I already see this is going to be zero but I\u0027ll write dy as 2tdt."},{"Start":"01:40.100 ","End":"01:44.200","Text":"We got lucky here because t squared minus t squared is 0."},{"Start":"01:44.200 ","End":"01:51.020","Text":"This whole thing can just be removed and we just have this integral to do."},{"Start":"01:51.020 ","End":"01:54.960","Text":"The integral of 2t is t squared,"},{"Start":"01:54.960 ","End":"01:59.125","Text":"integral of t squared is 1/3t cubed."},{"Start":"01:59.125 ","End":"02:11.129","Text":"I need to take this from 0-1 and when I plug in 1,"},{"Start":"02:11.129 ","End":"02:15.150","Text":"I get 1 squared plus 1,"},{"Start":"02:15.150 ","End":"02:26.295","Text":"I\u0027ll write it 1 squared plus 1/3 times 1 cubed minus 0 squared plus 1/3 0 cubed."},{"Start":"02:26.295 ","End":"02:31.840","Text":"This is 0 and this is just 1 plus 1/3,"},{"Start":"02:33.950 ","End":"02:42.400","Text":"1 plus a 1/3 is 1 and a 1/3 or 4/3 and that\u0027s the answer."}],"ID":8817},{"Watched":false,"Name":"Exercise 6 Part a","Duration":"4m 20s","ChapterTopicVideoID":8719,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8719.jpeg","UploadDate":"2017-02-13T05:44:22.8200000","DurationForVideoObject":"PT4M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.840","Text":"In this exercise, we want to compute this type 2 line integral"},{"Start":"00:06.840 ","End":"00:15.690","Text":"where the C here is a path from 0,0 to 2,4."},{"Start":"00:15.690 ","End":"00:20.130","Text":"Of course, this depends on the path C. In part A,"},{"Start":"00:20.130 ","End":"00:24.620","Text":"we\u0027ll take the straight line y equals 2x,"},{"Start":"00:24.620 ","End":"00:30.195","Text":"this will look something like this and that will be"},{"Start":"00:30.195 ","End":"00:38.785","Text":"the C. In the following clip, we\u0027ll go and do it along the parabola y equals x squared."},{"Start":"00:38.785 ","End":"00:40.770","Text":"This is our path C,"},{"Start":"00:40.770 ","End":"00:43.545","Text":"it goes from 0,0 to 2,4."},{"Start":"00:43.545 ","End":"00:48.445","Text":"What we want to do is parameterize this path."},{"Start":"00:48.445 ","End":"00:51.710","Text":"When y is given as a function of x,"},{"Start":"00:51.710 ","End":"00:56.820","Text":"then the easiest thing to do is to let x equals t,"},{"Start":"00:57.290 ","End":"01:01.980","Text":"and then y is just the same function,"},{"Start":"01:01.980 ","End":"01:03.570","Text":"instead of x you put t,"},{"Start":"01:03.570 ","End":"01:06.970","Text":"we\u0027ll have y equals 2t."},{"Start":"01:07.550 ","End":"01:11.360","Text":"Since we know the beginning and the endpoints,"},{"Start":"01:11.360 ","End":"01:14.710","Text":"you see that x goes from 0,"},{"Start":"01:14.710 ","End":"01:19.770","Text":"this 0 here for the x up to 2."},{"Start":"01:19.770 ","End":"01:23.700","Text":"So x goes between 0 and 2."},{"Start":"01:23.700 ","End":"01:34.040","Text":"Or let\u0027s replace x by t. Now we need to convert this to an integral in terms of t."},{"Start":"01:34.040 ","End":"01:40.850","Text":"Notice that from here, we can get that"},{"Start":"01:40.850 ","End":"01:51.305","Text":"dx is equal to dt and dy is equal to 2dt."},{"Start":"01:51.305 ","End":"01:56.305","Text":"A simple differentiation, derivative here is 1, here it\u0027s 2,"},{"Start":"01:56.305 ","End":"02:00.725","Text":"this is 1dt which is just dt,"},{"Start":"02:00.725 ","End":"02:05.160","Text":"1dx equals 1dt, 1dy equals 2dt."},{"Start":"02:05.870 ","End":"02:08.445","Text":"We just substitute in here,"},{"Start":"02:08.445 ","End":"02:14.210","Text":"instead of the curve we put the value of the parameter t."},{"Start":"02:14.210 ","End":"02:20.840","Text":"In other words we have t going from 0-2,"},{"Start":"02:21.170 ","End":"02:26.340","Text":"and then we plug in y is 2t."},{"Start":"02:26.340 ","End":"02:35.219","Text":"We have 2t and then dx is dt plus x squared"},{"Start":"02:35.219 ","End":"02:43.650","Text":"is t squared and dy is 2dt."},{"Start":"02:43.650 ","End":"02:46.695","Text":"I just want to bring the dt out,"},{"Start":"02:46.695 ","End":"02:48.555","Text":"it\u0027s just 1 integral,"},{"Start":"02:48.555 ","End":"02:51.950","Text":"just 1 time dt. What do we have here?"},{"Start":"02:51.950 ","End":"02:55.915","Text":"We have 2t plus 2t squared,"},{"Start":"02:55.915 ","End":"03:04.045","Text":"dt from 0-2."},{"Start":"03:04.045 ","End":"03:11.460","Text":"Straightforward integral, 2t gives me t squared, 2t squared gives,"},{"Start":"03:11.460 ","End":"03:16.215","Text":"see, it raises it to 1 to the power of 3 and divide by 3,"},{"Start":"03:16.215 ","End":"03:23.170","Text":"so 2/3t cubed this from 0-2."},{"Start":"03:23.890 ","End":"03:27.350","Text":"When we substitute t equals 0,"},{"Start":"03:27.350 ","End":"03:29.630","Text":"both of these are 0, forget about that."},{"Start":"03:29.630 ","End":"03:31.490","Text":"We just need to substitute 2,"},{"Start":"03:31.490 ","End":"03:36.990","Text":"so we get 2 squared is 4 plus 2/3,"},{"Start":"03:36.990 ","End":"03:39.615","Text":"2 cubed is 8."},{"Start":"03:39.615 ","End":"03:42.080","Text":"Let\u0027s see if we can simplify this."},{"Start":"03:42.080 ","End":"03:48.225","Text":"If I put it over 3, I\u0027ll get what?"},{"Start":"03:48.225 ","End":"03:54.735","Text":"12 over 3 plus 16 over 3,"},{"Start":"03:54.735 ","End":"04:03.015","Text":"that\u0027s 28 over 3 and that\u0027s the answer which I\u0027ll highlight."},{"Start":"04:03.015 ","End":"04:07.190","Text":"If you prefer mixed fractions,"},{"Start":"04:07.190 ","End":"04:09.320","Text":"then we could write this also as,"},{"Start":"04:09.320 ","End":"04:12.530","Text":"3 in 27 goes 9 remainder 1,"},{"Start":"04:12.530 ","End":"04:14.210","Text":"9 and 1/3 anyway."},{"Start":"04:14.210 ","End":"04:15.500","Text":"That\u0027s all there is to it."},{"Start":"04:15.500 ","End":"04:17.645","Text":"Now, on to part b,"},{"Start":"04:17.645 ","End":"04:20.760","Text":"in the next clip that is."}],"ID":8818},{"Watched":false,"Name":"Exercise 6 Part b","Duration":"2m 48s","ChapterTopicVideoID":8720,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8720.jpeg","UploadDate":"2017-02-13T05:44:57.5670000","DurationForVideoObject":"PT2M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"We just finished part A and we want to move on to part B."},{"Start":"00:03.660 ","End":"00:05.310","Text":"Let me erase what we don\u0027t need,"},{"Start":"00:05.310 ","End":"00:07.410","Text":"first of all, this stuff."},{"Start":"00:07.410 ","End":"00:15.750","Text":"The curve is going to be different and the function of t will be different."},{"Start":"00:15.750 ","End":"00:21.990","Text":"What we\u0027ll get in fact is y equals t squared"},{"Start":"00:21.990 ","End":"00:31.600","Text":"and dy therefore will be 2t dt."},{"Start":"00:31.600 ","End":"00:36.200","Text":"This part still stays and the sketch is different."},{"Start":"00:36.200 ","End":"00:38.870","Text":"It\u0027s a bit of a parabola."},{"Start":"00:38.870 ","End":"00:41.465","Text":"It doesn\u0027t have to be accurate."},{"Start":"00:41.465 ","End":"00:45.395","Text":"C, doesn\u0027t need a sketch at all really,"},{"Start":"00:45.395 ","End":"00:52.565","Text":"and now we\u0027ll go and do it with these equations."},{"Start":"00:52.565 ","End":"00:57.440","Text":"We get the integral from 0-2,"},{"Start":"00:57.440 ","End":"01:01.340","Text":"so it must be 4, y,"},{"Start":"01:01.340 ","End":"01:05.645","Text":"this time is t squared and dx is"},{"Start":"01:05.645 ","End":"01:13.830","Text":"dt and x squared is t squared,"},{"Start":"01:14.140 ","End":"01:25.780","Text":"and dy is 2t dt."},{"Start":"01:25.780 ","End":"01:30.450","Text":"Let\u0027s put it all with a single dt."},{"Start":"01:30.450 ","End":"01:39.250","Text":"What I have here is t squared plus 2t cubed dt."},{"Start":"01:40.070 ","End":"01:47.205","Text":"We get from here 1/3 t cubed, from here,"},{"Start":"01:47.205 ","End":"01:52.635","Text":"2t^4 over 4, 2 over 4 is easier to write as a half,"},{"Start":"01:52.635 ","End":"01:56.820","Text":"all this from 0-2."},{"Start":"01:56.820 ","End":"02:01.530","Text":"Now, 0 when we plug it in gives nothing so I only need to put in the 2."},{"Start":"02:01.530 ","End":"02:06.030","Text":"Here 2 cubed is 8 over 3,"},{"Start":"02:06.030 ","End":"02:17.040","Text":"and here t^4 is"},{"Start":"02:17.040 ","End":"02:24.400","Text":"16 over 2 is 8."},{"Start":"02:24.830 ","End":"02:29.625","Text":"This is 8 plus 2 and 2/3,"},{"Start":"02:29.625 ","End":"02:39.860","Text":"so I can write it as 10 and 2/3 and this is the answer as a mixed number."},{"Start":"02:39.860 ","End":"02:45.380","Text":"But if you prefer an improper fraction you can also write it as 32 over 3."},{"Start":"02:45.380 ","End":"02:48.240","Text":"Anyway, we are done."}],"ID":8819},{"Watched":false,"Name":"Exercise 7 Part a","Duration":"5m 40s","ChapterTopicVideoID":8696,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8696.jpeg","UploadDate":"2017-02-13T05:01:43.5470000","DurationForVideoObject":"PT5M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.805","Text":"In this exercise, we have to compute a line integral of type 2,"},{"Start":"00:05.805 ","End":"00:09.735","Text":"and it\u0027s in 2 dimensions is just x and y."},{"Start":"00:09.735 ","End":"00:13.740","Text":"Along each of the following curves, it\u0027s going to be 4 of them,"},{"Start":"00:13.740 ","End":"00:16.935","Text":"and I\u0027ll do them 1 at a time."},{"Start":"00:16.935 ","End":"00:20.400","Text":"This is not really precise notation,"},{"Start":"00:20.400 ","End":"00:25.020","Text":"what we really mean is the integral along a curve c,"},{"Start":"00:25.020 ","End":"00:28.095","Text":"and c will take us from here to here,"},{"Start":"00:28.095 ","End":"00:30.645","Text":"will make it as a parametrized curve,"},{"Start":"00:30.645 ","End":"00:35.850","Text":"and will be 4 different possibilities for each of these subsections."},{"Start":"00:35.850 ","End":"00:39.905","Text":"Let\u0027s start with a, y squared equals x."},{"Start":"00:39.905 ","End":"00:44.850","Text":"First of all, notice that these points, 1,"},{"Start":"00:44.850 ","End":"00:48.050","Text":"1, and 4, 2 are really on the curve."},{"Start":"00:48.050 ","End":"00:52.849","Text":"In fact, it\u0027s easier to write it as x equals y squared."},{"Start":"00:52.849 ","End":"00:59.490","Text":"You can see that 1 squared is 1 and 2 squared is 4,"},{"Start":"01:00.380 ","End":"01:05.410","Text":"so the parameter will go from 1-2."},{"Start":"01:05.900 ","End":"01:08.205","Text":"When x is y squared,"},{"Start":"01:08.205 ","End":"01:13.800","Text":"easiest thing to do is to take y as t. What we\u0027ll do is,"},{"Start":"01:13.800 ","End":"01:19.500","Text":"we\u0027ll describe c as x equals,"},{"Start":"01:19.500 ","End":"01:20.985","Text":"I\u0027ll write that in a moment,"},{"Start":"01:20.985 ","End":"01:26.060","Text":"y equals t. Since x is y squared,"},{"Start":"01:26.060 ","End":"01:32.270","Text":"then x is t squared and y equals t. The parameter goes from,"},{"Start":"01:32.270 ","End":"01:34.635","Text":"since y is the parameter t,"},{"Start":"01:34.635 ","End":"01:39.780","Text":"from 1-2, so I\u0027ll write it like that."},{"Start":"01:39.780 ","End":"01:44.630","Text":"Then in b, c, and d we\u0027ll have a different curve at each time,"},{"Start":"01:44.630 ","End":"01:47.105","Text":"but it will be the same expression,"},{"Start":"01:47.105 ","End":"01:49.440","Text":"line integral along c,"},{"Start":"01:50.150 ","End":"01:53.930","Text":"and each of them will take us from here to here."},{"Start":"01:53.930 ","End":"01:57.310","Text":"Let\u0027s get started with this 1 then."},{"Start":"01:57.310 ","End":"02:00.650","Text":"Now, this integral becomes, because it\u0027s parametrized,"},{"Start":"02:00.650 ","End":"02:04.430","Text":"we just take the upper and lower limits for the parameter,"},{"Start":"02:04.430 ","End":"02:06.185","Text":"so it\u0027s from 1-2."},{"Start":"02:06.185 ","End":"02:10.255","Text":"Maybe I\u0027ll emphasize it by putting t equals 1-2,"},{"Start":"02:10.255 ","End":"02:12.585","Text":"then we just substitute,"},{"Start":"02:12.585 ","End":"02:15.810","Text":"x is t squared,"},{"Start":"02:15.810 ","End":"02:22.215","Text":"y is t, and then we need dx."},{"Start":"02:22.215 ","End":"02:28.175","Text":"From here, if I differentiate I\u0027ll get 1 dx,"},{"Start":"02:28.175 ","End":"02:32.320","Text":"the d of both of them on here, I\u0027ll have 2tdt."},{"Start":"02:33.620 ","End":"02:36.060","Text":"As for y, well,"},{"Start":"02:36.060 ","End":"02:39.960","Text":"y equals t, so dy equals dt."},{"Start":"02:39.960 ","End":"02:44.350","Text":"Here I need dx, which is 2tdt,"},{"Start":"02:44.930 ","End":"02:51.014","Text":"so 2tdt plus y,"},{"Start":"02:51.014 ","End":"02:54.570","Text":"which is t minus x,"},{"Start":"02:54.570 ","End":"02:56.385","Text":"which is t squared,"},{"Start":"02:56.385 ","End":"03:00.405","Text":"and dy is just dt."},{"Start":"03:00.405 ","End":"03:10.095","Text":"What we want do is just write this as 1 single dt of integral from 1-2,"},{"Start":"03:10.095 ","End":"03:13.955","Text":"and we\u0027ll see what we can take out."},{"Start":"03:13.955 ","End":"03:20.160","Text":"Multiplying 2t times t squared is 2t cubed."},{"Start":"03:20.160 ","End":"03:24.630","Text":"Here, I have plus 2t squared,"},{"Start":"03:24.630 ","End":"03:29.060","Text":"but the 2t squared from here, minus the t squared"},{"Start":"03:29.060 ","End":"03:33.880","Text":"from here, will just leave us with 1t squared."},{"Start":"03:33.880 ","End":"03:38.880","Text":"Then what we\u0027re left with is still plus t,"},{"Start":"03:38.880 ","End":"03:41.950","Text":"and then all this is dt."},{"Start":"03:42.680 ","End":"03:49.320","Text":"Straightforward integral, 2t cubed becomes t^4,"},{"Start":"03:49.320 ","End":"03:53.640","Text":"divide by 4, 2 over 4 is a half."},{"Start":"03:53.640 ","End":"03:57.955","Text":"Next 1, I need t cubed divide by 3."},{"Start":"03:57.955 ","End":"04:02.915","Text":"Here I need t squared, and I divide by 2,"},{"Start":"04:02.915 ","End":"04:08.460","Text":"so this is what we have, and evaluate it from 1-2."},{"Start":"04:08.840 ","End":"04:11.520","Text":"Let\u0027s see what we get."},{"Start":"04:11.520 ","End":"04:13.950","Text":"If we put in 2,"},{"Start":"04:13.950 ","End":"04:20.265","Text":"we have 1/2 times 2^4 is"},{"Start":"04:20.265 ","End":"04:30.345","Text":"16 plus 1/3 times 8 plus 1/2 times 4,"},{"Start":"04:30.345 ","End":"04:32.475","Text":"that\u0027s for the 2."},{"Start":"04:32.475 ","End":"04:39.945","Text":"For the 1, we just get a 1/2 plus a 1/3 plus a 1/2."},{"Start":"04:39.945 ","End":"04:43.650","Text":"What does this come out to be?"},{"Start":"04:43.650 ","End":"04:54.350","Text":"The first 1, we\u0027ll get 8 plus 2 from here,"},{"Start":"04:54.350 ","End":"05:02.280","Text":"that\u0027s altogether 10, 8/3 is 2 2/3."},{"Start":"05:02.280 ","End":"05:04.370","Text":"Altogether, I\u0027ll do it in mixed numbers,"},{"Start":"05:04.370 ","End":"05:08.070","Text":"12 and 2/3 for the first 1,"},{"Start":"05:09.310 ","End":"05:15.090","Text":"1/2 and a 1/2 is 1, 1 1/3."},{"Start":"05:15.090 ","End":"05:23.390","Text":"Let\u0027s see, that comes out to be 11 and 1/3, could leave the answer like that,"},{"Start":"05:23.390 ","End":"05:27.170","Text":"or we could write it as an improper fraction,"},{"Start":"05:27.170 ","End":"05:33.780","Text":"11 times 3 plus 1 is 34, 34/3."},{"Start":"05:33.780 ","End":"05:34.920","Text":"Either 1 of these,"},{"Start":"05:34.920 ","End":"05:36.465","Text":"I\u0027ll just go with this 1,"},{"Start":"05:36.465 ","End":"05:40.260","Text":"that\u0027s the answer. We\u0027re done for part a."}],"ID":8795},{"Watched":false,"Name":"Exercise 7 Part b","Duration":"5m 28s","ChapterTopicVideoID":8697,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8697.jpeg","UploadDate":"2017-02-13T05:03:45.8470000","DurationForVideoObject":"PT5M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.050","Text":"That was part a. Now, let\u0027s get on to part b. I erased what I don\u0027t need."},{"Start":"00:07.050 ","End":"00:09.960","Text":"I don\u0027t need this."},{"Start":"00:09.960 ","End":"00:12.120","Text":"We\u0027re still going to have x as a function of t,"},{"Start":"00:12.120 ","End":"00:13.830","Text":"y as a function of t,"},{"Start":"00:13.830 ","End":"00:17.070","Text":"and t is going to go from something to something."},{"Start":"00:17.070 ","End":"00:22.110","Text":"I can already tell you that t is going from"},{"Start":"00:22.110 ","End":"00:28.880","Text":"0-1, because I\u0027m going to use the formula for a line segment joining 2 points."