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[{"Name":"Green\u0027s Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Green\u0027s Theorem","Duration":"9m 6s","ChapterTopicVideoID":8641,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8641.jpeg","UploadDate":"2020-02-26T12:27:13.5670000","DurationForVideoObject":"PT9M6S","Description":null,"MetaTitle":"Green\u0027s Theorem: Video + Workbook | Proprep","MetaDescription":"Green`s Theorem - Green\u0027s Theorem. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/green%60s-theorem/green%27s-theorem/vid8767","VideoComments":[],"Subtitles":[{"Start":"00:00.380 ","End":"00:02.655","Text":"We have a new topic now."},{"Start":"00:02.655 ","End":"00:05.460","Text":"We\u0027re still under line integrals,"},{"Start":"00:05.460 ","End":"00:08.925","Text":"but this is something called Green\u0027s theorem."},{"Start":"00:08.925 ","End":"00:11.100","Text":"Before I get into Green\u0027s theorem,"},{"Start":"00:11.100 ","End":"00:13.095","Text":"I\u0027ll just say something about Green,"},{"Start":"00:13.095 ","End":"00:16.395","Text":"for those who like historical background."},{"Start":"00:16.395 ","End":"00:20.700","Text":"He was a British mathematician"},{"Start":"00:20.700 ","End":"00:22.830","Text":"from Nottinghamshire in England,"},{"Start":"00:22.830 ","End":"00:27.850","Text":"and 1793-1841, died young."},{"Start":"00:27.850 ","End":"00:30.170","Text":"Couldn\u0027t find a photo of him,"},{"Start":"00:30.170 ","End":"00:32.704","Text":"but there\u0027s a mill,"},{"Start":"00:32.704 ","End":"00:36.555","Text":"that is in his name,"},{"Start":"00:36.555 ","End":"00:39.860","Text":"it was his father\u0027s, and now a science center,"},{"Start":"00:39.860 ","End":"00:42.170","Text":"and there\u0027s a picture of his grave."},{"Start":"00:42.170 ","End":"00:44.345","Text":"That\u0027s all for the history part."},{"Start":"00:44.345 ","End":"00:47.480","Text":"Now, before we get to the theorem,"},{"Start":"00:47.480 ","End":"00:50.975","Text":"I want to remind you of some terms we used."},{"Start":"00:50.975 ","End":"00:54.540","Text":"First of all, curves."},{"Start":"00:56.600 ","End":"00:58.770","Text":"When I say curves,"},{"Start":"00:58.770 ","End":"01:06.165","Text":"I usually mean a path which is parametrized."},{"Start":"01:06.165 ","End":"01:13.305","Text":"The terms that I want you to recall are the term smooth,"},{"Start":"01:13.305 ","End":"01:20.410","Text":"the term closed, and a new 1,"},{"Start":"01:20.890 ","End":"01:26.520","Text":"positively oriented, I\u0027ll explain in a moment."},{"Start":"01:27.050 ","End":"01:32.220","Text":"Smooth, we also talked about piecewise."},{"Start":"01:32.220 ","End":"01:34.700","Text":"Actually, we\u0027re going to talk about curves"},{"Start":"01:34.700 ","End":"01:37.325","Text":"which are piecewise,"},{"Start":"01:37.325 ","End":"01:41.760","Text":"smooth, closed, and positively oriented,"},{"Start":"01:41.760 ","End":"01:44.209","Text":"and I\u0027ll just draw a little sketch."},{"Start":"01:44.209 ","End":"01:48.545","Text":"But I forgot to say that we\u0027re working now in 2D,"},{"Start":"01:48.545 ","End":"01:50.360","Text":"and all this relates to 2D,"},{"Start":"01:50.360 ","End":"01:52.674","Text":"so it\u0027s a curve in the plane."