},{"Start":"00:28.880 ","End":"00:32.540","Text":"I\u0027ll remind you, in general, that if I have 2 points,"},{"Start":"00:32.540 ","End":"00:34.265","Text":"let say x naught,"},{"Start":"00:34.265 ","End":"00:39.200","Text":"y naught, and x_1, y_1,"},{"Start":"00:39.200 ","End":"00:41.555","Text":"which in our case will turn out,"},{"Start":"00:41.555 ","End":"00:42.890","Text":"I mean they are 1,"},{"Start":"00:42.890 ","End":"00:45.455","Text":"1, and 4, 2,"},{"Start":"00:45.455 ","End":"00:47.990","Text":"but I\u0027ll write the general formula, then"},{"Start":"00:47.990 ","End":"00:52.789","Text":"the parametric curve is given by the several formulas."},{"Start":"00:52.789 ","End":"01:02.780","Text":"One way is to say that x equals the first point plus t"},{"Start":"01:02.780 ","End":"01:11.860","Text":"times the difference between the axes x_1 minus x_0,"},{"Start":"01:11.860 ","End":"01:22.714","Text":"and y is the first y plus t times the difference, second y minus the first y,"},{"Start":"01:22.714 ","End":"01:30.040","Text":"in both values of the parameter is from 0-1."},{"Start":"01:30.920 ","End":"01:39.945","Text":"In our case, what we\u0027ll get is x"},{"Start":"01:39.945 ","End":"01:48.900","Text":"equals the first point is 1, and the difference,"},{"Start":"01:48.900 ","End":"01:57.630","Text":"4 minus 1 is 3 times t, so it\u0027s 1 plus 3t,"},{"Start":"01:57.630 ","End":"02:02.590","Text":"and y is equal to,"},{"Start":"02:03.230 ","End":"02:12.435","Text":"it\u0027s the first y, which is 1, plus the difference 2 minus 1 is 1 times"},{"Start":"02:12.435 ","End":"02:17.160","Text":"t. What I get is here,"},{"Start":"02:17.160 ","End":"02:24.625","Text":"1 plus 3t and here 1 plus t. We can check,"},{"Start":"02:24.625 ","End":"02:29.194","Text":"when t is 0, we get the point 1, 1,"},{"Start":"02:29.194 ","End":"02:33.124","Text":"which is fine/ When t is 1,"},{"Start":"02:33.124 ","End":"02:36.290","Text":"we get 1 plus 3 is 4."},{"Start":"02:36.290 ","End":"02:39.020","Text":"Here, 1 plus 1 is 2, 4,"},{"Start":"02:39.020 ","End":"02:44.499","Text":"2. Here, this works."},{"Start":"02:45.560 ","End":"02:49.545","Text":"I know for later on, I\u0027ll need dx and dy."},{"Start":"02:49.545 ","End":"02:55.235","Text":"Here dx, the derivative of this is just 3dt, and"},{"Start":"02:55.235 ","End":"03:03.250","Text":"dy will just equal 1dt or simply dt."},{"Start":"03:03.250 ","End":"03:06.965","Text":"I\u0027ve got everything I need now to do this."},{"Start":"03:06.965 ","End":"03:11.970","Text":"This time the parameter goes from 0-1."},{"Start":"03:11.970 ","End":"03:18.375","Text":"That\u0027s the t. x is 1 plus 3t,"},{"Start":"03:18.375 ","End":"03:22.335","Text":"y is 1 plus t,"},{"Start":"03:22.335 ","End":"03:24.915","Text":"all of this is dx,"},{"Start":"03:24.915 ","End":"03:34.980","Text":"3dt and the second one y is 1 plus t minus x. I\u0027ll just make them both minuses,"},{"Start":"03:34.980 ","End":"03:39.615","Text":"minus 1 minus 3t and dy."},{"Start":"03:39.615 ","End":"03:43.060","Text":"Here it is, it\u0027s just dt."},{"Start":"03:43.250 ","End":"03:51.655","Text":"I want to write this as just one integral from 0-1."},{"Start":"03:51.655 ","End":"03:55.175","Text":"I could have emphasized that this is t equals,"},{"Start":"03:55.175 ","End":"03:57.725","Text":"well, we know it\u0027s t. Let\u0027s see."},{"Start":"03:57.725 ","End":"04:00.665","Text":"I want to collect together everything."},{"Start":"04:00.665 ","End":"04:06.450","Text":"Let\u0027s collect the terms with t. Now remember, this is a 3 here."},{"Start":"04:06.450 ","End":"04:10.440","Text":"Well, 3t plus t is 4t,"},{"Start":"04:10.440 ","End":"04:15.705","Text":"4t times 3 is 12t."},{"Start":"04:15.705 ","End":"04:19.100","Text":"I have a 12t from here."},{"Start":"04:19.100 ","End":"04:21.395","Text":"Maybe I\u0027ll just do each one separately."},{"Start":"04:21.395 ","End":"04:26.255","Text":"12t and here 1 plus 1 is 2, times 3 is 6."},{"Start":"04:26.255 ","End":"04:28.310","Text":"This is 12t plus 6."},{"Start":"04:28.310 ","End":"04:33.195","Text":"The second bit is 1 minus 1 is nothing,"},{"Start":"04:33.195 ","End":"04:37.710","Text":"t minus 3t is minus 2t."},{"Start":"04:37.960 ","End":"04:41.545","Text":"Combining these, what do I get?"},{"Start":"04:41.545 ","End":"04:52.045","Text":"10t plus 6dt. This integral of 10t,"},{"Start":"04:52.045 ","End":"04:57.350","Text":"I raise the power to t squared divide by the 2,"},{"Start":"04:57.350 ","End":"05:00.065","Text":"so that\u0027s just 5, constant."},{"Start":"05:00.065 ","End":"05:04.795","Text":"So it gives me 6t from 0-1,"},{"Start":"05:04.795 ","End":"05:12.320","Text":"and if I put in 0, I don\u0027t get anything."},{"Start":"05:12.320 ","End":"05:17.690","Text":"If I put in the 1, I just get 5 plus 6."},{"Start":"05:17.690 ","End":"05:25.020","Text":"This is 11, and this is the answer to part b."},{"Start":"05:25.020 ","End":"05:28.360","Text":"On the next clip, we\u0027ll do part c."}],"ID":8796},{"Watched":false,"Name":"Exercise 7 Part c","Duration":"8m 23s","ChapterTopicVideoID":8698,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8698.jpeg","UploadDate":"2017-02-13T05:05:48.8530000","DurationForVideoObject":"PT8M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:09.840","Text":"Now, we come to part c. I\u0027ve included a sketch where the points 1,"},{"Start":"00:09.840 ","End":"00:12.240","Text":"1, and 4, 2 are."},{"Start":"00:12.240 ","End":"00:15.150","Text":"In the previous part b,"},{"Start":"00:15.150 ","End":"00:16.410","Text":"we took a line segment."},{"Start":"00:16.410 ","End":"00:17.760","Text":"It would be like I joined 1,"},{"Start":"00:17.760 ","End":"00:19.815","Text":"1 to 4, 2."},{"Start":"00:19.815 ","End":"00:24.960","Text":"In part c, it\u0027s like I make a stopover at the point 1, 2,"},{"Start":"00:24.960 ","End":"00:26.355","Text":"which is let see,"},{"Start":"00:26.355 ","End":"00:29.700","Text":"just on the same horizontal line as this,"},{"Start":"00:29.700 ","End":"00:32.850","Text":"on the same vertical line as this."},{"Start":"00:32.850 ","End":"00:35.310","Text":"I go from here to here,"},{"Start":"00:35.310 ","End":"00:37.605","Text":"and then from here to here."},{"Start":"00:37.605 ","End":"00:42.325","Text":"Vertical then horizontal and not directly."},{"Start":"00:42.325 ","End":"00:47.370","Text":"We\u0027ll get a different answer then in part b, probably."},{"Start":"00:47.370 ","End":"00:52.010","Text":"Let\u0027s see if we can parametrize each of these 2."},{"Start":"00:52.010 ","End":"00:55.610","Text":"What I\u0027m going to say is that this whole curve is c,"},{"Start":"00:55.610 ","End":"01:00.080","Text":"but this might be the curve C_1 and this might be the curve C_2."},{"Start":"01:00.080 ","End":"01:05.630","Text":"What we\u0027re going to do is break it up to the integral over C_1 of whatever it is,"},{"Start":"01:05.630 ","End":"01:10.965","Text":"plus the integral along C_2 of the same thing as here."},{"Start":"01:10.965 ","End":"01:14.345","Text":"Let\u0027s parametrize each one of them."},{"Start":"01:14.345 ","End":"01:21.065","Text":"Now, the easiest way to do it is to parametrize C_1 as follows."},{"Start":"01:21.065 ","End":"01:26.480","Text":"We just keep the same x and move with y,"},{"Start":"01:26.480 ","End":"01:28.070","Text":"let me just maybe mark some points here,"},{"Start":"01:28.070 ","End":"01:30.890","Text":"this is 1, this is 2,"},{"Start":"01:30.890 ","End":"01:35.230","Text":"this is 1, and this is 4."},{"Start":"01:35.230 ","End":"01:42.940","Text":"For C_1, I could take that x is the constant one,"},{"Start":"01:42.940 ","End":"01:46.515","Text":"and y equals t,"},{"Start":"01:46.515 ","End":"01:53.255","Text":"which is just the value of the y here, goes from 1 to 2."},{"Start":"01:53.255 ","End":"01:57.995","Text":"I can say t goes from 1 to 2."},{"Start":"01:57.995 ","End":"01:59.735","Text":"For the second part,"},{"Start":"01:59.735 ","End":"02:05.240","Text":"the horizontal part, x will be the one that\u0027s moving."},{"Start":"02:05.240 ","End":"02:10.970","Text":"X will be the t. Y will be constantly 2."},{"Start":"02:10.970 ","End":"02:13.055","Text":"The value of x, which is t,"},{"Start":"02:13.055 ","End":"02:15.890","Text":"will go from 1 to 4."},{"Start":"02:17.680 ","End":"02:21.980","Text":"It\u0027ll be a different t because it\u0027s different curve."},{"Start":"02:21.980 ","End":"02:25.100","Text":"Let\u0027s see if we can translate this."},{"Start":"02:25.100 ","End":"02:26.990","Text":"Well, let\u0027s do each one separately."},{"Start":"02:26.990 ","End":"02:30.975","Text":"The integral over C_1."},{"Start":"02:30.975 ","End":"02:33.855","Text":"Let\u0027s do that one first."},{"Start":"02:33.855 ","End":"02:40.115","Text":"Maybe I\u0027ll call this one asterisk and the other one double asterisk."},{"Start":"02:40.115 ","End":"02:43.685","Text":"Let\u0027s, first of all, do the asterisk."},{"Start":"02:43.685 ","End":"02:46.085","Text":"That\u0027s this one."},{"Start":"02:46.085 ","End":"02:54.710","Text":"We got the integral from t equals 1 to 2."},{"Start":"02:54.710 ","End":"03:02.990","Text":"X plus y is 1 plus t. We\u0027ll need dx and dy."},{"Start":"03:02.990 ","End":"03:04.520","Text":"Well, x is a constant,"},{"Start":"03:04.520 ","End":"03:08.615","Text":"so dx is just the derivative of this 0 dt,"},{"Start":"03:08.615 ","End":"03:17.240","Text":"which is just 0, and dy is equal to dt."},{"Start":"03:17.240 ","End":"03:20.120","Text":"For the first one, you might as well do the second one already."},{"Start":"03:20.120 ","End":"03:22.220","Text":"Well, I\u0027m differentiating here."},{"Start":"03:22.220 ","End":"03:30.860","Text":"Dx will equal dt and dy will equal 0,"},{"Start":"03:30.860 ","End":"03:34.889","Text":"0 dt, but it\u0027s 0."},{"Start":"03:36.320 ","End":"03:42.705","Text":"When we get here, we get 1 plus t. I\u0027ll write it,"},{"Start":"03:42.705 ","End":"03:51.900","Text":"0 dt that I\u0027m just omitting it so you can see where I\u0027m coming from plus y minus x,"},{"Start":"03:51.900 ","End":"03:56.805","Text":"which is at t minus 1."},{"Start":"03:56.805 ","End":"04:03.990","Text":"Dy along this curve is equal to just dt."},{"Start":"04:08.990 ","End":"04:12.935","Text":"I\u0027ll continue on the next line."},{"Start":"04:12.935 ","End":"04:16.175","Text":"This will equal, this is 0."},{"Start":"04:16.175 ","End":"04:25.590","Text":"It\u0027s just the integral from 1 to 2 of t minus 1 dt."},{"Start":"04:26.980 ","End":"04:33.590","Text":"This equals 1.5 t"},{"Start":"04:33.590 ","End":"04:39.030","Text":"squared minus t from 1 to 2."},{"Start":"04:39.100 ","End":"04:43.415","Text":"What we get, we plug in 2."},{"Start":"04:43.415 ","End":"04:48.000","Text":"We get 1/2 of t squared is 1/2 of 4,"},{"Start":"04:48.000 ","End":"04:52.780","Text":"is 2 minus 2 is 0."},{"Start":"04:53.030 ","End":"04:59.880","Text":"That it\u0027s 2 minus 2."},{"Start":"04:59.880 ","End":"05:02.520","Text":"I have to subtract."},{"Start":"05:02.520 ","End":"05:04.770","Text":"What happens when I plug in 1,"},{"Start":"05:04.770 ","End":"05:09.150","Text":"which is 1/2 minus 1?"},{"Start":"05:09.150 ","End":"05:12.675","Text":"This is 0 minus 1/2,"},{"Start":"05:12.675 ","End":"05:14.670","Text":"so this is 1/2."},{"Start":"05:14.670 ","End":"05:19.280","Text":"That\u0027s the first one and now the second one,"},{"Start":"05:19.280 ","End":"05:27.340","Text":"the double asterisk is the integral this time it\u0027s from"},{"Start":"05:27.340 ","End":"05:36.590","Text":"1 to 4 and we have the same thing,"},{"Start":"05:36.590 ","End":"05:39.590","Text":"x plus y, but it\u0027s different."},{"Start":"05:39.590 ","End":"05:43.890","Text":"X plus y here is t plus 2."},{"Start":"05:44.980 ","End":"05:48.530","Text":"Let me just say this is the asterisk and"},{"Start":"05:48.530 ","End":"05:51.380","Text":"this is the double asterisk so we\u0027re reading off here,"},{"Start":"05:51.380 ","End":"05:54.245","Text":"x plus y, t plus 2."},{"Start":"05:54.245 ","End":"05:59.960","Text":"Then dx is, where is it?"},{"Start":"05:59.960 ","End":"06:03.935","Text":"Here, dt plus."},{"Start":"06:03.935 ","End":"06:06.755","Text":"Then we need y minus x,"},{"Start":"06:06.755 ","End":"06:12.080","Text":"which is 2 minus"},{"Start":"06:12.080 ","End":"06:24.200","Text":"t. That is dy,"},{"Start":"06:24.200 ","End":"06:27.900","Text":"which is 0 dt."},{"Start":"06:28.850 ","End":"06:31.430","Text":"The second part is nothing."},{"Start":"06:31.430 ","End":"06:34.330","Text":"We just get the integral,"},{"Start":"06:34.330 ","End":"06:41.590","Text":"write it again, 1 to 4 of t plus 2 dt."},{"Start":"06:43.780 ","End":"06:50.180","Text":"This is equal to 1.5 t"},{"Start":"06:50.180 ","End":"06:56.300","Text":"squared plus 2t from 1 to 4."},{"Start":"06:56.300 ","End":"06:59.285","Text":"Let\u0027s see. When I plug in 4,"},{"Start":"06:59.285 ","End":"07:04.910","Text":"I get 4 squared over 2 is 8,"},{"Start":"07:04.910 ","End":"07:11.475","Text":"2 times 4 is 8 minus,"},{"Start":"07:11.475 ","End":"07:17.805","Text":"I plug in 1, I get 1.5 plus 2."},{"Start":"07:17.805 ","End":"07:26.535","Text":"What do I get? 16 minus 2.5 is 13 and 1/2."},{"Start":"07:26.535 ","End":"07:28.830","Text":"For those who like mixed numbers,"},{"Start":"07:28.830 ","End":"07:35.800","Text":"and if you like improper fractions, that\u0027s 27/2."},{"Start":"07:37.430 ","End":"07:42.920","Text":"Let\u0027s see, this is the answer for the asterisk."},{"Start":"07:42.920 ","End":"07:46.415","Text":"This is the answer for the double asterisk."},{"Start":"07:46.415 ","End":"07:48.560","Text":"I have to add them together."},{"Start":"07:48.560 ","End":"07:53.475","Text":"I\u0027ve got, maybe I\u0027ll use the 13.5,"},{"Start":"07:53.475 ","End":"07:56.190","Text":"doesn\u0027t matter, this plus this."},{"Start":"07:56.190 ","End":"08:02.250","Text":"13.5 plus 1.5 is equal"},{"Start":"08:02.250 ","End":"08:09.765","Text":"to 14 or if you did it as 28 over 2, it would still be 14."},{"Start":"08:09.765 ","End":"08:18.245","Text":"This is the answer for the integral along this broken path."},{"Start":"08:18.245 ","End":"08:22.800","Text":"That concludes part c. We still have part d to do."}],"ID":8797},{"Watched":false,"Name":"Exercise 7 Part d","Duration":"6m 7s","ChapterTopicVideoID":8699,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8699.jpeg","UploadDate":"2017-02-13T05:07:17.1570000","DurationForVideoObject":"PT6M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.410 ","End":"00:04.875","Text":"Now we come to part D of this exercise."},{"Start":"00:04.875 ","End":"00:11.175","Text":"In part D, in some ways it\u0027s easier because we already have it in parameterized form."},{"Start":"00:11.175 ","End":"00:15.360","Text":"In all the others we had to figure out parametric representation."},{"Start":"00:15.360 ","End":"00:19.560","Text":"Let\u0027s just check that when t is 0 and t is 1,"},{"Start":"00:19.560 ","End":"00:22.380","Text":"we really do get our 2 points. Let\u0027s see."},{"Start":"00:22.380 ","End":"00:24.300","Text":"When t is 0,"},{"Start":"00:24.300 ","End":"00:26.460","Text":"then what is x, y equal?"},{"Start":"00:26.460 ","End":"00:30.495","Text":"X when it\u0027s 0 this is 1,"},{"Start":"00:30.495 ","End":"00:32.760","Text":"and when t is 0,"},{"Start":"00:32.760 ","End":"00:35.385","Text":"y is 1 and that\u0027s okay."},{"Start":"00:35.385 ","End":"00:41.860","Text":"When t is 1, then we get 2 plus 1 plus 1."},{"Start":"00:41.930 ","End":"00:45.810","Text":"X, y is 4,"},{"Start":"00:45.810 ","End":"00:51.180","Text":"2 plus 1 plus 1. What else?"},{"Start":"00:51.180 ","End":"00:54.780","Text":"1 squared plus 1 is 2,"},{"Start":"00:54.780 ","End":"00:58.080","Text":"and that\u0027s also okay."},{"Start":"00:58.080 ","End":"01:01.620","Text":"We have from 1,1-4,2."},{"Start":"01:01.620 ","End":"01:06.165","Text":"What I will still need is the dx and the dy."},{"Start":"01:06.165 ","End":"01:08.760","Text":"I\u0027ll just write them under here."},{"Start":"01:08.760 ","End":"01:17.420","Text":"Dx is the derivative of this is 4t plus 1 and that\u0027s dt,"},{"Start":"01:17.420 ","End":"01:22.680","Text":"and dy is just 2tdt."},{"Start":"01:22.730 ","End":"01:25.710","Text":"Now we have everything we need."},{"Start":"01:25.710 ","End":"01:33.065","Text":"We write this as an integral according to parameter t from 0-1."},{"Start":"01:33.065 ","End":"01:40.065","Text":"I\u0027ll even stress that this is t, goes from 0-1."