},{"Start":"01:52.674 ","End":"01:56.390","Text":"Smooth, remember means that"},{"Start":"01:56.390 ","End":"02:01.765","Text":"the derivative is not 0, of the path."},{"Start":"02:01.765 ","End":"02:06.570","Text":"Piecewise smooth means that it may be smooth in pieces,"},{"Start":"02:06.570 ","End":"02:08.775","Text":"we have a 1 piece from here to here,"},{"Start":"02:08.775 ","End":"02:11.245","Text":"maybe this is the start point."},{"Start":"02:11.245 ","End":"02:14.000","Text":"It\u0027s a path, so it has a direction and it might be"},{"Start":"02:14.000 ","End":"02:17.130","Text":"a straight line piece here."},{"Start":"02:17.130 ","End":"02:22.420","Text":"Then it may go something like this and ends up here."},{"Start":"02:22.420 ","End":"02:25.589","Text":"The start and the end point are the same,"},{"Start":"02:25.589 ","End":"02:27.855","Text":"that makes it a closed curve."},{"Start":"02:27.855 ","End":"02:32.940","Text":"Positively oriented, means it goes around counterclockwise,"},{"Start":"02:32.940 ","End":"02:34.565","Text":"I\u0027ll just write CCW."},{"Start":"02:34.565 ","End":"02:38.150","Text":"In mathematics, this is the positive direction,"},{"Start":"02:38.150 ","End":"02:39.800","Text":"the counterclockwise."},{"Start":"02:39.800 ","End":"02:45.360","Text":"This will be the start and the end, and in 2D."},{"Start":"02:48.980 ","End":"02:51.705","Text":"It has to be simple,"},{"Start":"02:51.705 ","End":"02:53.055","Text":"not to cross itself,"},{"Start":"02:53.055 ","End":"02:56.115","Text":"and now I want a curve with all these 4 properties."},{"Start":"02:56.115 ","End":"03:00.130","Text":"Such a curve always has inside it a region,"},{"Start":"03:00.130 ","End":"03:03.889","Text":"and I\u0027ll call the region D. This is the curve C,"},{"Start":"03:03.889 ","End":"03:06.760","Text":"which goes around and D is the inside."},{"Start":"03:06.760 ","End":"03:11.205","Text":"If we have a curve that is all these 4 things,"},{"Start":"03:11.205 ","End":"03:12.680","Text":"it\u0027s a closed curve,"},{"Start":"03:12.680 ","End":"03:15.725","Text":"means it closes on itself, smooth, in piecewise,"},{"Start":"03:15.725 ","End":"03:20.140","Text":"may be in 3 pieces or however many, counterclockwise,"},{"Start":"03:20.140 ","End":"03:22.710","Text":"and doesn\u0027t cross itself,"},{"Start":"03:22.710 ","End":"03:26.930","Text":"that\u0027s part of the ingredients for Green\u0027s theorem,"},{"Start":"03:26.930 ","End":"03:28.750","Text":"but we\u0027re not done yet."},{"Start":"03:28.750 ","End":"03:30.510","Text":"I\u0027ll just summarize that."},{"Start":"03:30.510 ","End":"03:32.100","Text":"What we want is C,"},{"Start":"03:32.100 ","End":"03:34.260","Text":"is such a curve."},{"Start":"03:34.260 ","End":"03:35.460","Text":"I\u0027ll say such a curve,"},{"Start":"03:35.460 ","End":"03:40.800","Text":"I mean that it has all the above properties,"},{"Start":"03:40.800 ","End":"03:44.070","Text":"so that\u0027s ingredient number 1 to Green\u0027s theorem."},{"Start":"03:44.070 ","End":"03:46.615","Text":"Ingredient number 2 is we want D,"},{"Start":"03:46.615 ","End":"03:52.440","Text":"is the region or domain enclosed by the curve."