},{"Start":"01:40.065 ","End":"01:46.120","Text":"X plus y, I\u0027m just reading it off here,"},{"Start":"01:46.220 ","End":"01:53.310","Text":"x plus y is 2t squared plus t plus 1 is x,"},{"Start":"01:53.310 ","End":"01:57.330","Text":"and then y is t squared plus 1,"},{"Start":"01:57.330 ","End":"02:02.950","Text":"and dx is 4t plus 1dt."},{"Start":"02:03.530 ","End":"02:05.690","Text":"That\u0027s just the first part."},{"Start":"02:05.690 ","End":"02:08.630","Text":"Now the second part, y minus x,"},{"Start":"02:08.630 ","End":"02:12.620","Text":"t squared plus 1 minus all of these,"},{"Start":"02:12.620 ","End":"02:13.970","Text":"let\u0027s put them all with a minus,"},{"Start":"02:13.970 ","End":"02:18.815","Text":"minus 2t squared, minus t, minus 1."},{"Start":"02:18.815 ","End":"02:28.610","Text":"Here we have a dy which is 2tdt."},{"Start":"02:28.610 ","End":"02:33.365","Text":"You just have to do a bit of algebra here to collect like terms."},{"Start":"02:33.365 ","End":"02:35.975","Text":"We\u0027ll get the integral from 0-1."},{"Start":"02:35.975 ","End":"02:38.905","Text":"Let\u0027s see what we can do."},{"Start":"02:38.905 ","End":"02:43.510","Text":"Perhaps first of all simplify this."},{"Start":"02:44.180 ","End":"02:54.780","Text":"I\u0027ll just rewrite this as 3t squared plus t plus 2."},{"Start":"02:54.780 ","End":"02:59.210","Text":"This 1 we can collect like terms. What do we get?"},{"Start":"02:59.210 ","End":"03:07.620","Text":"T squared minus 2t squared is minus t squared,"},{"Start":"03:07.870 ","End":"03:13.885","Text":"and then the minus t and the plus 1, minus 1 cancels."},{"Start":"03:13.885 ","End":"03:17.265","Text":"Let\u0027s start multiplying out."},{"Start":"03:17.265 ","End":"03:21.875","Text":"I\u0027ll begin by multiplying 4t by all of these."},{"Start":"03:21.875 ","End":"03:31.620","Text":"I\u0027ve got 12t cubed plus 4t squared plus 8t, then plus 1."},{"Start":"03:31.620 ","End":"03:33.915","Text":"I just have to write this again,"},{"Start":"03:33.915 ","End":"03:38.590","Text":"plus 3t squared plus t plus 2."},{"Start":"03:38.590 ","End":"03:40.580","Text":"That\u0027s this times this."},{"Start":"03:40.580 ","End":"03:43.670","Text":"Now here I just have to multiply by 2t."},{"Start":"03:43.670 ","End":"03:45.215","Text":"Everything\u0027s going to be negative."},{"Start":"03:45.215 ","End":"03:49.865","Text":"It\u0027s minus 2t times t squared is 2t cubed,"},{"Start":"03:49.865 ","End":"03:59.640","Text":"and then minus 2t squared and all of this dt."},{"Start":"03:59.640 ","End":"04:04.155","Text":"Let\u0027s collect first of all the terms with t cubed, I have 1 here,"},{"Start":"04:04.155 ","End":"04:06.165","Text":"and I have 1 here,"},{"Start":"04:06.165 ","End":"04:10.800","Text":"so 12 minus 10t cubed."},{"Start":"04:10.800 ","End":"04:12.990","Text":"Next, let\u0027s go for t squared,"},{"Start":"04:12.990 ","End":"04:16.070","Text":"I have here, and I have here,"},{"Start":"04:16.070 ","End":"04:18.355","Text":"and I have here."},{"Start":"04:18.355 ","End":"04:27.220","Text":"That will give me 4 plus 3 minus 2, that\u0027s 5t squared."},{"Start":"04:27.220 ","End":"04:29.810","Text":"There\u0027s not much else."},{"Start":"04:29.810 ","End":"04:33.845","Text":"I have a t here and a t here,"},{"Start":"04:33.845 ","End":"04:39.109","Text":"8 plus 1 is 9t, and a constant,"},{"Start":"04:39.109 ","End":"04:42.080","Text":"I just have the 2dt"},{"Start":"04:42.080 ","End":"04:49.580","Text":"from 0-1."},{"Start":"04:49.580 ","End":"04:52.520","Text":"Next, just do the integration,"},{"Start":"04:52.520 ","End":"04:57.260","Text":"10 over 4 is like 5"},{"Start":"04:57.260 ","End":"05:06.810","Text":"over 2t^4,5 over 3t cubed,"},{"Start":"05:06.810 ","End":"05:12.630","Text":"9 over 2t squared and 2t."},{"Start":"05:12.630 ","End":"05:16.380","Text":"This from 0-1."},{"Start":"05:16.380 ","End":"05:19.615","Text":"The 0 doesn\u0027t give us anything, It\u0027s all 0."},{"Start":"05:19.615 ","End":"05:21.745","Text":"Just have to plug in the 1."},{"Start":"05:21.745 ","End":"05:24.760","Text":"I have a fraction, 5 over 2,"},{"Start":"05:24.760 ","End":"05:26.685","Text":"plus 5 over 3,"},{"Start":"05:26.685 ","End":"05:31.125","Text":"plus 9 over 2, plus 2."},{"Start":"05:31.125 ","End":"05:37.100","Text":"Let\u0027s see. If I take the 5 over 2 and 9 over 2,"},{"Start":"05:37.100 ","End":"05:39.725","Text":"I get 14 over 2, which is 7."},{"Start":"05:39.725 ","End":"05:43.410","Text":"7 plus 2 is 9,"},{"Start":"05:44.830 ","End":"05:48.980","Text":"and 5/3 is 1 and 2/3."},{"Start":"05:48.980 ","End":"05:52.680","Text":"I make that 9/2,10 and 2/3."},{"Start":"05:53.560 ","End":"05:55.850","Text":"Or if you prefer,"},{"Start":"05:55.850 ","End":"05:59.135","Text":"this could be written as 32/3."},{"Start":"05:59.135 ","End":"06:01.520","Text":"I\u0027ll choose this 1,"},{"Start":"06:01.520 ","End":"06:03.545","Text":"and we are done."},{"Start":"06:03.545 ","End":"06:07.260","Text":"That\u0027s part D, so that finishes this exercise."}],"ID":8798},{"Watched":false,"Name":"Exercise 8","Duration":"15m 13s","ChapterTopicVideoID":8700,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8700.jpeg","UploadDate":"2017-02-13T05:11:21.8430000","DurationForVideoObject":"PT15M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.370","Text":"In this exercise, we need to compute the type 2 line integral along the path C,"},{"Start":"00:08.370 ","End":"00:10.380","Text":"where C is described as in the picture."},{"Start":"00:10.380 ","End":"00:12.930","Text":"I guess I need to put an extra arrow here."},{"Start":"00:12.930 ","End":"00:20.310","Text":"C is this whole thing but obviously we\u0027re going to be breaking it up into 3 parts."},{"Start":"00:20.310 ","End":"00:23.595","Text":"This will be maybe C_1,"},{"Start":"00:23.595 ","End":"00:25.665","Text":"this will be C_2,"},{"Start":"00:25.665 ","End":"00:27.180","Text":"and this will be C_3."},{"Start":"00:27.180 ","End":"00:29.550","Text":"We can\u0027t do it all at once."},{"Start":"00:29.550 ","End":"00:33.585","Text":"We want a parametric form of each of them."},{"Start":"00:33.585 ","End":"00:36.105","Text":"Let\u0027s see."},{"Start":"00:36.105 ","End":"00:37.230","Text":"Let\u0027s mark some of the points."},{"Start":"00:37.230 ","End":"00:43.590","Text":"This is the origin so that\u0027s 00."},{"Start":"00:43.590 ","End":"00:45.665","Text":"Let me start with the more difficult one,"},{"Start":"00:45.665 ","End":"00:53.145","Text":"let\u0027s start with C_3. I\u0027ll mark it."},{"Start":"00:53.145 ","End":"00:59.040","Text":"This is 1, 2 and the origin is, I\u0027ll write it here."},{"Start":"00:59.040 ","End":"01:06.255","Text":"C_3 goes from 1, 2-0, 0."},{"Start":"01:06.255 ","End":"01:11.195","Text":"One way of parametrizing a line from 2 points is"},{"Start":"01:11.195 ","End":"01:16.970","Text":"just to say that what x equals and y equals,"},{"Start":"01:16.970 ","End":"01:22.019","Text":"we take the component from the first point,"},{"Start":"01:22.019 ","End":"01:24.150","Text":"in this case it\u0027s 1."},{"Start":"01:24.150 ","End":"01:30.980","Text":"Then we take t times the difference from the second minus the first."},{"Start":"01:30.980 ","End":"01:35.585","Text":"it\u0027s 0 minus 1 and the same thing for y."},{"Start":"01:35.585 ","End":"01:37.220","Text":"We take the 1st point,"},{"Start":"01:37.220 ","End":"01:43.500","Text":"which is 2 and put t times 0 minus 2."},{"Start":"01:44.360 ","End":"01:47.330","Text":"When we do it with this method,"},{"Start":"01:47.330 ","End":"01:51.360","Text":"it\u0027s always that t goes from 0-1."},{"Start":"01:51.560 ","End":"01:54.185","Text":"We just simplify this."},{"Start":"01:54.185 ","End":"01:56.180","Text":"What does this come out to be?"},{"Start":"01:56.180 ","End":"02:03.210","Text":"This comes out as x equals 1"},{"Start":"02:03.210 ","End":"02:12.370","Text":"minus t and y equals 2 minus 2t."},{"Start":"02:12.680 ","End":"02:17.700","Text":"You can always check by plugging in 0."},{"Start":"02:17.700 ","End":"02:20.450","Text":"When t is 0 we get 1, 2,"},{"Start":"02:20.450 ","End":"02:22.955","Text":"fine and when t is 1,"},{"Start":"02:22.955 ","End":"02:24.350","Text":"we get 0, 0."},{"Start":"02:24.350 ","End":"02:26.700","Text":"Yeah, that\u0027s all fine."},{"Start":"02:27.340 ","End":"02:29.900","Text":"That would be C_3."},{"Start":"02:29.900 ","End":"02:36.175","Text":"I might as well do all the curves first."},{"Start":"02:36.175 ","End":"02:39.300","Text":"I Should have been writing in the solution area."},{"Start":"02:39.300 ","End":"02:43.660","Text":"Well, forgive me for that. Let\u0027s go for C_1."},{"Start":"02:44.650 ","End":"02:49.640","Text":"Now when it\u0027s horizontal or vertical line,"},{"Start":"02:49.640 ","End":"02:53.825","Text":"we don\u0027t have to do it with this method because if it\u0027s horizontal,"},{"Start":"02:53.825 ","End":"02:56.269","Text":"then one of the variables is constant."},{"Start":"02:56.269 ","End":"03:04.670","Text":"In this case, we see that the y is always 0 and it\u0027s just x going from 0-1."},{"Start":"03:04.670 ","End":"03:08.870","Text":"Instead of saying x goes from 0-1,"},{"Start":"03:08.870 ","End":"03:14.465","Text":"I say x equals t and t goes from 0-1."},{"Start":"03:14.465 ","End":"03:23.795","Text":"As for C_2, then the x stays at"},{"Start":"03:23.795 ","End":"03:32.400","Text":"1 and y moves from 0-2."},{"Start":"03:32.400 ","End":"03:39.625","Text":"I let y equals t and let t go from 0-2."},{"Start":"03:39.625 ","End":"03:43.280","Text":"I find this is just a bit easier when I have a vertical or horizontal line."},{"Start":"03:43.280 ","End":"03:47.740","Text":"Of course you could do it the same system as we did here."},{"Start":"03:47.740 ","End":"03:52.610","Text":"Now what we\u0027re going to do is say that the integral over C,"},{"Start":"03:52.610 ","End":"03:56.030","Text":"this one here, we break it up into 3 pieces."},{"Start":"03:56.030 ","End":"04:01.710","Text":"The integral over C_1 plus the"},{"Start":"04:01.710 ","End":"04:10.480","Text":"integral along C_2 plus the integral along C_3 of whatever it is here."},{"Start":"04:11.540 ","End":"04:14.345","Text":"Let\u0027s do them one by one."},{"Start":"04:14.345 ","End":"04:20.070","Text":"The first one, the integral of C_1,"},{"Start":"04:21.040 ","End":"04:23.750","Text":"well, maybe I\u0027ll copy it."},{"Start":"04:23.750 ","End":"04:34.180","Text":"X squared ydx plus xdy is equal to,"},{"Start":"04:34.180 ","End":"04:39.090","Text":"we have C_1 we\u0027re talking about,"},{"Start":"04:39.090 ","End":"04:41.685","Text":"t goes from 0 to 1."},{"Start":"04:41.685 ","End":"04:44.885","Text":"We put 0, 1 if you want to emphasize it,"},{"Start":"04:44.885 ","End":"04:49.390","Text":"like t equals and then we start substituting."},{"Start":"04:49.390 ","End":"04:53.205","Text":"We\u0027re going to need dx and dy."},{"Start":"04:53.205 ","End":"04:56.635","Text":"Let me add those here."},{"Start":"04:56.635 ","End":"05:03.660","Text":"If x is t, then I have that dx is equal to"},{"Start":"05:03.660 ","End":"05:11.780","Text":"dt and dy is equal to 0 dt which is just 0."},{"Start":"05:11.780 ","End":"05:15.180","Text":"I might as well continue for all of them."},{"Start":"05:16.460 ","End":"05:20.040","Text":"Maybe I\u0027ll write them outside."},{"Start":"05:20.040 ","End":"05:22.340","Text":"It was getting a bit crowded."},{"Start":"05:22.340 ","End":"05:29.590","Text":"Here, we\u0027ll get that dx is also 0 because the derivative of 1 is 0,"},{"Start":"05:29.590 ","End":"05:34.490","Text":"0dt and dy will equal dt."},{"Start":"05:35.120 ","End":"05:38.050","Text":"For the curve C_3,"},{"Start":"05:38.050 ","End":"05:39.605","Text":"I\u0027ll do it from here."},{"Start":"05:39.605 ","End":"05:44.840","Text":"We have that dx is equal to minus 1dt,"},{"Start":"05:44.840 ","End":"05:52.345","Text":"which is just minus dt and dy is minus 2dt."},{"Start":"05:52.345 ","End":"05:58.035","Text":"I think we\u0027re all set up for doing this integral."},{"Start":"05:58.035 ","End":"06:02.230","Text":"Back here, we\u0027re doing number 1."},{"Start":"06:07.550 ","End":"06:14.680","Text":"Y is 0, so x squared y dx is 0,"},{"Start":"06:15.200 ","End":"06:19.590","Text":"and dy is also 0."},{"Start":"06:19.590 ","End":"06:25.180","Text":"Basically we get 0."},{"Start":"06:25.970 ","End":"06:30.965","Text":"I\u0027ll write it a bit more in full because I just want to say 0."},{"Start":"06:30.965 ","End":"06:40.500","Text":"The x squared y is 0 because y is 0 and that the dx is just dt."},{"Start":"06:42.080 ","End":"06:50.270","Text":"Then we have that x is equal to t,"},{"Start":"06:50.270 ","End":"06:58.880","Text":"but dy is 0 or even 0dt."},{"Start":"06:58.880 ","End":"07:02.280","Text":"I could write the dt here, I suppose."},{"Start":"07:02.280 ","End":"07:07.115","Text":"It doesn\u0027t hurt. Altogether this is 0."},{"Start":"07:07.115 ","End":"07:12.045","Text":"It\u0027s just the integral of 0, which is 0."},{"Start":"07:12.045 ","End":"07:14.280","Text":"That\u0027s the first one done."},{"Start":"07:14.280 ","End":"07:19.620","Text":"Now let\u0027s go to the next one."},{"Start":"07:19.620 ","End":"07:25.430","Text":"The integral of C_2 of the same thing. Just copy it."},{"Start":"07:25.430 ","End":"07:32.850","Text":"X squared ydx plus xdy and that will equal the integral."},{"Start":"07:32.850 ","End":"07:40.595","Text":"For C_2 I have from 0-2 dt."},{"Start":"07:40.595 ","End":"07:46.765","Text":"Let\u0027s see, x squared y. X squared y is 1 squared t,"},{"Start":"07:46.765 ","End":"07:50.560","Text":"which is t. Then dx."},{"Start":"07:50.560 ","End":"07:56.680","Text":"But dx is 0dt,"},{"Start":"07:56.680 ","End":"08:01.585","Text":"and the other 1 xdy, x is 1,"},{"Start":"08:01.585 ","End":"08:07.420","Text":"dy is dt, 1dt."},{"Start":"08:07.420 ","End":"08:10.840","Text":"Altogether, it\u0027s the integral of just 1dt."},{"Start":"08:10.840 ","End":"08:18.040","Text":"The integral of 1 is just t taken from 0-2,"},{"Start":"08:18.040 ","End":"08:21.985","Text":"just 2 minus 0, which is 2."},{"Start":"08:21.985 ","End":"08:28.420","Text":"Maybe I should highlight the subtotals, C_1, C_2, C_3."},{"Start":"08:28.420 ","End":"08:32.065","Text":"That\u0027s C_1, that\u0027s C_2,"},{"Start":"08:32.065 ","End":"08:34.030","Text":"at the end we have to add all 3 of them up."},{"Start":"08:34.030 ","End":"08:36.055","Text":"Now, we have 1 more."},{"Start":"08:36.055 ","End":"08:42.820","Text":"We have the integral along C_3 of the same thing,"},{"Start":"08:42.820 ","End":"08:46.120","Text":"x squared ydx plus xdy."},{"Start":"08:46.120 ","End":"08:52.554","Text":"For C_3, we have,"},{"Start":"08:52.554 ","End":"08:58.095","Text":"let\u0027s see, there it is, we just see it."},{"Start":"08:58.095 ","End":"09:03.075","Text":"X squared y is going to be"},{"Start":"09:03.075 ","End":"09:09.610","Text":"1 minus t squared times y,"},{"Start":"09:09.610 ","End":"09:11.755","Text":"which is 2 minus 2t."},{"Start":"09:11.755 ","End":"09:19.400","Text":"I need to write that this goes from 0-1."},{"Start":"09:19.710 ","End":"09:23.470","Text":"Then I need the dx,"},{"Start":"09:23.470 ","End":"09:26.750","Text":"which is minus dt."},{"Start":"09:27.210 ","End":"09:30.175","Text":"Then we need the xdy,"},{"Start":"09:30.175 ","End":"09:41.810","Text":"x is 1 minus t. Then dy is minus 2dt."},{"Start":"09:44.700 ","End":"09:48.620","Text":"Let\u0027s see if we can simplify this."},{"Start":"09:57.090 ","End":"10:05.200","Text":"What we have here is 2 minus t. I can take a 2 out."},{"Start":"10:05.200 ","End":"10:08.095","Text":"I\u0027m going to continue on the next line."},{"Start":"10:08.095 ","End":"10:10.255","Text":"I can take a 2 out."},{"Start":"10:10.255 ","End":"10:13.825","Text":"I\u0027ve got the integral from 0-1,"},{"Start":"10:13.825 ","End":"10:25.375","Text":"twice 1 minus t. Then 1 minus t with 1 minus t squared will be 1 minus t cubed."},{"Start":"10:25.375 ","End":"10:31.460","Text":"I have twice 1 minus t cubed,"},{"Start":"10:31.980 ","End":"10:35.260","Text":"and it\u0027s a minus."},{"Start":"10:35.260 ","End":"10:44.410","Text":"Minus this, dt, and put brackets here because you have to put everything dt,"},{"Start":"10:44.410 ","End":"10:50.845","Text":"and here I have minus"},{"Start":"10:50.845 ","End":"10:58.100","Text":"2 plus 2t, and that dt."},{"Start":"10:59.760 ","End":"11:02.859","Text":"I\u0027ll compute this using the formula."},{"Start":"11:02.859 ","End":"11:03.985","Text":"Write it at the side."