},{"Start":"03:52.580 ","End":"03:56.460","Text":"The other ingredients is 2 functions,"},{"Start":"03:56.460 ","End":"04:02.190","Text":"P of x, y and Q of x, y,"},{"Start":"04:02.190 ","End":"04:05.450","Text":"but they\u0027re not just any functions of 2 variables,"},{"Start":"04:05.450 ","End":"04:07.950","Text":"well that\u0027s defined on D,"},{"Start":"04:08.620 ","End":"04:12.010","Text":"but there\u0027s an extra condition,"},{"Start":"04:12.010 ","End":"04:14.360","Text":"P and Q have to have continuous"},{"Start":"04:14.360 ","End":"04:16.565","Text":"first-order partial derivatives,"},{"Start":"04:16.565 ","End":"04:18.200","Text":"meaning dP by dx,"},{"Start":"04:18.200 ","End":"04:21.780","Text":"dP by dy, Q with respect to x,"},{"Start":"04:21.780 ","End":"04:26.510","Text":"Q with respect to y, all the 4 first-order partial derivatives are"},{"Start":"04:26.510 ","End":"04:31.520","Text":"continuous still on D. Now that we have all the ingredients,"},{"Start":"04:31.520 ","End":"04:36.620","Text":"let me say what Green\u0027s theorem says, and here it is."},{"Start":"04:36.620 ","End":"04:37.790","Text":"This is Green\u0027s theorem,"},{"Start":"04:37.790 ","End":"04:43.100","Text":"it says that the line integral over the curve C, well,"},{"Start":"04:43.100 ","End":"04:44.420","Text":"since it\u0027s a closed curve,"},{"Start":"04:44.420 ","End":"04:48.620","Text":"we should put a little circle around it."},{"Start":"04:48.620 ","End":"04:51.800","Text":"Sometimes, the circle even has a tiny arrow"},{"Start":"04:51.800 ","End":"04:55.790","Text":"on it to show that it\u0027s counterclockwise."},{"Start":"04:55.790 ","End":"04:59.270","Text":"Anyway, the line integral over this closed curve"},{"Start":"04:59.270 ","End":"05:01.605","Text":"is equal to the double integral,"},{"Start":"05:01.605 ","End":"05:06.260","Text":"and perhaps it\u0027s time to review your double integrals over"},{"Start":"05:06.260 ","End":"05:11.960","Text":"this domain or region D of the following expression."},{"Start":"05:11.960 ","End":"05:14.960","Text":"The partial derivative of Q with respect to x"},{"Start":"05:14.960 ","End":"05:19.820","Text":"minus the partial derivative of P with respect to y."},{"Start":"05:19.820 ","End":"05:23.385","Text":"This is dA, the element of area, although in practice,"},{"Start":"05:23.385 ","End":"05:29.724","Text":"dA will turn out to be either dx dy or dy dx,"},{"Start":"05:29.724 ","End":"05:32.240","Text":"depending on which we\u0027re integrating first"},{"Start":"05:32.240 ","End":"05:34.670","Text":"with respect to x or with respect to y."},{"Start":"05:34.670 ","End":"05:38.020","Text":"Anyway, this is the theorem,"},{"Start":"05:38.020 ","End":"05:41.270","Text":"and it relates a line integral on the boundary of"},{"Start":"05:41.270 ","End":"05:43.550","Text":"a region to the double integral"},{"Start":"05:43.550 ","End":"05:46.320","Text":"of something inside the region."},{"Start":"05:46.810 ","End":"05:51.940","Text":"Let me now do an example."},{"Start":"05:52.130 ","End":"05:55.785","Text":"Just clear some space here."},{"Start":"05:55.785 ","End":"06:01.715","Text":"As an example, let\u0027s take the following line integral."},{"Start":"06:01.715 ","End":"06:08.215","Text":"The integral of y squared dx"},{"Start":"06:08.215 ","End":"06:17.650","Text":"plus 3xy dy over the curve C, the closed curve."