},{"Start":"11:03.985 ","End":"11:09.130","Text":"That a minus b cubed is a cubed minus 3a"},{"Start":"11:09.130 ","End":"11:15.685","Text":"squared b plus 3ab squared minus b cubed."},{"Start":"11:15.685 ","End":"11:24.380","Text":"So we will get the integral from 0-1 of minus 2,"},{"Start":"11:27.900 ","End":"11:37.000","Text":"1 minus 3t plus 3t squared minus t"},{"Start":"11:37.000 ","End":"11:47.440","Text":"cubed minus 2 plus 2t dt."},{"Start":"11:47.440 ","End":"11:54.820","Text":"We can now compute this in our heads."},{"Start":"11:54.820 ","End":"12:01.355","Text":"Just collect together minus 2,"},{"Start":"12:01.355 ","End":"12:05.880","Text":"with minus 2, will give me minus 4."},{"Start":"12:05.880 ","End":"12:12.080","Text":"It\u0027s from 0-1. We have here,"},{"Start":"12:12.080 ","End":"12:19.590","Text":"plus 6t plus 2t, is plus 8t."},{"Start":"12:19.590 ","End":"12:25.150","Text":"Then t squared, I\u0027ve got minus 6 of those"},{"Start":"12:25.150 ","End":"12:32.830","Text":"and plus 2t cubed, all this dt."},{"Start":"12:33.540 ","End":"12:36.835","Text":"Let\u0027s see what we get."},{"Start":"12:36.835 ","End":"12:45.205","Text":"We get minus 4t plus 8t squared over 2,"},{"Start":"12:45.205 ","End":"12:50.185","Text":"makes it 4t squared here,"},{"Start":"12:50.185 ","End":"12:53.305","Text":"minus 6 over 3t cubed,"},{"Start":"12:53.305 ","End":"12:58.430","Text":"minus 2t cubed, and here,"},{"Start":"13:00.060 ","End":"13:04.360","Text":"t^4 over 4 times 2"},{"Start":"13:04.360 ","End":"13:13.240","Text":"is 1/2t^4, from 0-1."},{"Start":"13:13.240 ","End":"13:15.520","Text":"I plug in 0, everything 0."},{"Start":"13:15.520 ","End":"13:19.450","Text":"So I just need to plug in the 1,"},{"Start":"13:19.450 ","End":"13:25.510","Text":"so I get minus 4 plus 4,"},{"Start":"13:25.510 ","End":"13:29.875","Text":"minus 2 plus a 1/2,"},{"Start":"13:29.875 ","End":"13:35.725","Text":"altogether, that is minus 1.5."},{"Start":"13:35.725 ","End":"13:38.740","Text":"I\u0027ll highlight this."},{"Start":"13:38.740 ","End":"13:42.940","Text":"Now, I go back here."},{"Start":"13:42.940 ","End":"13:53.785","Text":"I can write this as 0 plus 2 plus minus 1 and 1/2,"},{"Start":"13:53.785 ","End":"13:58.370","Text":"and that is just equal to 1/2."},{"Start":"13:58.470 ","End":"14:01.030","Text":"That is the answer."},{"Start":"14:01.030 ","End":"14:06.760","Text":"But I Just like to say something else as an alternative method for doing the integral."},{"Start":"14:06.760 ","End":"14:09.985","Text":"We could have done the integral,"},{"Start":"14:09.985 ","End":"14:16.345","Text":"I\u0027ll just take this part of 1 minus t, all cubed."},{"Start":"14:16.345 ","End":"14:19.480","Text":"Let\u0027s just say we\u0027re taking the indefinite integral."},{"Start":"14:19.480 ","End":"14:21.145","Text":"What we could\u0027ve done,"},{"Start":"14:21.145 ","End":"14:23.965","Text":"is because it\u0027s a linear function of t,"},{"Start":"14:23.965 ","End":"14:28.585","Text":"treated as if 1 minus t was just a variable,"},{"Start":"14:28.585 ","End":"14:35.965","Text":"and we could have said this is 1 minus t^4 over 4."},{"Start":"14:35.965 ","End":"14:39.430","Text":"Then because it isn\u0027t t,"},{"Start":"14:39.430 ","End":"14:40.690","Text":"it\u0027s 1 minus t,"},{"Start":"14:40.690 ","End":"14:43.300","Text":"the inner derivative is minus 1,"},{"Start":"14:43.300 ","End":"14:46.615","Text":"we could have divided by minus 1."},{"Start":"14:46.615 ","End":"14:51.340","Text":"Dividing by minus 1 is like multiplying by minus 1,"},{"Start":"14:51.340 ","End":"14:55.070","Text":"and we could have just put a minus here."},{"Start":"14:56.190 ","End":"14:59.335","Text":"I just wanted to mention that from this point,"},{"Start":"14:59.335 ","End":"15:06.355","Text":"you could have used this instead of using the formula for the cube of a binomial,"},{"Start":"15:06.355 ","End":"15:12.740","Text":"would have got the same answer. We\u0027re done."}],"ID":8799},{"Watched":false,"Name":"Exercise 9","Duration":"10m 16s","ChapterTopicVideoID":8701,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8701.jpeg","UploadDate":"2017-02-13T05:14:05.1600000","DurationForVideoObject":"PT10M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.520","Text":"In this exercise, we\u0027re asked to compute the line integral of type 2 here,"},{"Start":"00:08.520 ","End":"00:12.900","Text":"and C is the closed path,"},{"Start":"00:12.900 ","End":"00:14.535","Text":"as in the picture."},{"Start":"00:14.535 ","End":"00:16.875","Text":"Maybe we should mark the points."},{"Start":"00:16.875 ","End":"00:21.495","Text":"This one, this one, and this one."},{"Start":"00:21.495 ","End":"00:25.065","Text":"The origin is obviously 0,0."},{"Start":"00:25.065 ","End":"00:27.000","Text":"Here, we\u0027re given that y is 1,"},{"Start":"00:27.000 ","End":"00:31.060","Text":"so this is the point 0,1."},{"Start":"00:31.160 ","End":"00:35.055","Text":"If y equals x squared and y equals 1,"},{"Start":"00:35.055 ","End":"00:38.895","Text":"then this has got to be the point 1,1."},{"Start":"00:38.895 ","End":"00:43.200","Text":"We go in a circular way."},{"Start":"00:43.200 ","End":"00:49.520","Text":"All this is C, the counterclockwise path."},{"Start":"00:49.520 ","End":"00:51.320","Text":"But to do the integral,"},{"Start":"00:51.320 ","End":"00:54.430","Text":"we want to break it up into 3 parts."},{"Start":"00:54.430 ","End":"00:58.470","Text":"Let\u0027s call this bit C_1,"},{"Start":"00:58.470 ","End":"01:00.300","Text":"and then we go along here,"},{"Start":"01:00.300 ","End":"01:04.665","Text":"that\u0027s C_2, and here, C_3."},{"Start":"01:04.665 ","End":"01:09.050","Text":"Then what we\u0027ll do is instead of computing the integral over"},{"Start":"01:09.050 ","End":"01:13.085","Text":"the whole of C of this thing, whatever it is,"},{"Start":"01:13.085 ","End":"01:16.215","Text":"we\u0027ll break it up into the integral over"},{"Start":"01:16.215 ","End":"01:22.220","Text":"C_1 plus the integral over C_2 of whatever it is,"},{"Start":"01:22.220 ","End":"01:27.515","Text":"plus the integral over C_3, 3 separate bits."},{"Start":"01:27.515 ","End":"01:34.500","Text":"We\u0027d like to parametrize each of the 3 separate pieces."},{"Start":"01:34.500 ","End":"01:38.915","Text":"I\u0027ll do it over here even though it says solution here."},{"Start":"01:38.915 ","End":"01:43.375","Text":"Let\u0027s take C_1. Now, C_1,"},{"Start":"01:43.375 ","End":"01:47.480","Text":"we\u0027re going from here to here along the curve y equals x squared,"},{"Start":"01:47.480 ","End":"01:48.920","Text":"and y is a function of x,"},{"Start":"01:48.920 ","End":"01:51.200","Text":"we just let x be the parameter t."},{"Start":"01:51.200 ","End":"01:54.405","Text":"We\u0027ll say x equals t,"},{"Start":"01:54.405 ","End":"01:58.385","Text":"and y which is x squared is now t squared."},{"Start":"01:58.385 ","End":"02:03.495","Text":"If you look at it, the x goes from 0 to 1,"},{"Start":"02:03.495 ","End":"02:08.580","Text":"so that\u0027s t goes from 0 to 1."},{"Start":"02:08.580 ","End":"02:10.740","Text":"That\u0027s C_1."},{"Start":"02:10.740 ","End":"02:13.860","Text":"Well, let\u0027s see, C_2."},{"Start":"02:13.860 ","End":"02:16.970","Text":"When it\u0027s a horizontal or vertical line,"},{"Start":"02:16.970 ","End":"02:22.580","Text":"I don\u0027t use the usual formula for a line segment between 2 points."},{"Start":"02:22.580 ","End":"02:27.440","Text":"If it\u0027s horizontal, then one of them is constant."},{"Start":"02:27.440 ","End":"02:29.570","Text":"I mean, the y is constant."},{"Start":"02:29.570 ","End":"02:32.330","Text":"We know here that y equals 1."},{"Start":"02:32.330 ","End":"02:39.150","Text":"But x goes from"},{"Start":"02:40.160 ","End":"02:43.560","Text":"1 down to 0,"},{"Start":"02:43.560 ","End":"02:52.305","Text":"so I can\u0027t let x equals t. I\u0027ll do a take 2 on the last bit."},{"Start":"02:52.305 ","End":"02:58.200","Text":"Here, I use the formula that x is the x of"},{"Start":"02:58.200 ","End":"03:06.830","Text":"the start point which is 1 plus t times the difference in the x\u0027s,"},{"Start":"03:06.830 ","End":"03:12.175","Text":"the destination minus the source, 0 minus 1,"},{"Start":"03:12.175 ","End":"03:16.590","Text":"and same thing for y but"},{"Start":"03:16.590 ","End":"03:23.490","Text":"y is just equal to 1 so we can take a shortcut here and just say y equals 1."},{"Start":"03:24.010 ","End":"03:33.585","Text":"I can rewrite this as just 1 minus t,"},{"Start":"03:33.585 ","End":"03:36.555","Text":"and that will be simpler."},{"Start":"03:36.555 ","End":"03:41.160","Text":"Then for C_3, something similar,"},{"Start":"03:41.160 ","End":"03:48.090","Text":"I forgot to say, t goes from 0 to 1."},{"Start":"03:48.090 ","End":"03:51.930","Text":"For C_3, similar idea."},{"Start":"03:51.930 ","End":"03:54.350","Text":"Only here, the x,"},{"Start":"03:54.350 ","End":"04:01.305","Text":"we don\u0027t have to mess with because it\u0027s always 0 along the y axis,"},{"Start":"04:01.305 ","End":"04:05.979","Text":"and y will equal the"},{"Start":"04:06.020 ","End":"04:13.880","Text":"source y which is 1 plus t times the end minus the start y,"},{"Start":"04:13.880 ","End":"04:17.405","Text":"also 0 minus 1."},{"Start":"04:17.405 ","End":"04:21.630","Text":"Again, t goes from 0 to 1."},{"Start":"04:21.630 ","End":"04:23.550","Text":"I\u0027ll save a line."},{"Start":"04:23.550 ","End":"04:25.770","Text":"This comes out to be 1 minus t,"},{"Start":"04:25.770 ","End":"04:31.185","Text":"I\u0027ll just cross this out and put that y is"},{"Start":"04:31.185 ","End":"04:40.090","Text":"1 minus t. Let\u0027s see what we get."},{"Start":"04:40.250 ","End":"04:43.710","Text":"Let\u0027s start with the C_1 part."},{"Start":"04:43.710 ","End":"04:46.410","Text":"For this part, we have the integral,"},{"Start":"04:46.410 ","End":"04:49.215","Text":"t goes from 0 to 1."},{"Start":"04:49.215 ","End":"04:54.190","Text":"We want x minus y squared which is"},{"Start":"04:54.190 ","End":"05:00.950","Text":"t minus y squared is t squared squared which is t^4."},{"Start":"05:00.950 ","End":"05:03.635","Text":"We don\u0027t have dx and dy,"},{"Start":"05:03.635 ","End":"05:05.885","Text":"so let\u0027s fill those in."},{"Start":"05:05.885 ","End":"05:08.435","Text":"dx is just dt,"},{"Start":"05:08.435 ","End":"05:13.200","Text":"and dy would be 2tdt."},{"Start":"05:15.130 ","End":"05:18.745","Text":"Here, I need dx so that\u0027s dt,"},{"Start":"05:18.745 ","End":"05:23.830","Text":"and then I need dy and just copy 2tdt."},{"Start":"05:28.010 ","End":"05:32.660","Text":"Let\u0027s see. I just combine this."},{"Start":"05:32.660 ","End":"05:35.060","Text":"I\u0027ll continue on the same line."},{"Start":"05:35.060 ","End":"05:38.695","Text":"The integral from 0 to 1."},{"Start":"05:38.695 ","End":"05:41.460","Text":"I\u0027ll just combine it all dt."},{"Start":"05:41.460 ","End":"05:44.940","Text":"I have t and another 2t is"},{"Start":"05:44.940 ","End":"05:51.830","Text":"3t minus t^4 dt."},{"Start":"05:51.830 ","End":"05:54.660","Text":"I\u0027ll just keep going on the same line."},{"Start":"05:56.890 ","End":"06:02.465","Text":"This is 3 over 2t squared."},{"Start":"06:02.465 ","End":"06:07.225","Text":"This is 1/5 t^5."},{"Start":"06:07.225 ","End":"06:11.070","Text":"This has to be taken from 0 to 1."},{"Start":"06:11.070 ","End":"06:14.789","Text":"I just need the 1 because 0 gives 0,"},{"Start":"06:14.789 ","End":"06:23.140","Text":"so it\u0027s the fraction 3/2 minus 1/5."},{"Start":"06:23.140 ","End":"06:25.255","Text":"If we put it all over 10,"},{"Start":"06:25.255 ","End":"06:28.030","Text":"that will be what?"},{"Start":"06:28.030 ","End":"06:34.505","Text":"3 times 5 is 15 minus 2 over 10."},{"Start":"06:34.505 ","End":"06:37.270","Text":"This is 13 over 10."},{"Start":"06:37.270 ","End":"06:41.630","Text":"That\u0027s C_1."},{"Start":"06:41.630 ","End":"06:48.000","Text":"Now C_2."},{"Start":"06:48.000 ","End":"06:50.650","Text":"C_2, again, I\u0027ll need the dx and dy."},{"Start":"06:50.780 ","End":"06:57.650","Text":"Here, dx, the derivative of this is minus 1dt,"},{"Start":"06:57.650 ","End":"06:59.750","Text":"so it\u0027s just minus dt."},{"Start":"06:59.750 ","End":"07:06.005","Text":"dy is just 0."},{"Start":"07:06.005 ","End":"07:08.690","Text":"We\u0027ll put it as 0dt."},{"Start":"07:08.690 ","End":"07:11.900","Text":"I might as well continue already with the dx\u0027s."},{"Start":"07:11.900 ","End":"07:15.905","Text":"Here, dx is just 0."},{"Start":"07:15.905 ","End":"07:18.500","Text":"We\u0027ll, write it as 0dt,"},{"Start":"07:18.500 ","End":"07:22.940","Text":"and dy is also because of the minus 1,"},{"Start":"07:22.940 ","End":"07:25.070","Text":"I\u0027ve got minus dt."},{"Start":"07:25.070 ","End":"07:28.205","Text":"Now, we\u0027ll continue over here."},{"Start":"07:28.205 ","End":"07:32.375","Text":"Let\u0027s see, C_2."},{"Start":"07:32.375 ","End":"07:36.110","Text":"All of them are going to be from 0 to 1."},{"Start":"07:36.110 ","End":"07:43.265","Text":"In fact, I might as well just work on the 3 of them simultaneously or partially, 0 to 1."},{"Start":"07:43.265 ","End":"07:45.305","Text":"Now, let\u0027s see. For C_2,"},{"Start":"07:45.305 ","End":"07:48.395","Text":"I need x minus y squared."},{"Start":"07:48.395 ","End":"07:57.225","Text":"It\u0027s x is 1 minus t minus y squared minus 1 squared."},{"Start":"07:57.225 ","End":"08:02.385","Text":"All this is dx which is minus dt."},{"Start":"08:02.385 ","End":"08:08.490","Text":"Then I need plus dy plus 0dt."},{"Start":"08:08.490 ","End":"08:10.895","Text":"If I combine all this,"},{"Start":"08:10.895 ","End":"08:13.879","Text":"1 minus 1 squared is just nothing."},{"Start":"08:13.879 ","End":"08:19.595","Text":"I\u0027ve just got minus t and minus dt so it\u0027s tdt."},{"Start":"08:19.595 ","End":"08:22.085","Text":"This part gives nothing."},{"Start":"08:22.085 ","End":"08:29.849","Text":"This comes out to be 1/2 t squared from 0 to 1,"},{"Start":"08:29.849 ","End":"08:33.820","Text":"and this is just a 1/2."},{"Start":"08:34.250 ","End":"08:38.270","Text":"The last one, let\u0027s see."},{"Start":"08:38.270 ","End":"08:42.995","Text":"Again, I need the x minus y squared,"},{"Start":"08:42.995 ","End":"08:49.095","Text":"so x minus y squared is just 0"},{"Start":"08:49.095 ","End":"08:55.685","Text":"minus 1 minus t squared."},{"Start":"08:55.685 ","End":"09:00.990","Text":"Then dx which is 0dt."},{"Start":"09:00.990 ","End":"09:04.890","Text":"I didn\u0027t have to bother with this but anyway."},{"Start":"09:04.890 ","End":"09:10.795","Text":"Then plus dy, dy is minus dt which is minus dt."},{"Start":"09:10.795 ","End":"09:19.050","Text":"It\u0027s just the integral from 0 to 1 of minus dt or minus 1dt."},{"Start":"09:19.050 ","End":"09:25.420","Text":"This is minus t from 0 to 1,"},{"Start":"09:25.520 ","End":"09:30.910","Text":"and this comes out to be just minus 1."},{"Start":"09:31.180 ","End":"09:34.535","Text":"Now, I can go back here."},{"Start":"09:34.535 ","End":"09:37.760","Text":"This part is the C_1 part."},{"Start":"09:37.760 ","End":"09:39.905","Text":"Here, I have C_2,"},{"Start":"09:39.905 ","End":"09:42.840","Text":"here, I have C_3."},{"Start":"09:45.920 ","End":"09:48.814","Text":"You know what? Maybe I\u0027ll do it in decimal."},{"Start":"09:48.814 ","End":"09:52.530","Text":"This will be 1.3"},{"Start":"09:54.320 ","End":"10:01.060","Text":"plus 0.5 minus 1."},{"Start":"10:01.060 ","End":"10:02.910","Text":"What do I get?"},{"Start":"10:02.910 ","End":"10:07.435","Text":"1.8 minus 1 is 0.8."},{"Start":"10:07.435 ","End":"10:09.499","Text":"Really, we should do it as a fraction,"},{"Start":"10:09.499 ","End":"10:13.970","Text":"so I\u0027ll write it as 4/5,"},{"Start":"10:13.970 ","End":"10:16.530","Text":"and that\u0027s the answer."}],"ID":8800},{"Watched":false,"Name":"Exercise 10 Part a","Duration":"9m 1s","ChapterTopicVideoID":8702,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8702.jpeg","UploadDate":"2017-02-13T05:16:29.1130000","DurationForVideoObject":"PT9M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:04.710","Text":"This exercise is made up of 3 parts."},{"Start":"00:04.710 ","End":"00:06.285","Text":"But in each part,"},{"Start":"00:06.285 ","End":"00:14.220","Text":"we\u0027re given a function f of 3 variables and it\u0027s a vector function."},{"Start":"00:14.220 ","End":"00:18.150","Text":"These are the i, j, and k components."},{"Start":"00:18.150 ","End":"00:22.590","Text":"It\u0027s in 3D and we have to compute the line"},{"Start":"00:22.590 ","End":"00:31.340","Text":"integral F.dr from this point to this point along each of 3 different paths."