},{"Start":"06:18.650 ","End":"06:23.430","Text":"The curve C, as in the illustration,"},{"Start":"06:23.430 ","End":"06:25.830","Text":"I better put some axis in."},{"Start":"06:25.830 ","End":"06:28.710","Text":"This is the y-axis,"},{"Start":"06:28.710 ","End":"06:30.420","Text":"this is the x-axis,"},{"Start":"06:30.420 ","End":"06:32.265","Text":"and this is half the unit circle."},{"Start":"06:32.265 ","End":"06:34.410","Text":"This is 1, this is minus 1,"},{"Start":"06:34.410 ","End":"06:40.635","Text":"this is 1, and certainly this is a piecewise."},{"Start":"06:40.635 ","End":"06:45.940","Text":"Let\u0027s go take a look again at the things the curve has to be."},{"Start":"06:45.940 ","End":"06:48.640","Text":"It has to be smooth piecewise."},{"Start":"06:48.640 ","End":"06:51.580","Text":"Well, this upper half a circle, certainly smooth,"},{"Start":"06:51.580 ","End":"06:53.530","Text":"and the straight line is smooth."},{"Start":"06:53.530 ","End":"06:55.450","Text":"It\u0027s closed."},{"Start":"06:55.450 ","End":"07:01.260","Text":"We don\u0027t even need to know the start and end point."},{"Start":"07:01.260 ","End":"07:03.890","Text":"It\u0027s going around counterclockwise,"},{"Start":"07:03.890 ","End":"07:05.960","Text":"positively oriented, and it\u0027s simple,"},{"Start":"07:05.960 ","End":"07:08.495","Text":"doesn\u0027t cross itself, so that\u0027s fine."},{"Start":"07:08.495 ","End":"07:11.480","Text":"We also have the conditions on the"},{"Start":"07:11.480 ","End":"07:14.180","Text":"partial derivatives because these are polynomials,"},{"Start":"07:14.180 ","End":"07:15.350","Text":"and they\u0027re going to have continuous"},{"Start":"07:15.350 ","End":"07:18.175","Text":"partial derivatives of any order."},{"Start":"07:18.175 ","End":"07:22.550","Text":"Now that we\u0027ve satisfied the conditions of Green\u0027s theorem,"},{"Start":"07:22.550 ","End":"07:29.600","Text":"what we can do is evaluate it using Green\u0027s theorem."},{"Start":"07:29.600 ","End":"07:31.820","Text":"Turns out it\u0027s not so easy to do directly."},{"Start":"07:31.820 ","End":"07:34.595","Text":"We don\u0027t need Green\u0027s theorem to evaluate this,"},{"Start":"07:34.595 ","End":"07:37.100","Text":"doing it mostly for practice,"},{"Start":"07:37.100 ","End":"07:39.530","Text":"but it actually turns out to be easier to"},{"Start":"07:39.530 ","End":"07:43.020","Text":"compute the double integral than the line integral."},{"Start":"07:45.760 ","End":"07:48.984","Text":"I\u0027ll call this thing P,"},{"Start":"07:48.984 ","End":"07:54.450","Text":"and this bit here will be Q of x and y,"},{"Start":"07:54.450 ","End":"07:57.860","Text":"and I need the parametrization of the curve."},{"Start":"07:57.860 ","End":"07:59.645","Text":"Well, I\u0027ll do it in 2 bits."},{"Start":"07:59.645 ","End":"08:02.810","Text":"The semicircle bit will be,"},{"Start":"08:02.810 ","End":"08:07.340","Text":"if I use the parametric rather than"},{"Start":"08:07.340 ","End":"08:09.380","Text":"the vector form, almost the same."},{"Start":"08:09.380 ","End":"08:16.720","Text":"The straight line can just be where y is 0,"},{"Start":"08:16.720 ","End":"08:22.655","Text":"and x will just be a parameter t from minus 1 to 1,"},{"Start":"08:22.655 ","End":"08:28.590","Text":"so x equals t, and minus 1 less than or equal t,"},{"Start":"08:28.590 ","End":"08:30.300","Text":"less than or equal to 1."},{"Start":"08:30.300 ","End":"08:32.565","Text":"That\u0027s the line bit."},{"Start":"08:32.565 ","End":"08:37.605","Text":"We\u0027ll call this C_1 and this C_2,"},{"Start":"08:37.605 ","End":"08:40.715","Text":"so this is a parametrization for C_1."},{"Start":"08:40.715 ","End":"08:44.705","Text":"For C_2, what we can do is the half circle,"},{"Start":"08:44.705 ","End":"08:49.540","Text":"the usual parametrization is x equals cosine t,"},{"Start":"08:49.540 ","End":"08:52.845","Text":"y equals sine t,"},{"Start":"08:52.845 ","End":"08:59.680","Text":"and t goes from 0 to Pi."},{"Start":"08:59.680 ","End":"09:02.330","Text":"I think this is good time for a break"},{"Start":"09:02.330 ","End":"09:07.020","Text":"and we\u0027ll continue to solve this afterwards."}],"ID":8767},{"Watched":false,"Name":"Green\u0027s Theorem Cont","Duration":"14m 41s","ChapterTopicVideoID":8642,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"Let\u0027s try it first the more difficult way"},{"Start":"00:04.140 ","End":"00:08.310","Text":"using the line integral as if we didn\u0027t have Green\u0027s theorem."},{"Start":"00:08.310 ","End":"00:11.085","Text":"In that case, what we would do is break the integral up."},{"Start":"00:11.085 ","End":"00:15.990","Text":"The integral of a curve c would be equal"},{"Start":"00:15.990 ","End":"00:20.805","Text":"to the integral of a curve c_1 plus the integral of the curve c_2,"},{"Start":"00:20.805 ","End":"00:27.180","Text":"because c_1 and c_2 together makeup c. Let\u0027s see what we get."},{"Start":"00:27.180 ","End":"00:29.460","Text":"I\u0027ll start with the easier of the 2,"},{"Start":"00:29.460 ","End":"00:31.230","Text":"which will be c_1."},{"Start":"00:31.230 ","End":"00:35.615","Text":"What we get is the integral,"},{"Start":"00:35.615 ","End":"00:38.635","Text":"write it as the integral of a c_1,"},{"Start":"00:38.635 ","End":"00:47.160","Text":"we get the integral and the parameter for c_1 is from minus 1-1."},{"Start":"00:47.330 ","End":"00:52.550","Text":"Here\u0027s where we get lucky because look, y equals 0."},{"Start":"00:52.550 ","End":"00:57.210","Text":"Here we have y, and here we have y."},{"Start":"00:57.370 ","End":"01:01.020","Text":"This whole thing, we don\u0027t even have to"},{"Start":"01:01.100 ","End":"01:06.470","Text":"compute because we can straight away say that this is equal to 0."},{"Start":"01:06.470 ","End":"01:08.960","Text":"Because y is 0, and here we have y,"},{"Start":"01:08.960 ","End":"01:10.655","Text":"and here we have y."},{"Start":"01:10.655 ","End":"01:12.785","Text":"Everything is just 0."},{"Start":"01:12.785 ","End":"01:15.950","Text":"So that just leaves us with c_2."},{"Start":"01:15.950 ","End":"01:20.330","Text":"The answer for c_2 will just be the answer for c as well."},{"Start":"01:20.330 ","End":"01:29.595","Text":"We\u0027re left with computing the integral this time from 0 to Pi of,"},{"Start":"01:29.595 ","End":"01:38.870","Text":"now let\u0027s see, y squared is sine squared t. What is dx?"},{"Start":"01:38.870 ","End":"01:42.200","Text":"Well, I should really write what dx, dy are."},{"Start":"01:42.200 ","End":"01:49.410","Text":"Dx, the derivative of cosine is minus sine t"},{"Start":"01:49.410 ","End":"01:57.490","Text":"with dt and dy is cosine t, also dt."