},{"Start":"00:31.340 ","End":"00:35.100","Text":"Just want to point out that, strictly speaking,"},{"Start":"00:35.100 ","End":"00:41.350","Text":"should really be writing vectors with an arrow or a bar on top."},{"Start":"00:41.350 ","End":"00:43.760","Text":"When it\u0027s in bold phase like this,"},{"Start":"00:43.760 ","End":"00:45.770","Text":"then we understand that it\u0027s a vector."},{"Start":"00:45.770 ","End":"00:52.320","Text":"It\u0027s another way of doing it and actually dr is also a vector."},{"Start":"00:52.320 ","End":"00:58.485","Text":"In fact, dr is equal to,"},{"Start":"00:58.485 ","End":"01:01.620","Text":"if I do it in the i, j, k form,"},{"Start":"01:01.620 ","End":"01:08.085","Text":"is just dx times i plus"},{"Start":"01:08.085 ","End":"01:15.420","Text":"dy times j plus dz times k,"},{"Start":"01:15.420 ","End":"01:18.050","Text":"or if you want it in the angular bracket form,"},{"Start":"01:18.050 ","End":"01:24.560","Text":"it\u0027s dx, dy, dz."},{"Start":"01:24.560 ","End":"01:25.790","Text":"For all 3 parts,"},{"Start":"01:25.790 ","End":"01:29.780","Text":"we can actually rewrite this integral because we have an i,"},{"Start":"01:29.780 ","End":"01:33.635","Text":"j, k here, dot with an i, j, k here."},{"Start":"01:33.635 ","End":"01:37.530","Text":"What we get, really and this,"},{"Start":"01:37.530 ","End":"01:44.930","Text":"as I say for all 3 cases is the integral along some path from here to here,"},{"Start":"01:44.930 ","End":"01:49.380","Text":"we can just informally write from 0,"},{"Start":"01:49.380 ","End":"01:51.890","Text":"0, 0 to 1, 1, 1."},{"Start":"01:51.890 ","End":"01:54.695","Text":"Although of course this depends on the path,"},{"Start":"01:54.695 ","End":"02:00.950","Text":"but we just sometimes informally write like this of this dx."},{"Start":"02:00.950 ","End":"02:10.760","Text":"So 3x squared minus 6yz dx plus the next component,"},{"Start":"02:10.760 ","End":"02:24.410","Text":"the y component, 2y plus 3xz dy."},{"Start":"02:24.410 ","End":"02:26.435","Text":"Just made some more room here."},{"Start":"02:26.435 ","End":"02:31.370","Text":"Plus the last 1 is 1 minus"},{"Start":"02:31.370 ","End":"02:38.370","Text":"4xyz squared times dz."},{"Start":"02:39.130 ","End":"02:42.530","Text":"This we\u0027ll compute in 3 different ways and in"},{"Start":"02:42.530 ","End":"02:45.560","Text":"each case we\u0027ll get a different parameterized path."},{"Start":"02:45.560 ","End":"02:53.130","Text":"This will be the integral along a path c and differently for a,"},{"Start":"02:53.130 ","End":"02:55.850","Text":"b, and c. Let\u0027s start with part a."},{"Start":"02:55.850 ","End":"03:00.305","Text":"Now in part a, we already have the parametric representation."},{"Start":"03:00.305 ","End":"03:05.270","Text":"We have that the curve c is given by x equals t,"},{"Start":"03:05.270 ","End":"03:07.685","Text":"y equals t squared,"},{"Start":"03:07.685 ","End":"03:10.130","Text":"z equals t cubed."},{"Start":"03:10.130 ","End":"03:12.409","Text":"But there is something missing,"},{"Start":"03:12.409 ","End":"03:15.520","Text":"we have to know where t goes from and to."},{"Start":"03:15.520 ","End":"03:18.665","Text":"T goes from something to something."},{"Start":"03:18.665 ","End":"03:21.335","Text":"They\u0027ve left that out or have they?"},{"Start":"03:21.335 ","End":"03:28.070","Text":"Well, we can easily deduce this because we know that we start from the point 0,"},{"Start":"03:28.070 ","End":"03:33.620","Text":"0, 0 so what value of t could I possibly put to get 0, 0, 0?"},{"Start":"03:33.620 ","End":"03:36.760","Text":"I think it\u0027s clear that that\u0027s 0."},{"Start":"03:36.760 ","End":"03:40.340","Text":"Likewise, to get the point 1, 1,"},{"Start":"03:40.340 ","End":"03:44.045","Text":"1, if I let t equals 1, that will do it."},{"Start":"03:44.045 ","End":"03:48.890","Text":"Now we have the path parameterized and we have the range of values"},{"Start":"03:48.890 ","End":"03:53.900","Text":"for t. Now we can convert this integral."},{"Start":"03:53.900 ","End":"03:56.780","Text":"The integral along the curve c,"},{"Start":"03:56.780 ","End":"04:04.770","Text":"becomes just the integral of the parameter t from 0 to 1."},{"Start":"04:05.510 ","End":"04:10.100","Text":"Then we just have to interpret this because we have x, y,"},{"Start":"04:10.100 ","End":"04:15.185","Text":"and z in terms of t. What we don\u0027t have,"},{"Start":"04:15.185 ","End":"04:20.010","Text":"we don\u0027t have dx, dy, and dz."},{"Start":"04:20.540 ","End":"04:22.920","Text":"I could write these at the side."},{"Start":"04:22.920 ","End":"04:25.470","Text":"Let\u0027s see we want to know what dx equals,"},{"Start":"04:25.470 ","End":"04:30.740","Text":"what dy equals, and what dz equals."},{"Start":"04:30.740 ","End":"04:36.245","Text":"If x equals t, then just dx equals dt. No problem."},{"Start":"04:36.245 ","End":"04:38.254","Text":"Y equals t squared,"},{"Start":"04:38.254 ","End":"04:43.460","Text":"take the derivative of d of each of these."},{"Start":"04:43.460 ","End":"04:51.200","Text":"I\u0027ve got 1dy or dy equals 2t dt."},{"Start":"04:51.200 ","End":"04:55.290","Text":"Here dz is just 3t squared,"},{"Start":"04:55.290 ","End":"04:58.980","Text":"derivative of t cubed dt."},{"Start":"04:58.980 ","End":"05:05.820","Text":"I think now we\u0027re ready to substitute into this expression or this expression."},{"Start":"05:05.860 ","End":"05:08.270","Text":"This 1 I mean."},{"Start":"05:08.270 ","End":"05:18.930","Text":"3x squared is 3t squared and I need a bracket minus 6yz."},{"Start":"05:19.430 ","End":"05:30.170","Text":"Y times z is t^5 so I have minus 6t^5 and then dx is dt,"},{"Start":"05:30.170 ","End":"05:33.525","Text":"and that\u0027s just the first part."},{"Start":"05:33.525 ","End":"05:43.070","Text":"Then continuing 2y is 2t squared plus 3xz,"},{"Start":"05:43.140 ","End":"05:49.600","Text":"3 and xz is t^4."},{"Start":"05:49.600 ","End":"05:53.750","Text":"Here we have dy, which is 2tdt."},{"Start":"05:55.910 ","End":"06:05.925","Text":"The last piece, the third component is 1 minus 4xyz squared."},{"Start":"06:05.925 ","End":"06:09.905","Text":"Now xyz squared is t,"},{"Start":"06:09.905 ","End":"06:13.955","Text":"t squared and t^6."},{"Start":"06:13.955 ","End":"06:18.935","Text":"Let\u0027s see, do the arithmetic t times t squared is t cubed,"},{"Start":"06:18.935 ","End":"06:26.955","Text":"times t^6 that should be t^9."},{"Start":"06:26.955 ","End":"06:33.340","Text":"Then dz is 3t squared dt."},{"Start":"06:33.340 ","End":"06:37.040","Text":"Now we just have an integral in t,"},{"Start":"06:37.040 ","End":"06:42.480","Text":"the first thing I\u0027ll do is just tidy it up."},{"Start":"06:42.480 ","End":"06:46.305","Text":"You want to have something dt."},{"Start":"06:46.305 ","End":"06:51.885","Text":"First step, I\u0027ll just open up brackets here,"},{"Start":"06:51.885 ","End":"06:58.920","Text":"3t squared minus 6t^5 and here multiplying by 2t."},{"Start":"06:58.920 ","End":"07:05.890","Text":"So it\u0027s 4t cubed plus 6t^5."},{"Start":"07:06.500 ","End":"07:11.175","Text":"Then this 1, 3t squared times each of these."},{"Start":"07:11.175 ","End":"07:13.635","Text":"So 3t squared"},{"Start":"07:13.635 ","End":"07:22.890","Text":"minus 12t^11 dt."},{"Start":"07:22.890 ","End":"07:26.375","Text":"Let\u0027s see, do I have any like terms to collect?"},{"Start":"07:26.375 ","End":"07:28.920","Text":"Well, yes."},{"Start":"07:29.000 ","End":"07:35.120","Text":"I have a 3t squared and I have another 3t squared."},{"Start":"07:35.120 ","End":"07:40.280","Text":"That will be like 6t squared and what else can I combine?"},{"Start":"07:40.280 ","End":"07:44.330","Text":"I have a t^5 and a t^5,"},{"Start":"07:44.330 ","End":"07:46.550","Text":"these just cancel each other out."},{"Start":"07:46.550 ","End":"07:49.560","Text":"It\u0027s a plus 6 and minus 6."},{"Start":"07:51.730 ","End":"07:56.300","Text":"I\u0027ll also cross these 2 out just so I don\u0027t count them twice because"},{"Start":"07:56.300 ","End":"08:01.440","Text":"I\u0027ve already got it here in the 6t squared."},{"Start":"08:01.630 ","End":"08:05.700","Text":"The integral is here,"},{"Start":"08:05.700 ","End":"08:08.625","Text":"I get 6 over 3t cubed,"},{"Start":"08:08.625 ","End":"08:13.530","Text":"6 over 3 is 2t cubed."},{"Start":"08:13.530 ","End":"08:17.865","Text":"From here 4t^4 over 4,"},{"Start":"08:17.865 ","End":"08:22.170","Text":"so that\u0027s just t^4."},{"Start":"08:22.170 ","End":"08:27.979","Text":"Then finally t^12 over 12."},{"Start":"08:27.979 ","End":"08:30.200","Text":"But the 12 with the 12 cancels,"},{"Start":"08:30.200 ","End":"08:34.715","Text":"so it\u0027s just minus t^12."},{"Start":"08:34.715 ","End":"08:39.065","Text":"All this taken from 0-1."},{"Start":"08:39.065 ","End":"08:42.530","Text":"When I put in 0, I get nothing,"},{"Start":"08:42.530 ","End":"08:45.425","Text":"so I just have to plug in 1."},{"Start":"08:45.425 ","End":"08:55.540","Text":"All we get is 2 plus 1 minus 1 and so the answer to part A is 2."},{"Start":"08:55.540 ","End":"09:01.990","Text":"I\u0027ll now highlight it and then we\u0027ll move on to part B."}],"ID":8801},{"Watched":false,"Name":"Exercise 10 Part b","Duration":"12m 29s","ChapterTopicVideoID":8703,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8703.jpeg","UploadDate":"2017-02-13T05:19:42.5870000","DurationForVideoObject":"PT12M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.515","Text":"Here we are in part B. I just left some of the stuff from part A that we can reuse."},{"Start":"00:07.515 ","End":"00:14.640","Text":"The path C this time is still the path from 0,0,0 to 1,1,1."},{"Start":"00:14.640 ","End":"00:17.760","Text":"But here we\u0027re going in 3 straight lines."},{"Start":"00:17.760 ","End":"00:20.685","Text":"We\u0027re starting from here and making a stop over"},{"Start":"00:20.685 ","End":"00:24.105","Text":"here and here until we finally end up where we want to go."},{"Start":"00:24.105 ","End":"00:26.715","Text":"That\u0027s like 3 straight lines."},{"Start":"00:26.715 ","End":"00:28.470","Text":"I\u0027m not going to draw it in 3D."},{"Start":"00:28.470 ","End":"00:32.280","Text":"I\u0027ll just represent the curve C of the picture."},{"Start":"00:32.280 ","End":"00:35.715","Text":"We go from this point to some other point,"},{"Start":"00:35.715 ","End":"00:37.710","Text":"and then to some other point,"},{"Start":"00:37.710 ","End":"00:42.630","Text":"and then finally to our destination."},{"Start":"00:42.630 ","End":"00:47.565","Text":"This would be the point 0,0,0,"},{"Start":"00:47.565 ","End":"00:56.670","Text":"this is the point 0,0,1,"},{"Start":"00:56.670 ","End":"01:00.600","Text":"then the point 0,1,1,"},{"Start":"01:00.600 ","End":"01:03.810","Text":"and then we end up at 1,1,1."},{"Start":"01:03.810 ","End":"01:09.595","Text":"This whole thing will be the path C,"},{"Start":"01:09.595 ","End":"01:14.665","Text":"but it\u0027s going to be made up of 3 separate paths."},{"Start":"01:14.665 ","End":"01:18.430","Text":"We\u0027ll call the first portion C_1,"},{"Start":"01:18.430 ","End":"01:21.645","Text":"the second portion C_2,"},{"Start":"01:21.645 ","End":"01:24.960","Text":"and the third portion C_3."},{"Start":"01:24.960 ","End":"01:30.965","Text":"What we do when we have a piecewise path is we just take the sum."},{"Start":"01:30.965 ","End":"01:36.695","Text":"The integral over C of all this stuff, blah, blah, blah,"},{"Start":"01:36.695 ","End":"01:41.075","Text":"is equal to the integral over C_1 of whatever it is,"},{"Start":"01:41.075 ","End":"01:44.570","Text":"plus the integral over C_2 of whatever it is,"},{"Start":"01:44.570 ","End":"01:48.690","Text":"plus the integral over C_3 of whatever it is."},{"Start":"01:48.690 ","End":"01:54.590","Text":"We make 3 separate calculations and then we add the 3 results together."},{"Start":"01:54.590 ","End":"01:58.310","Text":"Let\u0027s start with the path C_1."},{"Start":"01:58.310 ","End":"02:01.420","Text":"Now I need to parameterize it."},{"Start":"02:01.420 ","End":"02:05.495","Text":"I\u0027m going to use a shortcut because if you look at it,"},{"Start":"02:05.495 ","End":"02:11.390","Text":"you see that the x and the y are the same for the start and end."},{"Start":"02:11.390 ","End":"02:15.530","Text":"0,0 and 0,0 are what x and y are."},{"Start":"02:15.530 ","End":"02:19.315","Text":"Only the z is changing from 0 to 1."},{"Start":"02:19.315 ","End":"02:24.515","Text":"When you get a simple case like this where only 1 of them is moving,"},{"Start":"02:24.515 ","End":"02:30.410","Text":"then we can easily say that x equals 0,"},{"Start":"02:30.410 ","End":"02:34.940","Text":"y equals 0, and z is the only one that depends on the"},{"Start":"02:34.940 ","End":"02:41.180","Text":"parameter t. We\u0027re going from 0 to 1,"},{"Start":"02:41.180 ","End":"02:45.710","Text":"and so we can say that z equals t and"},{"Start":"02:45.710 ","End":"02:53.280","Text":"then t goes from 0 to 1."},{"Start":"02:55.130 ","End":"02:58.020","Text":"We know we\u0027re going to need dx,"},{"Start":"02:58.020 ","End":"03:01.800","Text":"dy, and dz, so let\u0027s just write those."},{"Start":"03:01.800 ","End":"03:06.285","Text":"dx is just also 0,"},{"Start":"03:06.285 ","End":"03:10.970","Text":"I\u0027d like to write it as 0 dt to remind myself there is a dt there."},{"Start":"03:10.970 ","End":"03:15.860","Text":"Similarly, y is just a derivative of 0 is 0 dt,"},{"Start":"03:15.860 ","End":"03:18.080","Text":"and z equals t,"},{"Start":"03:18.080 ","End":"03:21.620","Text":"so dz equals dt."},{"Start":"03:21.620 ","End":"03:24.530","Text":"That\u0027s fairly straightforward."},{"Start":"03:24.530 ","End":"03:28.280","Text":"The integral will be,"},{"Start":"03:28.280 ","End":"03:34.355","Text":"we take the parameter range that t goes from 0 to 1,"},{"Start":"03:34.355 ","End":"03:36.140","Text":"and then we do this expression."},{"Start":"03:36.140 ","End":"03:45.505","Text":"But of course, since dx and dy are both 0,"},{"Start":"03:45.505 ","End":"03:48.409","Text":"I don\u0027t need this because that\u0027s 0."},{"Start":"03:48.409 ","End":"03:51.745","Text":"I don\u0027t need this, that\u0027s 0."},{"Start":"03:51.745 ","End":"03:55.275","Text":"I only need the last bit then."},{"Start":"03:55.275 ","End":"04:02.910","Text":"I have 1 minus 4xyz squared."},{"Start":"04:02.910 ","End":"04:10.845","Text":"But xyz squared is 0 already,"},{"Start":"04:10.845 ","End":"04:13.710","Text":"so it\u0027s just 1 minus 0,"},{"Start":"04:13.710 ","End":"04:17.220","Text":"I\u0027m going to leave the 0 there to show you that I haven\u0027t forgotten it,"},{"Start":"04:17.220 ","End":"04:20.560","Text":"dz which is dt."},{"Start":"04:22.040 ","End":"04:25.540","Text":"This is now very straightforward."},{"Start":"04:25.540 ","End":"04:30.980","Text":"The integral of 1 from 0 to 1."},{"Start":"04:30.980 ","End":"04:32.710","Text":"Whenever you have the integral of 1,"},{"Start":"04:32.710 ","End":"04:34.960","Text":"it\u0027s always the upper minus the lower,"},{"Start":"04:34.960 ","End":"04:36.760","Text":"it comes out to be just 1."},{"Start":"04:36.760 ","End":"04:40.510","Text":"If you want, you can write the integral of 1 is t. When t is 1,"},{"Start":"04:40.510 ","End":"04:42.590","Text":"it\u0027s 1, when t is 0, it\u0027s 0."},{"Start":"04:42.590 ","End":"04:46.610","Text":"It\u0027s so simple, I\u0027m just writing the answer 1."},{"Start":"04:46.610 ","End":"04:50.540","Text":"Maybe I\u0027ll just write it below here,"},{"Start":"04:50.540 ","End":"04:53.005","Text":"I write that this is 1."},{"Start":"04:53.005 ","End":"04:55.470","Text":"Then we\u0027ll write what this is and this is,"},{"Start":"04:55.470 ","End":"04:57.370","Text":"and then we\u0027ll do the addition at the end."},{"Start":"04:57.370 ","End":"05:00.650","Text":"Let\u0027s move on to C_2 now."},{"Start":"05:01.140 ","End":"05:05.380","Text":"Want to keep this in view. Let\u0027s see."},{"Start":"05:05.380 ","End":"05:14.320","Text":"For C_2, I\u0027m also going to use a shortcut because when I look at C_2,"},{"Start":"05:14.320 ","End":"05:22.510","Text":"you see that the x and the z don\u0027t change."},{"Start":"05:22.730 ","End":"05:29.250","Text":"I\u0027ve got 1 here,"},{"Start":"05:29.250 ","End":"05:32.325","Text":"and I have 1 here,"},{"Start":"05:32.325 ","End":"05:34.650","Text":"and I have 0 here,"},{"Start":"05:34.650 ","End":"05:37.900","Text":"and I have 0 here."},{"Start":"05:38.210 ","End":"05:41.670","Text":"Only the middle bit, the y bit,"},{"Start":"05:41.670 ","End":"05:44.460","Text":"is changing from 0 to 1."