},{"Start":"01:57.620 ","End":"02:03.389","Text":"We get sine squared t from the y squared from the dx,"},{"Start":"02:03.389 ","End":"02:06.030","Text":"we have minus sine t dt,"},{"Start":"02:06.030 ","End":"02:11.945","Text":"we have minus this and then we have sine t. You know what?"},{"Start":"02:11.945 ","End":"02:13.685","Text":"The dt, I won\u0027t write,"},{"Start":"02:13.685 ","End":"02:17.390","Text":"I\u0027ll write it at the end because this whole thing is going to be in"},{"Start":"02:17.390 ","End":"02:21.350","Text":"the brackets and I\u0027ll put the dt at the end."},{"Start":"02:21.350 ","End":"02:23.855","Text":"Well, I just guess how much space I need here."},{"Start":"02:23.855 ","End":"02:26.165","Text":"Plus 3xy."},{"Start":"02:26.165 ","End":"02:34.380","Text":"So 3xy, x is cosine t,"},{"Start":"02:35.170 ","End":"02:40.230","Text":"y is sine t,"},{"Start":"02:41.470 ","End":"02:50.040","Text":"and dy is cosine t. Also,"},{"Start":"02:50.180 ","End":"03:00.385","Text":"I can just put a squared here instead of an extra cosine and then dt."},{"Start":"03:00.385 ","End":"03:05.325","Text":"Fine. I want to simplify this a bit."},{"Start":"03:05.325 ","End":"03:12.600","Text":"What I can do is use the famous formula that sine"},{"Start":"03:12.600 ","End":"03:21.670","Text":"squared Alpha plus cosine squared Alpha a is equal to 1."},{"Start":"03:25.100 ","End":"03:29.360","Text":"Let me first of all take sine t outside the brackets,"},{"Start":"03:29.360 ","End":"03:31.650","Text":"getting ahead of myself."},{"Start":"03:32.210 ","End":"03:36.290","Text":"This is equal to the integral from 0 to Pi."},{"Start":"03:36.290 ","End":"03:40.960","Text":"Now, I\u0027ll take sine t outside the brackets."},{"Start":"03:40.960 ","End":"03:48.840","Text":"What I get is minus sine squared t plus"},{"Start":"03:48.840 ","End":"03:57.185","Text":"3 cosine squared t. At this point I\u0027m going to use this formula."},{"Start":"03:57.185 ","End":"04:03.160","Text":"What I\u0027m going to do is, I\u0027m going to add sine squared and cosine squared and subtract 1."},{"Start":"04:03.160 ","End":"04:05.929","Text":"Actually, what I could do here,"},{"Start":"04:05.929 ","End":"04:14.295","Text":"I could write this as this plus this minus 1 is equal to 0."},{"Start":"04:14.295 ","End":"04:18.850","Text":"If that\u0027s equal to 0, I can add it in here."},{"Start":"04:18.850 ","End":"04:20.410","Text":"Well, let me just copy this bit,"},{"Start":"04:20.410 ","End":"04:21.985","Text":"that\u0027s not going to change,"},{"Start":"04:21.985 ","End":"04:25.390","Text":"sine t. Here\u0027s where I\u0027m doing the trick."},{"Start":"04:25.390 ","End":"04:28.555","Text":"I\u0027m adding sine squared that will be nothing."},{"Start":"04:28.555 ","End":"04:29.920","Text":"I\u0027m adding cosine squared,"},{"Start":"04:29.920 ","End":"04:32.515","Text":"so I\u0027ll get 4 cosine squared."},{"Start":"04:32.515 ","End":"04:35.420","Text":"Then I\u0027ll be subtracting 1dt."},{"Start":"04:37.280 ","End":"04:41.890","Text":"At this point, I would break it up into 2 separate integrals,"},{"Start":"04:41.890 ","End":"04:43.810","Text":"this bit and the minus this bit."},{"Start":"04:43.810 ","End":"04:49.865","Text":"The first bit would be the integral from 0 to Pi this times this."},{"Start":"04:49.865 ","End":"04:51.710","Text":"Let me take the 4 out front."},{"Start":"04:51.710 ","End":"04:59.295","Text":"I\u0027ve got 4 times the integral of cosine squared t,"},{"Start":"04:59.295 ","End":"05:02.985","Text":"sine t. I just changed the order,"},{"Start":"05:02.985 ","End":"05:06.810","Text":"dt, that\u0027s the first bit."},{"Start":"05:06.810 ","End":"05:12.710","Text":"The second bit is just minus the integral from 0 to Pi."},{"Start":"05:12.710 ","End":"05:19.965","Text":"We\u0027re just left with minus sine t, dt."},{"Start":"05:19.965 ","End":"05:21.710","Text":"Neither of these is very difficult."},{"Start":"05:21.710 ","End":"05:25.010","Text":"Actually, it\u0027s turning out easier than I expected for the line integral."},{"Start":"05:25.010 ","End":"05:28.595","Text":"I thought it was going to be a nightmare. It isn\u0027t."},{"Start":"05:28.595 ","End":"05:31.295","Text":"If I look at this,"},{"Start":"05:31.295 ","End":"05:35.825","Text":"I\u0027d like to do it by substitution, but mentally."},{"Start":"05:35.825 ","End":"05:37.770","Text":"I\u0027m thinking that I have something squared,"},{"Start":"05:37.770 ","End":"05:41.905","Text":"it\u0027s cosine squared and I have the derivative of cosine next to it."},{"Start":"05:41.905 ","End":"05:46.550","Text":"Well, not quite because the derivative of cosine is minus sine."},{"Start":"05:46.550 ","End":"05:51.910","Text":"How about I fix it a bit by adding a minus here,"},{"Start":"05:51.910 ","End":"05:55.625","Text":"and then I\u0027ll compensate by putting a minus here also,"},{"Start":"05:55.625 ","End":"05:57.365","Text":"then I\u0027ll be okay."},{"Start":"05:57.365 ","End":"06:03.224","Text":"We get, this equals minus 4,"},{"Start":"06:03.224 ","End":"06:07.765","Text":"and then the integral of cosine squared,"},{"Start":"06:07.765 ","End":"06:11.825","Text":"something squared would be 1/3 of that thing cubed."},{"Start":"06:11.825 ","End":"06:17.195","Text":"So it\u0027s cosine cubed t. Instead of putting over 3 here,"},{"Start":"06:17.195 ","End":"06:20.410","Text":"why don\u0027t I just write the over 3 here,"},{"Start":"06:20.410 ","End":"06:23.375","Text":"and really I have the derivative outside."},{"Start":"06:23.375 ","End":"06:26.795","Text":"This is all this is. Now, if you\u0027re in doubt differentiate this."},{"Start":"06:26.795 ","End":"06:28.850","Text":"You\u0027ll see we get 3 cancels with"},{"Start":"06:28.850 ","End":"06:33.335","Text":"the 3 cosine squared t times the inner derivative is minus sine t,"},{"Start":"06:33.335 ","End":"06:38.260","Text":"which will cover up with this minus."},{"Start":"06:41.750 ","End":"06:49.805","Text":"All this has to be evaluated between 0 and Pi in a moment."},{"Start":"06:49.805 ","End":"06:51.935","Text":"Then for the second bit,"},{"Start":"06:51.935 ","End":"06:54.185","Text":"the integral of sine is minus cosine,"},{"Start":"06:54.185 ","End":"06:57.275","Text":"but the integral of minus sine is cosine."},{"Start":"06:57.275 ","End":"07:03.535","Text":"It\u0027s plus cosine t evaluated from 0 to Pi,"},{"Start":"07:03.535 ","End":"07:06.125","Text":"because the minus sine t gives me the cosine of"},{"Start":"07:06.125 ","End":"07:11.735","Text":"t. Now we just need to do some substitutions."