},{"Start":"05:44.460 ","End":"05:46.010","Text":"Just like we did before,"},{"Start":"05:46.010 ","End":"05:50.610","Text":"we don\u0027t need to crank out all this formula that"},{"Start":"05:50.610 ","End":"05:55.460","Text":"we used to do with the first point plus t times the second minus the first and all that,"},{"Start":"05:55.460 ","End":"05:57.860","Text":"we can just straight away say,"},{"Start":"05:57.860 ","End":"06:01.715","Text":"x is constantly 0,"},{"Start":"06:01.715 ","End":"06:05.730","Text":"z is constantly 1,"},{"Start":"06:05.730 ","End":"06:08.460","Text":"and only the y is changing."},{"Start":"06:08.460 ","End":"06:14.260","Text":"So it\u0027s like t, but t goes from 0 to 1."},{"Start":"06:14.260 ","End":"06:17.480","Text":"This is a fairly standard shortcut."},{"Start":"06:17.480 ","End":"06:19.730","Text":"This is because if we do it in 3D,"},{"Start":"06:19.730 ","End":"06:21.500","Text":"we\u0027re going parallel to the axes."},{"Start":"06:21.500 ","End":"06:24.335","Text":"Here we\u0027re going up in the z direction,"},{"Start":"06:24.335 ","End":"06:27.335","Text":"here we\u0027re going in the y direction,"},{"Start":"06:27.335 ","End":"06:29.845","Text":"parallel to the y-axis."},{"Start":"06:29.845 ","End":"06:33.850","Text":"Of course, the last bit we\u0027re going to be moving in the x direction anyway,"},{"Start":"06:33.850 ","End":"06:35.735","Text":"let\u0027s not get ahead of ourselves."},{"Start":"06:35.735 ","End":"06:39.800","Text":"As before, we know we\u0027re going to need dx."},{"Start":"06:39.800 ","End":"06:42.050","Text":"Did I miss a d here?"},{"Start":"06:42.050 ","End":"06:46.875","Text":"Sorry. Yeah, too late now, but never mind."},{"Start":"06:46.875 ","End":"06:49.905","Text":"Here also we\u0027ll need dx,"},{"Start":"06:49.905 ","End":"06:53.880","Text":"we\u0027ll need dy, and we\u0027ll need dz."},{"Start":"06:53.880 ","End":"06:59.145","Text":"Let\u0027s see, again, x and z are constants."},{"Start":"06:59.145 ","End":"07:02.250","Text":"These are both going to be 0,"},{"Start":"07:02.250 ","End":"07:05.150","Text":"this is also going to be 0."},{"Start":"07:05.150 ","End":"07:07.970","Text":"Only the y is moving and y is t,"},{"Start":"07:07.970 ","End":"07:10.980","Text":"so dy is dt."},{"Start":"07:11.690 ","End":"07:16.270","Text":"In the case of the integral,"},{"Start":"07:18.530 ","End":"07:20.760","Text":"the dx is still 0,"},{"Start":"07:20.760 ","End":"07:24.830","Text":"dy no longer, and the dz will be 0."},{"Start":"07:24.830 ","End":"07:28.865","Text":"So we just need the middle bit, the dy."},{"Start":"07:28.865 ","End":"07:39.840","Text":"We get the integral from 0 to 1 of 2y plus 3xz."},{"Start":"07:40.090 ","End":"07:49.620","Text":"2y is 2t plus 3xz."},{"Start":"07:49.620 ","End":"07:51.570","Text":"Well, x is 0,"},{"Start":"07:51.570 ","End":"07:58.750","Text":"so that makes this 0 dt."},{"Start":"07:59.150 ","End":"08:03.000","Text":"The integral of 2t is t squared,"},{"Start":"08:03.000 ","End":"08:06.495","Text":"so it\u0027s t squared from 0 to 1."},{"Start":"08:06.495 ","End":"08:12.760","Text":"It\u0027s just 1 squared minus 0 squared is just 1."},{"Start":"08:12.760 ","End":"08:15.300","Text":"That also came out as 1,"},{"Start":"08:15.300 ","End":"08:19.800","Text":"and I\u0027ll write that here as 1."},{"Start":"08:19.800 ","End":"08:27.080","Text":"That\u0027s 2 down and 1 to go, just C_3."},{"Start":"08:27.080 ","End":"08:31.150","Text":"I just copied the formula from here"},{"Start":"08:31.150 ","End":"08:35.450","Text":"to here because I\u0027m going to scroll now and I\u0027ll lose it."},{"Start":"08:35.550 ","End":"08:40.575","Text":"What we need to do really is keep this here."},{"Start":"08:40.575 ","End":"08:42.840","Text":"I\u0027ve got what I need."},{"Start":"08:42.840 ","End":"08:46.845","Text":"This time we need to parameterize C_3,"},{"Start":"08:46.845 ","End":"08:49.845","Text":"which is this bit here."},{"Start":"08:49.845 ","End":"08:52.675","Text":"In this bit here,"},{"Start":"08:52.675 ","End":"08:54.190","Text":"if I look at the numbers,"},{"Start":"08:54.190 ","End":"08:58.210","Text":"notice that the y and z are not changing,"},{"Start":"08:58.210 ","End":"09:03.050","Text":"I have 1 and 1, and here I have 1 and 1."},{"Start":"09:03.050 ","End":"09:08.075","Text":"Only the x is changing from 0 to 1,"},{"Start":"09:08.075 ","End":"09:14.704","Text":"so we don\u0027t need to crank out the whole formula for segment between 2 points."},{"Start":"09:14.704 ","End":"09:24.065","Text":"We can easily see that the curve C_3 can be parameterized by y is 1,"},{"Start":"09:24.065 ","End":"09:27.455","Text":"z is 1, that\u0027s the 1 and the 1 here,"},{"Start":"09:27.455 ","End":"09:29.530","Text":"and only x is moving,"},{"Start":"09:29.530 ","End":"09:31.650","Text":"so it goes from 0 to 1,"},{"Start":"09:31.650 ","End":"09:39.240","Text":"so I\u0027ll write it as x equals t and we\u0027ll let t go from 0 to 1."},{"Start":"09:39.240 ","End":"09:43.060","Text":"As before, we need dx, dy, and dz."},{"Start":"09:43.160 ","End":"09:48.000","Text":"dx from here is just dt,"},{"Start":"09:48.000 ","End":"09:52.890","Text":"and dy, that y is a constant,"},{"Start":"09:52.890 ","End":"09:56.285","Text":"so dy is just 0dt,"},{"Start":"09:56.285 ","End":"09:59.250","Text":"and dz is also,"},{"Start":"09:59.250 ","End":"10:03.360","Text":"z is a constant, it\u0027s also 0dt, it\u0027s just 0."},{"Start":"10:03.360 ","End":"10:13.880","Text":"Basically what we get is that this part is going to be 0 and this part is going to be 0."},{"Start":"10:13.880 ","End":"10:17.820","Text":"So I\u0027ll only need the integral of the first bit."},{"Start":"10:19.250 ","End":"10:26.640","Text":"What we will get is the integral from 0 to 1,"},{"Start":"10:26.640 ","End":"10:34.170","Text":"that\u0027s for t, of we just need 3x squared minus 6yz."},{"Start":"10:34.170 ","End":"10:40.425","Text":"Let\u0027s see, 3x squared is 3t squared"},{"Start":"10:40.425 ","End":"10:49.840","Text":"minus 6 times y is 1 and z is 1."},{"Start":"11:04.130 ","End":"11:08.025","Text":"We need dx, of course,"},{"Start":"11:08.025 ","End":"11:13.080","Text":"and dx is equal to dt."},{"Start":"11:13.080 ","End":"11:16.185","Text":"I don\u0027t know why I said dz before, sorry,"},{"Start":"11:16.185 ","End":"11:22.380","Text":"dx which is dt."},{"Start":"11:22.380 ","End":"11:24.330","Text":"I can just even do it over here,"},{"Start":"11:24.330 ","End":"11:26.680","Text":"I don\u0027t want to lose this bit."},{"Start":"11:27.590 ","End":"11:34.620","Text":"For 3t squared I get 3t cubed over 3 is just t cubed,"},{"Start":"11:34.620 ","End":"11:38.620","Text":"6 just gives me 6t."},{"Start":"11:38.720 ","End":"11:43.365","Text":"Then this from 0 to 1, 0 gives nothing,"},{"Start":"11:43.365 ","End":"11:47.820","Text":"1 gives me 1 minus 6,"},{"Start":"11:47.820 ","End":"11:51.070","Text":"so that\u0027s minus 5."},{"Start":"11:51.380 ","End":"11:59.290","Text":"That is the integral over the third part is minus 5."},{"Start":"11:59.290 ","End":"12:02.230","Text":"If I add all these 3 together,"},{"Start":"12:02.230 ","End":"12:06.820","Text":"I get 1 plus 1."},{"Start":"12:06.820 ","End":"12:14.070","Text":"Well, I\u0027ll just write it, it\u0027s 1 plus 1 plus negative 5,"},{"Start":"12:14.070 ","End":"12:16.574","Text":"and that\u0027s 2 minus 5,"},{"Start":"12:16.574 ","End":"12:19.485","Text":"so that\u0027s minus 3."},{"Start":"12:19.485 ","End":"12:24.015","Text":"That\u0027s the answer to part B."},{"Start":"12:24.015 ","End":"12:28.090","Text":"Next clip, we\u0027ll do part C."}],"ID":8802},{"Watched":false,"Name":"Exercise 11 Part a","Duration":"7m 44s","ChapterTopicVideoID":8704,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8704.jpeg","UploadDate":"2017-02-13T05:21:20.7870000","DurationForVideoObject":"PT7M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"In this exercise, we have to compute a type 2 line integral."},{"Start":"00:05.970 ","End":"00:10.230","Text":"It\u0027s in 2 variables in the plane x,y."},{"Start":"00:10.230 ","End":"00:14.280","Text":"But it\u0027s not written in the usual customary"},{"Start":"00:14.280 ","End":"00:17.580","Text":"form that you\u0027re familiar with."},{"Start":"00:17.580 ","End":"00:20.440","Text":"We\u0027re going to rewrite it a bit."},{"Start":"00:21.590 ","End":"00:25.220","Text":"We usually write vectors with an arrow,"},{"Start":"00:25.220 ","End":"00:28.860","Text":"so let me add arrows everywhere."},{"Start":"00:29.630 ","End":"00:32.625","Text":"Here this is a vector,"},{"Start":"00:32.625 ","End":"00:35.460","Text":"F is a vector,"},{"Start":"00:35.460 ","End":"00:39.220","Text":"r is a vector."},{"Start":"00:42.470 ","End":"00:48.235","Text":"What does this mean when we write r in this form?"},{"Start":"00:48.235 ","End":"00:50.525","Text":"Just straightforward."},{"Start":"00:50.525 ","End":"00:57.260","Text":"We have that r which is like x,y,"},{"Start":"00:57.260 ","End":"01:16.020","Text":"r is just x,y, and dr is just dx, dy."},{"Start":"01:16.020 ","End":"01:18.790","Text":"So the integral of"},{"Start":"01:18.790 ","End":"01:27.320","Text":"F.dr could be written in the more familiar form,"},{"Start":"01:27.320 ","End":"01:29.555","Text":"I\u0027ll use this particular F,"},{"Start":"01:29.555 ","End":"01:35.420","Text":"it\u0027s the dot product of this vector with this vector,"},{"Start":"01:35.420 ","End":"01:38.765","Text":"it\u0027s just integral."},{"Start":"01:38.765 ","End":"01:46.455","Text":"X squared y cubed goes with dx plus,"},{"Start":"01:46.455 ","End":"01:49.620","Text":"and then this with dy,"},{"Start":"01:49.620 ","End":"01:54.200","Text":"it\u0027s negative, I\u0027ll just change that plus to a minus,"},{"Start":"01:54.200 ","End":"02:02.670","Text":"and then we have y square root of x times dy."},{"Start":"02:02.670 ","End":"02:04.680","Text":"This is a more familiar form,"},{"Start":"02:04.680 ","End":"02:06.989","Text":"and as for the curve C,"},{"Start":"02:06.989 ","End":"02:08.955","Text":"r also defines that."},{"Start":"02:08.955 ","End":"02:13.265","Text":"We can write our curve C in parametrized form."},{"Start":"02:13.265 ","End":"02:17.960","Text":"I could write it as x equals, and y equals."},{"Start":"02:17.960 ","End":"02:21.250","Text":"This vector is x,"},{"Start":"02:21.250 ","End":"02:25.515","Text":"y, so component y is x is t squared,"},{"Start":"02:25.515 ","End":"02:28.095","Text":"y is minus t cubed,"},{"Start":"02:28.095 ","End":"02:33.780","Text":"and we have the parameter range where it goes from 0-1."},{"Start":"02:33.780 ","End":"02:39.560","Text":"Now we can even write this in terms of t"},{"Start":"02:39.560 ","End":"02:41.285","Text":"and get a regular integral,"},{"Start":"02:41.285 ","End":"02:45.995","Text":"we get the integral from 0-1,"},{"Start":"02:45.995 ","End":"02:53.490","Text":"which I sometimes emphasize it\u0027s t. Just take it 1 at a time,"},{"Start":"02:53.490 ","End":"02:57.135","Text":"x, I look up is t squared."},{"Start":"02:57.135 ","End":"03:08.010","Text":"T squared, squared, y cubed is minus t cubed, cubed."},{"Start":"03:08.010 ","End":"03:11.175","Text":"All this is dx."},{"Start":"03:11.175 ","End":"03:14.850","Text":"We still don\u0027t have dx,"},{"Start":"03:14.850 ","End":"03:18.000","Text":"dy for our particular case."},{"Start":"03:18.000 ","End":"03:19.230","Text":"Let me just write that,"},{"Start":"03:19.230 ","End":"03:23.100","Text":"dx equals and dy equals."},{"Start":"03:23.100 ","End":"03:26.350","Text":"This will be 2tdt,"},{"Start":"03:27.230 ","End":"03:33.690","Text":"and this will be minus 3t squared dt."},{"Start":"03:33.690 ","End":"03:36.165","Text":"I can get back here now,"},{"Start":"03:36.165 ","End":"03:39.285","Text":"and write times 2tdt."},{"Start":"03:39.285 ","End":"03:43.665","Text":"That was the dx which is here."},{"Start":"03:43.665 ","End":"03:46.200","Text":"That\u0027s just the first part,"},{"Start":"03:46.200 ","End":"03:48.330","Text":"now we have a minus."},{"Start":"03:48.330 ","End":"03:54.400","Text":"We need y which is minus t cubed,"},{"Start":"03:56.090 ","End":"04:00.624","Text":"and square root of x."},{"Start":"04:00.624 ","End":"04:05.795","Text":"Just like to go over a small important technical point."},{"Start":"04:05.795 ","End":"04:12.020","Text":"The square root of x is the square root of t squared."},{"Start":"04:12.020 ","End":"04:17.870","Text":"Now, the square root of t squared is not just automatically t,"},{"Start":"04:17.870 ","End":"04:21.050","Text":"it\u0027s the absolute value of t if you remember."},{"Start":"04:21.050 ","End":"04:25.294","Text":"But because our t is in the non-negative range,"},{"Start":"04:25.294 ","End":"04:30.170","Text":"in our case it is equal to t. Getting back here,"},{"Start":"04:30.170 ","End":"04:33.380","Text":"the square root of x is square root of t squared,"},{"Start":"04:33.380 ","End":"04:38.900","Text":"in our case is t. Finally the dy from here"},{"Start":"04:38.900 ","End":"04:44.220","Text":"is minus 3t squared dt."},{"Start":"04:44.890 ","End":"04:48.865","Text":"Now we want to simplify this a bit."},{"Start":"04:48.865 ","End":"04:54.870","Text":"It\u0027s the integral, it\u0027s still from 0-1. Let\u0027s see."},{"Start":"04:54.870 ","End":"05:00.265","Text":"The first 1 is an exercise is an exponents."},{"Start":"05:00.265 ","End":"05:08.100","Text":"Certainly we have 2, in fact, even we have a minus 2,"},{"Start":"05:08.660 ","End":"05:14.940","Text":"everything else is powers of t. This would be t^4."},{"Start":"05:14.940 ","End":"05:18.720","Text":"Here we have a t^4, here t cubed,"},{"Start":"05:18.720 ","End":"05:21.120","Text":"cubed is t^9,"},{"Start":"05:21.120 ","End":"05:26.475","Text":"and here we\u0027ll have just a t^1."},{"Start":"05:26.475 ","End":"05:30.485","Text":"If I take 4 plus 9 plus 1,"},{"Start":"05:30.485 ","End":"05:33.175","Text":"that makes it 14."},{"Start":"05:33.175 ","End":"05:36.915","Text":"So t to the power of 14."},{"Start":"05:36.915 ","End":"05:43.040","Text":"Here, I have 3 minuses, minus, minus, minus."},{"Start":"05:43.040 ","End":"05:47.445","Text":"It\u0027s going to be minus constant,"},{"Start":"05:47.445 ","End":"05:49.935","Text":"the numbers I have are 3."},{"Start":"05:49.935 ","End":"05:55.290","Text":"Everything else is just powers of t, t cubed,"},{"Start":"05:55.290 ","End":"05:58.860","Text":"t to the 1, t to the 2,"},{"Start":"05:58.860 ","End":"06:02.805","Text":"3 and 1 and 2 is 6."},{"Start":"06:02.805 ","End":"06:12.120","Text":"It\u0027s t to the 6 and dt."},{"Start":"06:12.120 ","End":"06:15.100","Text":"Let\u0027s do this integral."},{"Start":"06:15.680 ","End":"06:20.730","Text":"Raise the power by 1 is 15 and divide by that."},{"Start":"06:20.730 ","End":"06:25.330","Text":"I\u0027ve got minus 2/15t^15,"},{"Start":"06:26.900 ","End":"06:35.010","Text":"and here it\u0027s 7 divide by the 7 minus 3/7t^7."},{"Start":"06:35.010 ","End":"06:43.890","Text":"Evaluate this from 0-1."},{"Start":"06:43.890 ","End":"06:46.380","Text":"At 0, I don\u0027t get anything."},{"Start":"06:46.380 ","End":"06:54.060","Text":"At 1, we just get the fraction minus 2/15 minus 3/7,"},{"Start":"06:54.060 ","End":"06:56.150","Text":"why don\u0027t I just put a bracket and make"},{"Start":"06:56.150 ","End":"07:02.550","Text":"that a plus 3/7, exercise in fractions."},{"Start":"07:03.460 ","End":"07:10.605","Text":"A common denominator has to be 15 times 7, nothing less."},{"Start":"07:10.605 ","End":"07:15.300","Text":"That\u0027s 105, 15 into a 105 goes 7 times,"},{"Start":"07:15.300 ","End":"07:18.765","Text":"7 times 2 is 14,"},{"Start":"07:18.765 ","End":"07:22.230","Text":"7 into 105 goes 15 times,"},{"Start":"07:22.230 ","End":"07:26.865","Text":"15 times 3 is 45."},{"Start":"07:26.865 ","End":"07:34.560","Text":"I\u0027ve kept the negative here, 14 and 45 is 59."},{"Start":"07:34.560 ","End":"07:39.580","Text":"The final answer is minus 59/105."},{"Start":"07:39.940 ","End":"07:43.340","Text":"I like to highlight it,"},{"Start":"07:43.340 ","End":"07:45.810","Text":"and we are done."}],"ID":8803},{"Watched":false,"Name":"Exercise 11 Part b","Duration":"9m 17s","ChapterTopicVideoID":8705,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8705.jpeg","UploadDate":"2017-02-13T05:23:32.8100000","DurationForVideoObject":"PT9M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.075","Text":"In this exercise, we have a line integral of type 2 over a curve C,"},{"Start":"00:09.075 ","End":"00:11.415","Text":"function f it\u0027s in 3 dimensions."},{"Start":"00:11.415 ","End":"00:13.470","Text":"We have x, y, and z."},{"Start":"00:13.470 ","End":"00:19.230","Text":"Actually, I think vector should be written with an arrow on top."},{"Start":"00:19.230 ","End":"00:21.135","Text":"F is a vector,"},{"Start":"00:21.135 ","End":"00:25.320","Text":"and this r is a vector. Let me explain."},{"Start":"00:25.320 ","End":"00:27.660","Text":"This is not the standard notation,"},{"Start":"00:27.660 ","End":"00:30.840","Text":"r or r of t,"},{"Start":"00:30.840 ","End":"00:33.660","Text":"but in general, r is the vector."},{"Start":"00:33.660 ","End":"00:36.270","Text":"I use the bracket form of a vector."},{"Start":"00:36.270 ","End":"00:39.375","Text":"It\u0027s x, y, z,"},{"Start":"00:39.375 ","End":"00:48.790","Text":"and dr also a vector is just dx, dy, dz."},{"Start":"00:49.040 ","End":"00:54.160","Text":"Now we can interpret this dot-product F.dr."},{"Start":"00:54.160 ","End":"01:02.315","Text":"It\u0027s this vector, dot with this vector and we just multiply component-wise and add."},{"Start":"01:02.315 ","End":"01:06.290","Text":"What we have is the integral along the curve"},{"Start":"01:06.290 ","End":"01:13.915","Text":"C of sine x times dx,"},{"Start":"01:13.915 ","End":"01:17.480","Text":"plus cosine y from here,"},{"Start":"01:17.480 ","End":"01:25.640","Text":"times dy, plus 1/3 component, xz times dz."},{"Start":"01:26.540 ","End":"01:29.335","Text":"What is this curve C?"},{"Start":"01:29.335 ","End":"01:32.314","Text":"Well, it\u0027s given in parametric form."},{"Start":"01:32.314 ","End":"01:38.405","Text":"C is just, if I write it with the curly bracket,"},{"Start":"01:38.405 ","End":"01:42.500","Text":"this is x, y, z as functions of t,"},{"Start":"01:42.500 ","End":"01:45.900","Text":"but it\u0027s x of t,"},{"Start":"01:45.900 ","End":"01:52.040","Text":"I don\u0027t bother to write the x of t is just t cubed, y of t,"},{"Start":"01:52.040 ","End":"01:55.055","Text":"or just y is minus t squared,"},{"Start":"01:55.055 ","End":"01:58.085","Text":"and z of t is just t,"},{"Start":"01:58.085 ","End":"02:00.950","Text":"and of course we need to know where the parameter goes from and to,"},{"Start":"02:00.950 ","End":"02:04.825","Text":"and it\u0027s just written here from 0-1."},{"Start":"02:04.825 ","End":"02:08.510","Text":"Now we can convert this to a regular integral in terms of"},{"Start":"02:08.510 ","End":"02:14.915","Text":"the parameter t. We have t goes from 0-1."},{"Start":"02:14.915 ","End":"02:17.660","Text":"I would like to emphasize the parameter,"},{"Start":"02:17.660 ","End":"02:20.945","Text":"and then we just translate everything."},{"Start":"02:20.945 ","End":"02:28.260","Text":"I see x, then I see that x is t cubed along the curve,"},{"Start":"02:28.260 ","End":"02:34.055","Text":"so I have sine of t cubed and then dx,"},{"Start":"02:34.055 ","End":"02:36.595","Text":"that we don\u0027t have."},{"Start":"02:36.595 ","End":"02:38.490","Text":"Let\u0027s just write those,"},{"Start":"02:38.490 ","End":"02:40.580","Text":"let\u0027s see what dx equals,"},{"Start":"02:40.580 ","End":"02:44.635","Text":"what dy equals and dz equals."},{"Start":"02:44.635 ","End":"02:47.150","Text":"Basically just the derivative of this times dt."},{"Start":"02:47.150 ","End":"02:52.850","Text":"3t squared dt minus"},{"Start":"02:52.850 ","End":"02:59.700","Text":"2t dt and just 1 dt."},{"Start":"02:59.700 ","End":"03:10.125","Text":"Getting back here, now I can write that dx is 3t squared dt plus cosine."},{"Start":"03:10.125 ","End":"03:15.190","Text":"I look up y, y is minus t squared,"},{"Start":"03:15.350 ","End":"03:24.810","Text":"and dy is minus 2t dt,"},{"Start":"03:24.810 ","End":"03:26.925","Text":"and the third component,"},{"Start":"03:26.925 ","End":"03:33.585","Text":"xz is t cubed times t,"},{"Start":"03:33.585 ","End":"03:37.095","Text":"and dz is just dt."},{"Start":"03:37.095 ","End":"03:40.905","Text":"We\u0027re going to simplify this expression a bit,"},{"Start":"03:40.905 ","End":"03:47.180","Text":"and I\u0027d like to just write this with a single dt."},{"Start":"03:47.180 ","End":"03:57.500","Text":"I\u0027ve got the integral from 0-1 of sine of t cubed times 3t"},{"Start":"03:57.500 ","End":"04:03.215","Text":"squared plus cosine of minus"},{"Start":"04:03.215 ","End":"04:09.120","Text":"t squared times minus 2t plus,"},{"Start":"04:09.120 ","End":"04:11.685","Text":"and this I can write as t^4,"},{"Start":"04:11.685 ","End":"04:15.090","Text":"and all this is dt."},{"Start":"04:15.090 ","End":"04:18.705","Text":"Now I have 3 integrals to do."},{"Start":"04:18.705 ","End":"04:23.945","Text":"These they look difficult to actually simpler than you think,"},{"Start":"04:23.945 ","End":"04:26.674","Text":"they were cooked up to be simple."},{"Start":"04:26.674 ","End":"04:29.600","Text":"Let me do an aside here."},{"Start":"04:29.600 ","End":"04:36.995","Text":"If I have a function sine of something with x in it,"},{"Start":"04:36.995 ","End":"04:39.275","Text":"and I take its derivative,"},{"Start":"04:39.275 ","End":"04:43.250","Text":"what I would get would be cosine of that thing."},{"Start":"04:43.250 ","End":"04:46.820","Text":"But then I\u0027d also have to multiply by the inner derivative,"},{"Start":"04:46.820 ","End":"04:50.255","Text":"which is say, box prime."},{"Start":"04:50.255 ","End":"04:59.465","Text":"I can do this backwards and say if I have the integral of cosine of some function of x,"},{"Start":"04:59.465 ","End":"05:03.125","Text":"and I have the derivative of that thing alongside,"},{"Start":"05:03.125 ","End":"05:08.150","Text":"then this is just sine of this."},{"Start":"05:08.150 ","End":"05:10.550","Text":"If we\u0027re doing indefinite integrals,"},{"Start":"05:10.550 ","End":"05:14.335","Text":"would have to add a plus c for definite integrals, we wouldn\u0027t need it."},{"Start":"05:14.335 ","End":"05:21.070","Text":"Similarly, the integral, if we have the sine of something,"},{"Start":"05:21.070 ","End":"05:25.005","Text":"maybe I better put this in brackets,"},{"Start":"05:25.005 ","End":"05:31.930","Text":"sine of this, this, this."},{"Start":"05:36.410 ","End":"05:41.380","Text":"I have also the derivative alongside,"},{"Start":"05:41.380 ","End":"05:44.460","Text":"and maybe it\u0027s dx here,"},{"Start":"05:44.460 ","End":"05:49.025","Text":"say the box is a function of x,"},{"Start":"05:49.025 ","End":"05:55.740","Text":"then the integral of sine is minus cosine of whatever this was,"},{"Start":"05:55.740 ","End":"05:58.380","Text":"and if we\u0027re doing indefinite integral we would"},{"Start":"05:58.380 ","End":"06:01.625","Text":"add plus c. Now why I\u0027m mentioning all this?"},{"Start":"06:01.625 ","End":"06:08.689","Text":"Because I happened to notice that if I take t cubed to be the box,"},{"Start":"06:08.689 ","End":"06:13.570","Text":"then this 3t squared is exactly box prime."},{"Start":"06:13.570 ","End":"06:17.795","Text":"The question was cooked up to come out easy and similarly,"},{"Start":"06:17.795 ","End":"06:22.830","Text":"if minus t squared is what I call box,"},{"Start":"06:22.830 ","End":"06:27.845","Text":"then minus 2t is the derivative of that."},{"Start":"06:27.845 ","End":"06:30.965","Text":"Here I worked with x, but it could have worked just as well with"},{"Start":"06:30.965 ","End":"06:36.705","Text":"t, doesn\u0027t really matter."},{"Start":"06:36.705 ","End":"06:40.070","Text":"What I can say is that this integral using"},{"Start":"06:40.070 ","End":"06:46.470","Text":"this scheme comes out to be using the one for sine."},{"Start":"06:46.580 ","End":"06:50.985","Text":"Then I get minus cosine,"},{"Start":"06:50.985 ","End":"06:57.070","Text":"so here I have minus cosine of t cubed,"},{"Start":"06:57.070 ","End":"07:01.730","Text":"and I don\u0027t need the plus c because it\u0027s going to be a definite integral."},{"Start":"07:01.730 ","End":"07:07.720","Text":"The next bit for the cosine, I\u0027m using these 2."},{"Start":"07:07.720 ","End":"07:11.130","Text":"This was just a way to show you how I got to this."},{"Start":"07:14.530 ","End":"07:20.370","Text":"The integral of cosine is just sine,"},{"Start":"07:20.370 ","End":"07:26.285","Text":"so I have sine of minus t squared,"},{"Start":"07:26.285 ","End":"07:29.420","Text":"and the last one is just a polynomial,"},{"Start":"07:29.420 ","End":"07:33.140","Text":"so it\u0027s 1/5 t^5."},{"Start":"07:33.140 ","End":"07:39.630","Text":"Now all this has to be evaluated between 0 and 1."},{"Start":"07:40.160 ","End":"07:42.935","Text":"Let\u0027s see what we get."},{"Start":"07:42.935 ","End":"07:46.800","Text":"First of all, we want to substitute 1,"},{"Start":"07:48.830 ","End":"07:51.260","Text":"1 cubed is 1,"},{"Start":"07:51.260 ","End":"07:56.360","Text":"so we get minus cosine of 1."},{"Start":"07:56.360 ","End":"08:01.400","Text":"Then we have plus sine of minus 1."},{"Start":"08:01.400 ","End":"08:03.590","Text":"But sine is an odd function."},{"Start":"08:03.590 ","End":"08:06.220","Text":"The sine of a minus I can bring the minus in front,"},{"Start":"08:06.220 ","End":"08:09.080","Text":"so this is minus sine 1."},{"Start":"08:09.080 ","End":"08:11.530","Text":"Make a note that that\u0027s an odd function."},{"Start":"08:11.530 ","End":"08:13.955","Text":"I can take the minus in front,"},{"Start":"08:13.955 ","End":"08:22.080","Text":"and here I get plus 1/5,"},{"Start":"08:22.080 ","End":"08:25.055","Text":"and all this is the 1 part."},{"Start":"08:25.055 ","End":"08:28.325","Text":"Now I need to subtract the 0 part."},{"Start":"08:28.325 ","End":"08:35.525","Text":"I get minus cosine 0 and cosine 0 is 1."},{"Start":"08:35.525 ","End":"08:42.920","Text":"That\u0027s minus 1, and then sine of 0 is 0,"},{"Start":"08:42.920 ","End":"08:49.155","Text":"so I have just 0 and here also 0."},{"Start":"08:49.155 ","End":"08:52.475","Text":"If I add it all up together,"},{"Start":"08:52.475 ","End":"08:57.160","Text":"let\u0027s see the numbers without the trigonometric stuff."},{"Start":"08:57.160 ","End":"09:04.470","Text":"Fifth minus minus 1 is 1 and 1/5 or 6/5 whichever,"},{"Start":"09:04.470 ","End":"09:12.805","Text":"and then minus cosine 1 minus sine 1,"},{"Start":"09:12.805 ","End":"09:17.050","Text":"and this is the answer to the question. We\u0027re done."}],"ID":8804},{"Watched":false,"Name":"Exercise 12 Part a","Duration":"9m 40s","ChapterTopicVideoID":8706,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8706.jpeg","UploadDate":"2017-02-13T05:25:54.3100000","DurationForVideoObject":"PT9M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"This exercise is really from physics"},{"Start":"00:03.540 ","End":"00:07.815","Text":"but don\u0027t worry, I\u0027ll give you all the formulas that you need."},{"Start":"00:07.815 ","End":"00:12.045","Text":"We have to compute the work done by a force field,"},{"Start":"00:12.045 ","End":"00:14.205","Text":"which is given as follows."},{"Start":"00:14.205 ","End":"00:18.285","Text":"On a particle which moves along the parabola,"},{"Start":"00:18.285 ","End":"00:21.420","Text":"y equals x squared from this point to this point."},{"Start":"00:21.420 ","End":"00:25.515","Text":"Let\u0027s just worry about part a first and we\u0027ll get to part b."},{"Start":"00:25.515 ","End":"00:32.385","Text":"Now, I brought in the formula or the theory that we need for doing this problem."},{"Start":"00:32.385 ","End":"00:35.630","Text":"I found this on the Internet and I copied it."},{"Start":"00:35.630 ","End":"00:41.470","Text":"It says that as a particle moves through a force field along the path C,"},{"Start":"00:41.470 ","End":"00:44.900","Text":"the work done by the force is the line integral,"},{"Start":"00:44.900 ","End":"00:47.495","Text":"and that\u0027s the chapter we\u0027re on, line integrals."},{"Start":"00:47.495 ","End":"00:50.995","Text":"Given by W, W is the work,"},{"Start":"00:50.995 ","End":"00:53.385","Text":"F is the force field,"},{"Start":"00:53.385 ","End":"00:57.825","Text":"and what we don\u0027t have here is C,"},{"Start":"00:57.825 ","End":"01:02.185","Text":"and I need to explain again what is dr."},{"Start":"01:02.185 ","End":"01:04.665","Text":"I\u0027d like to start with C,"},{"Start":"01:04.665 ","End":"01:09.635","Text":"and C is this path along the parabola from minus 2,4 to 1,1."},{"Start":"01:09.635 ","End":"01:10.955","Text":"We don\u0027t need a sketch,"},{"Start":"01:10.955 ","End":"01:13.340","Text":"but it might just help."},{"Start":"01:13.340 ","End":"01:16.625","Text":"Here\u0027s a pair of axis,"},{"Start":"01:16.625 ","End":"01:18.950","Text":"and let\u0027s get 1,1."},{"Start":"01:18.950 ","End":"01:21.200","Text":"Let\u0027s say this is 1 unit,"},{"Start":"01:21.200 ","End":"01:24.340","Text":"1 unit somewhere around here,"},{"Start":"01:24.340 ","End":"01:28.995","Text":"and minus 2,4, let\u0027s say it\u0027s about 2."},{"Start":"01:28.995 ","End":"01:31.755","Text":"It doesn\u0027t really matter, somewhere here."},{"Start":"01:31.755 ","End":"01:35.035","Text":"Certainly, the parabola goes through the origin."},{"Start":"01:35.035 ","End":"01:38.190","Text":"We get something like this."},{"Start":"01:38.190 ","End":"01:41.495","Text":"Over here, we get something like this."},{"Start":"01:41.495 ","End":"01:44.765","Text":"It really doesn\u0027t have to be exact"},{"Start":"01:44.765 ","End":"01:47.285","Text":"but you don\u0027t need a sketch at all."},{"Start":"01:47.285 ","End":"01:55.310","Text":"The point is that the x goes from minus 2 to 1."},{"Start":"01:55.310 ","End":"02:00.240","Text":"The curve, just this part, I\u0027ll highlight it,"},{"Start":"02:00.240 ","End":"02:07.275","Text":"we just want this path and going in this direction."},{"Start":"02:07.275 ","End":"02:09.680","Text":"That\u0027s going to be our curve C,"},{"Start":"02:09.680 ","End":"02:12.575","Text":"just from here to here."},{"Start":"02:12.575 ","End":"02:17.260","Text":"Now, I would like to have C as parametrized curve."},{"Start":"02:17.260 ","End":"02:22.270","Text":"This is easy to do because we have y as a function of x,"},{"Start":"02:22.270 ","End":"02:26.730","Text":"and we have where x goes from and to."},{"Start":"02:26.730 ","End":"02:35.389","Text":"We just let x be t. Here\u0027s some curly braces and then x equals t,"},{"Start":"02:35.389 ","End":"02:40.325","Text":"and then y being x squared is just t squared."},{"Start":"02:40.325 ","End":"02:43.220","Text":"The range of t, which is x,"},{"Start":"02:43.220 ","End":"02:49.395","Text":"is from minus 2 up to 1."},{"Start":"02:49.395 ","End":"02:51.930","Text":"That\u0027s the parametrized, the"},{"Start":"02:51.930 ","End":"03:00.630","Text":"C. I know that we\u0027ll also need later dx and dy,"},{"Start":"03:00.630 ","End":"03:03.105","Text":"and we always do."},{"Start":"03:03.105 ","End":"03:09.450","Text":"From here, I can get that dx is equal to dt."},{"Start":"03:09.450 ","End":"03:13.680","Text":"From here, I can see that dy j,"},{"Start":"03:13.680 ","End":"03:17.380","Text":"ust the derivative 2t squared, which is 2tdt."},{"Start":"03:18.530 ","End":"03:21.840","Text":"That\u0027s the curve C. Now,"},{"Start":"03:21.840 ","End":"03:25.120","Text":"what\u0027s this r and dr?"},{"Start":"03:25.910 ","End":"03:33.080","Text":"Well, the vector r is the position vector of a point along"},{"Start":"03:33.080 ","End":"03:36.545","Text":"our curve C. It\u0027s"},{"Start":"03:36.545 ","End":"03:45.680","Text":"just x times the vector i plus y times vector j,"},{"Start":"03:45.680 ","End":"03:48.950","Text":"where x and y are the x and y of a point on the curve."},{"Start":"03:48.950 ","End":"03:53.020","Text":"We could also write it in the angular brackets notation."},{"Start":"03:53.020 ","End":"03:55.710","Text":"You could write it x, y,"},{"Start":"03:55.710 ","End":"03:57.600","Text":"but since f is given with the i,"},{"Start":"03:57.600 ","End":"03:59.580","Text":"j, I\u0027m sticking with the i,"},{"Start":"03:59.580 ","End":"04:02.005","Text":"j and I\u0027ll erase this."},{"Start":"04:02.005 ","End":"04:05.370","Text":"In general, I deliberately like to sometimes use the i,"},{"Start":"04:05.370 ","End":"04:08.255","Text":"j notation and sometimes the brackets"},{"Start":"04:08.255 ","End":"04:13.025","Text":"notation for vectors because they\u0027re both used in the literature."},{"Start":"04:13.025 ","End":"04:16.625","Text":"You should know both. That\u0027s that,"},{"Start":"04:16.625 ","End":"04:20.250","Text":"and as I was saying, we also need to know what is dr."},{"Start":"04:20.990 ","End":"04:24.240","Text":"This is also a vector,"},{"Start":"04:24.240 ","End":"04:26.505","Text":"and that\u0027s just dx,"},{"Start":"04:26.505 ","End":"04:30.920","Text":"I\u0027ll write it as dx,dy or dx times vector i,"},{"Start":"04:30.920 ","End":"04:35.260","Text":"plus dy times vector j."},{"Start":"04:35.260 ","End":"04:38.620","Text":"Here, I have a dot-product."},{"Start":"04:38.720 ","End":"04:45.270","Text":"What we have if I do this dot product, the f part,"},{"Start":"04:45.270 ","End":"04:48.310","Text":"I can take from here,"},{"Start":"04:48.890 ","End":"04:55.035","Text":"and the dr, which is this one here,"},{"Start":"04:55.035 ","End":"04:57.855","Text":"I can take from here."},{"Start":"04:57.855 ","End":"05:04.220","Text":"The dot-product is just we take the i components and multiply them,"},{"Start":"05:04.220 ","End":"05:06.935","Text":"the j components and multiply them, and then add."},{"Start":"05:06.935 ","End":"05:08.905","Text":"I\u0027ll write that here."},{"Start":"05:08.905 ","End":"05:14.045","Text":"That W is just the integral along the curve."},{"Start":"05:14.045 ","End":"05:17.690","Text":"That\u0027s the type 2 line integral."},{"Start":"05:18.960 ","End":"05:26.230","Text":"We need x cubed y times dx"},{"Start":"05:26.230 ","End":"05:34.770","Text":"plus then x minus y times dy."},{"Start":"05:34.770 ","End":"05:38.940","Text":"This is a more familiar form of the line integral."},{"Start":"05:38.940 ","End":"05:42.410","Text":"Now, we use the standard methods since C is"},{"Start":"05:42.410 ","End":"05:46.700","Text":"parametrized and we have the complete parametrization,"},{"Start":"05:46.700 ","End":"05:52.430","Text":"with x and y is in the range of t. We convert this to a regular"},{"Start":"05:52.430 ","End":"06:00.569","Text":"integral in terms of t. We say that t goes from minus 2 to 1,"},{"Start":"06:00.680 ","End":"06:04.355","Text":"and then we substitute everything."},{"Start":"06:04.355 ","End":"06:08.480","Text":"The x and the y here are the x and the y along the curve,"},{"Start":"06:08.480 ","End":"06:10.595","Text":"and that\u0027s this here."},{"Start":"06:10.595 ","End":"06:17.690","Text":"So x cubed y is t cubed times t squared."},{"Start":"06:17.690 ","End":"06:24.760","Text":"Then dx, here it is, dt,"},{"Start":"06:24.760 ","End":"06:28.200","Text":"plus x minus y,"},{"Start":"06:28.200 ","End":"06:30.869","Text":"which is t minus t squared,"},{"Start":"06:30.869 ","End":"06:35.620","Text":"and dy is here, 2tdt."},{"Start":"06:36.830 ","End":"06:43.880","Text":"Now, I\u0027ll just arrange this so that I have just everything dt."},{"Start":"06:43.880 ","End":"06:46.685","Text":"Let\u0027s open the bracket and let\u0027s see what we have."},{"Start":"06:46.685 ","End":"06:50.305","Text":"From here, we get t^5."},{"Start":"06:50.305 ","End":"06:56.010","Text":"This times this gives me plus 2t squared,"},{"Start":"06:56.010 ","End":"07:01.665","Text":"and then minus 2t cubed dt,"},{"Start":"07:01.665 ","End":"07:04.740","Text":"and everything is completely routine."},{"Start":"07:04.740 ","End":"07:09.100","Text":"Here, we get t^6 over 6."},{"Start":"07:10.400 ","End":"07:16.950","Text":"Here, I have 2t cubed over 3,"},{"Start":"07:16.950 ","End":"07:19.650","Text":"so it\u0027s 2/3t cubed."},{"Start":"07:19.650 ","End":"07:21.600","Text":"Here, I have minus,"},{"Start":"07:21.600 ","End":"07:24.375","Text":"it\u0027s t^4, and here,"},{"Start":"07:24.375 ","End":"07:26.610","Text":"minus 2 over 4."},{"Start":"07:26.610 ","End":"07:30.530","Text":"So 2/4 is a 1/2, and all this,"},{"Start":"07:30.530 ","End":"07:34.445","Text":"we substitute from minus 2 to 1 this,"},{"Start":"07:34.445 ","End":"07:36.930","Text":"and then this and subtract."},{"Start":"07:38.330 ","End":"07:41.655","Text":"Let\u0027s see, if I plug in 1,"},{"Start":"07:41.655 ","End":"07:48.345","Text":"I get 1/6 plus 2/3 minus 1/2."},{"Start":"07:48.345 ","End":"07:53.590","Text":"If I plug in minus 2, let\u0027s see."},{"Start":"07:53.590 ","End":"07:55.490","Text":"It\u0027s an even power,"},{"Start":"07:55.490 ","End":"07:56.675","Text":"so it\u0027s like 2^6,"},{"Start":"07:56.675 ","End":"08:03.850","Text":"which is 64/6, I\u0027ll write it as 32/3."},{"Start":"08:03.850 ","End":"08:08.285","Text":"Next, it\u0027s going to be negative because it\u0027s an odd power."},{"Start":"08:08.285 ","End":"08:12.500","Text":"We have minus 8 times 2/3."},{"Start":"08:12.500 ","End":"08:21.610","Text":"That minus 8 times 2/3, 16/3."},{"Start":"08:21.610 ","End":"08:24.030","Text":"Then here, the minus 2_4,"},{"Start":"08:24.030 ","End":"08:26.625","Text":"so it\u0027s plus 16,"},{"Start":"08:26.625 ","End":"08:28.455","Text":"but it\u0027s minus a 1/2."},{"Start":"08:28.455 ","End":"08:31.545","Text":"So it\u0027s minus 8."},{"Start":"08:31.545 ","End":"08:38.775","Text":"Let\u0027s see now, 1/6 and 2/3 is 5/6, minus 1/2,"},{"Start":"08:38.775 ","End":"08:45.990","Text":"which is 3/6, so altogether 2/6, this is 1/3."},{"Start":"08:45.990 ","End":"08:48.270","Text":"As for the other one, this minus,"},{"Start":"08:48.270 ","End":"08:55.860","Text":"this is just 16/3."},{"Start":"08:55.860 ","End":"09:04.450","Text":"16/3 is 5 and 1/3, minus 8."},{"Start":"09:04.970 ","End":"09:08.685","Text":"I\u0027ll just write it as 5 and 1/3 minus 8,"},{"Start":"09:08.685 ","End":"09:10.485","Text":"I\u0027ll compute it later."},{"Start":"09:10.485 ","End":"09:19.485","Text":"Let\u0027s see, 1/3 minus 5/3 is just minus 5,"},{"Start":"09:19.485 ","End":"09:22.995","Text":"and then plus 8."},{"Start":"09:22.995 ","End":"09:26.560","Text":"I make the answer as 3."},{"Start":"09:26.560 ","End":"09:29.015","Text":"This is the answer to part a."},{"Start":"09:29.015 ","End":"09:36.485","Text":"This is the work performed by the force on the particle that travels along the curve."},{"Start":"09:36.485 ","End":"09:40.830","Text":"That concludes part a and part b in the next clip."}],"ID":8805},{"Watched":false,"Name":"Exercise 12 Part b","Duration":"2m 14s","ChapterTopicVideoID":8707,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8707.jpeg","UploadDate":"2017-02-13T05:26:29.1630000","DurationForVideoObject":"PT2M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"Now in part b, they\u0027re asking us,"},{"Start":"00:03.585 ","End":"00:10.545","Text":"how would the answer change if the particle moved from this point,"},{"Start":"00:10.545 ","End":"00:16.890","Text":"this is the 1,1 to minus 2,4,"},{"Start":"00:16.890 ","End":"00:20.910","Text":"implying it\u0027s along the same curve."},{"Start":"00:20.910 ","End":"00:25.050","Text":"In fact, if we go in the opposite direction,"},{"Start":"00:25.050 ","End":"00:27.210","Text":"there is a name for this curve."},{"Start":"00:27.210 ","End":"00:30.600","Text":"We call this curve minus c,"},{"Start":"00:30.600 ","End":"00:34.480","Text":"same thing but to the opposite direction."},{"Start":"00:35.780 ","End":"00:40.395","Text":"In fact, there\u0027s a theorem that relates to that."},{"Start":"00:40.395 ","End":"00:46.010","Text":"That in general, if we take the integral along"},{"Start":"00:46.010 ","End":"00:54.200","Text":"the opposite curve of some line integral of type 2 F.dr in this case,"},{"Start":"00:54.200 ","End":"01:02.380","Text":"that this is just equal to minus the regular direction F.dr."},{"Start":"01:05.050 ","End":"01:10.610","Text":"Another way of expressing this is that suppose we give them names."},{"Start":"01:10.610 ","End":"01:15.140","Text":"Maybe this is the point M and this is the point N,"},{"Start":"01:15.140 ","End":"01:17.374","Text":"just pick a pair of letters."},{"Start":"01:17.374 ","End":"01:27.125","Text":"As you could say that the integral from M to N, I won\u0027t repeat it, of the same thing"},{"Start":"01:27.125 ","End":"01:32.700","Text":"is equal to minus the"},{"Start":"01:32.700 ","End":"01:40.665","Text":"integral from N to M. Or perhaps I meant backwards."},{"Start":"01:40.665 ","End":"01:46.055","Text":"The 1 I want in terms of the 1 I have, but anyway, in both cases,"},{"Start":"01:46.055 ","End":"01:51.950","Text":"we have the extra minus here if we reverse the curve or reverse the start and end points."},{"Start":"01:51.950 ","End":"01:55.940","Text":"In our case, the answer would just make it a minus."},{"Start":"01:55.940 ","End":"02:01.580","Text":"In part b, the answer would just become then minus 3."},{"Start":"02:01.580 ","End":"02:02.885","Text":"Let\u0027s do it with 3."},{"Start":"02:02.885 ","End":"02:06.320","Text":"This is for part a in the regular direction and this is"},{"Start":"02:06.320 ","End":"02:10.955","Text":"the answer for part b. I guess I\u0027ll highlight this 1 also."},{"Start":"02:10.955 ","End":"02:13.600","Text":"That concludes part b."}],"ID":8806},{"Watched":false,"Name":"Exercise 13","Duration":"6m 35s","ChapterTopicVideoID":8708,"CourseChapterTopicPlaylistID":57502,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8708.jpeg","UploadDate":"2017-02-13T05:28:03.0430000","DurationForVideoObject":"PT6M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"Here we have a question from physics involving the work done by"},{"Start":"00:03.660 ","End":"00:07.650","Text":"a force field on a particle which moves along a path."},{"Start":"00:07.650 ","End":"00:09.930","Text":"We\u0027ve had 1 of these before,"},{"Start":"00:09.930 ","End":"00:11.460","Text":"but in case you haven\u0027t,"},{"Start":"00:11.460 ","End":"00:14.910","Text":"I brought the formula again."},{"Start":"00:14.910 ","End":"00:19.485","Text":"Essentially, all it says here is that to calculate the work,"},{"Start":"00:19.485 ","End":"00:21.870","Text":"you just have to compute the line integral,"},{"Start":"00:21.870 ","End":"00:28.140","Text":"the type 2 line integral of F dot product with dr."},{"Start":"00:28.140 ","End":"00:35.665","Text":"In general, we use the vector r to be an abbreviation for"},{"Start":"00:35.665 ","End":"00:45.065","Text":"x in the i direction plus y in the j direction,"},{"Start":"00:45.065 ","End":"00:51.485","Text":"plus z in the k direction."},{"Start":"00:51.485 ","End":"00:54.320","Text":"If you\u0027re using brackets notation,"},{"Start":"00:54.320 ","End":"00:59.240","Text":"you would just write x, y, z,"},{"Start":"00:59.240 ","End":"01:05.165","Text":"and in general also as r could be a function of t,"},{"Start":"01:05.165 ","End":"01:10.114","Text":"which case we\u0027d have r of t is x of t, y of t,"},{"Start":"01:10.114 ","End":"01:13.620","Text":"z of t, and I\u0027m going to use the i,"},{"Start":"01:13.620 ","End":"01:15.810","Text":"j, k so I\u0027ll erase this,"},{"Start":"01:15.810 ","End":"01:20.195","Text":"and I can rewrite this now in our case because we have F,"},{"Start":"01:20.195 ","End":"01:24.200","Text":"which is given as this."},{"Start":"01:25.280 ","End":"01:30.374","Text":"I didn\u0027t say what is dr. Well, in general,"},{"Start":"01:30.374 ","End":"01:34.890","Text":"dr is just dx in"},{"Start":"01:34.890 ","End":"01:45.795","Text":"the i direction plus dyj plus dzk,"},{"Start":"01:45.795 ","End":"01:51.260","Text":"and so now I do have this which I can replace by this."},{"Start":"01:51.260 ","End":"01:55.190","Text":"Now I need the dot product of this with this."},{"Start":"01:55.190 ","End":"01:56.810","Text":"You know how to do dot-product,"},{"Start":"01:56.810 ","End":"02:00.050","Text":"we just multiply the i components,"},{"Start":"02:00.050 ","End":"02:04.245","Text":"the j components, and the k components and add them."},{"Start":"02:04.245 ","End":"02:10.850","Text":"We would get that w is the integral along C,"},{"Start":"02:10.850 ","End":"02:16.435","Text":"and we still haven\u0027t quite said what is C. We\u0027ll get to that."},{"Start":"02:16.435 ","End":"02:24.090","Text":"Let\u0027s see, we have yzdx,"},{"Start":"02:24.090 ","End":"02:25.650","Text":"for the first component."},{"Start":"02:25.650 ","End":"02:27.000","Text":"For the second component,"},{"Start":"02:27.000 ","End":"02:37.080","Text":"I have xz with dy and xy with dz."},{"Start":"02:37.720 ","End":"02:40.520","Text":"This is a more familiar form."},{"Start":"02:40.520 ","End":"02:46.355","Text":"Now, the curve C or the path is just what\u0027s given by the definition of"},{"Start":"02:46.355 ","End":"02:55.160","Text":"r. We could write it in parametric form that the path C is given by,"},{"Start":"02:55.160 ","End":"02:57.470","Text":"let\u0027s see, x equals,"},{"Start":"02:57.470 ","End":"03:00.200","Text":"y equals, z equals."},{"Start":"03:00.200 ","End":"03:04.249","Text":"X is just t, the first component,"},{"Start":"03:04.249 ","End":"03:09.905","Text":"y is t squared and z is t cubed,"},{"Start":"03:09.905 ","End":"03:12.110","Text":"and we also need to know what the parameter"},{"Start":"03:12.110 ","End":"03:14.765","Text":"goes from and to and that\u0027s what\u0027s written here."},{"Start":"03:14.765 ","End":"03:17.810","Text":"0 less than or equal to t,"},{"Start":"03:17.810 ","End":"03:20.430","Text":"less than or equal to 1."},{"Start":"03:20.770 ","End":"03:23.780","Text":"We need to know what dx,"},{"Start":"03:23.780 ","End":"03:26.704","Text":"dy, and dz in our case."},{"Start":"03:26.704 ","End":"03:28.850","Text":"Well, if x is t,"},{"Start":"03:28.850 ","End":"03:31.730","Text":"then dx is just dt."},{"Start":"03:31.730 ","End":"03:38.900","Text":"Here, dy is just the derivative of this dt so 2t, dt,"},{"Start":"03:38.900 ","End":"03:46.405","Text":"and dz is 3t squared dt."},{"Start":"03:46.405 ","End":"03:50.060","Text":"I\u0027ve got everything, the x, y,"},{"Start":"03:50.060 ","End":"03:52.490","Text":"and z here are all the x, y,"},{"Start":"03:52.490 ","End":"03:55.850","Text":"and z along the curve so that\u0027s these."},{"Start":"03:55.850 ","End":"04:01.520","Text":"Just substitute everything and instead of integral along the curve,"},{"Start":"04:01.520 ","End":"04:08.550","Text":"we just get a regular integral for t going from 0 to 1, 0 to 1."},{"Start":"04:08.550 ","End":"04:13.930","Text":"Let\u0027s see, yz is t squared,"},{"Start":"04:13.930 ","End":"04:18.780","Text":"t cubed and dx is dt."},{"Start":"04:18.780 ","End":"04:22.185","Text":"The next bit, x is t,"},{"Start":"04:22.185 ","End":"04:29.820","Text":"z is t cubed and dy is 2tdt."},{"Start":"04:29.820 ","End":"04:37.080","Text":"What\u0027s here? Lastly, xy is t,"},{"Start":"04:37.080 ","End":"04:42.135","Text":"t squared from here and here and the dz from here"},{"Start":"04:42.135 ","End":"04:48.210","Text":"is 3t squared dt."},{"Start":"04:48.210 ","End":"04:50.360","Text":"Just want to simplify this,"},{"Start":"04:50.360 ","End":"04:55.520","Text":"just get it as a single dt so integral from 0 to 1."},{"Start":"04:55.520 ","End":"04:57.455","Text":"Now, what do we have here?"},{"Start":"04:57.455 ","End":"05:01.450","Text":"T squared times t cubed is t^5."},{"Start":"05:01.450 ","End":"05:04.440","Text":"From here we have t,"},{"Start":"05:04.440 ","End":"05:06.180","Text":"t cubed and t,"},{"Start":"05:06.180 ","End":"05:08.445","Text":"give me t^5 with the 2."},{"Start":"05:08.445 ","End":"05:13.410","Text":"That\u0027s 2t^5 and this bit,"},{"Start":"05:13.410 ","End":"05:17.630","Text":"I also get t^5 because it\u0027s 1 plus 2 plus 2."},{"Start":"05:17.630 ","End":"05:24.940","Text":"But there\u0027s a 3 here, 3t^5 dt."},{"Start":"05:24.940 ","End":"05:28.080","Text":"They\u0027re all t^5 so I could just add them up."},{"Start":"05:28.080 ","End":"05:30.885","Text":"1 plus 2 plus 3 is 6."},{"Start":"05:30.885 ","End":"05:37.580","Text":"In fact, I can take the 6 in front of the integral so I get 6 times the integral"},{"Start":"05:37.580 ","End":"05:41.525","Text":"from 0 to 1"},{"Start":"05:41.525 ","End":"05:49.140","Text":"of t^5 dt and I\u0027ve changed my mind,"},{"Start":"05:49.140 ","End":"05:52.905","Text":"I\u0027m going to put the 6 back in here, you\u0027ll see why."},{"Start":"05:52.905 ","End":"05:55.290","Text":"The reason I put the 6 back here,"},{"Start":"05:55.290 ","End":"05:57.775","Text":"is I realized that 6t^5,"},{"Start":"05:57.775 ","End":"06:02.675","Text":"the integral of that is exactly t^6"},{"Start":"06:02.675 ","End":"06:09.950","Text":"because you raise the power by 1 and divide or if you just look at it the other way,"},{"Start":"06:09.950 ","End":"06:13.685","Text":"if I differentiate t^6 I get 6t^5."},{"Start":"06:13.685 ","End":"06:18.740","Text":"This is t^6, taken from 0 to 1."},{"Start":"06:18.740 ","End":"06:21.770","Text":"When I put in 0, I get nothing."},{"Start":"06:21.770 ","End":"06:27.630","Text":"When I put in 1, I just get 1 so the answer is nice and simple."},{"Start":"06:27.630 ","End":"06:35.670","Text":"It\u0027s just 1 unit of work which I shall highlight and that\u0027s all there is to it."}],"ID":8807}],"Thumbnail":null,"ID":57502}]

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