},{"Start":"07:11.735 ","End":"07:16.530","Text":"I\u0027ll remind you that cosine of"},{"Start":"07:16.530 ","End":"07:22.594","Text":"Pi is minus 1 and cosine of 0 is 1."},{"Start":"07:22.594 ","End":"07:25.260","Text":"In both cases, we need the cosine."},{"Start":"07:26.450 ","End":"07:29.770","Text":"What we get is,"},{"Start":"07:29.770 ","End":"07:32.690","Text":"let\u0027s leave the minus 4/3 for the moment."},{"Start":"07:32.690 ","End":"07:38.280","Text":"Cosine cubed t is minus 1 cubed. You know what?"},{"Start":"07:38.880 ","End":"07:41.305","Text":"I can take the minus 4/3 out."},{"Start":"07:41.305 ","End":"07:43.870","Text":"Minus 1 cubed is minus 1."},{"Start":"07:43.870 ","End":"07:47.155","Text":"If I plug in 0, it\u0027s 1 cubed,"},{"Start":"07:47.155 ","End":"07:49.690","Text":"but I\u0027m subtracting 1."},{"Start":"07:49.690 ","End":"07:52.435","Text":"Then the next bit,"},{"Start":"07:52.435 ","End":"07:54.760","Text":"what I get is, if I put in Pi,"},{"Start":"07:54.760 ","End":"07:56.470","Text":"I\u0027ve got minus 1,"},{"Start":"07:56.470 ","End":"07:57.640","Text":"and then take away,"},{"Start":"07:57.640 ","End":"07:59.020","Text":"what happens when I put in 0,"},{"Start":"07:59.020 ","End":"08:01.585","Text":"which is I\u0027m taking away 1."},{"Start":"08:01.585 ","End":"08:06.580","Text":"Now, what this gives me is this is going to"},{"Start":"08:06.580 ","End":"08:15.400","Text":"be 8/3 minus 2 is 2,"},{"Start":"08:15.400 ","End":"08:16.660","Text":"and 2/3 minus 2,"},{"Start":"08:16.660 ","End":"08:20.215","Text":"it just comes out to be 2/3."},{"Start":"08:20.215 ","End":"08:22.780","Text":"Since I don\u0027t have to add the 0,"},{"Start":"08:22.780 ","End":"08:25.730","Text":"this is really the final answer."},{"Start":"08:26.400 ","End":"08:28.855","Text":"Maybe I\u0027ll write it over here."},{"Start":"08:28.855 ","End":"08:30.100","Text":"This is equal to,"},{"Start":"08:30.100 ","End":"08:37.645","Text":"the first integral turned out to be 0."},{"Start":"08:37.645 ","End":"08:41.065","Text":"The second integral turned out to be 2/3."},{"Start":"08:41.065 ","End":"08:45.670","Text":"0 plus 2/3 equals 2/3."},{"Start":"08:45.670 ","End":"08:48.950","Text":"We\u0027ll highlight it."},{"Start":"08:49.800 ","End":"08:52.495","Text":"Wasn\u0027t as bad as I thought."},{"Start":"08:52.495 ","End":"08:56.815","Text":"The line integral method gives us 2/3."},{"Start":"08:56.815 ","End":"09:05.420","Text":"Now let\u0027s do it using the double integral using Green\u0027s theorem."},{"Start":"09:05.640 ","End":"09:09.175","Text":"Let me clear the board a bit."},{"Start":"09:09.175 ","End":"09:14.365","Text":"There we are. Let\u0027s just go back up here."},{"Start":"09:14.365 ","End":"09:16.990","Text":"Just need to see the formula."},{"Start":"09:16.990 ","End":"09:20.995","Text":"This time we\u0027re going to do it not by the line integral,"},{"Start":"09:20.995 ","End":"09:28.360","Text":"but using the double integral over the region D. As usual,"},{"Start":"09:28.360 ","End":"09:31.030","Text":"you have the dilemma of whether you\u0027re going to slice it in"},{"Start":"09:31.030 ","End":"09:34.015","Text":"vertical slices or horizontal slices,"},{"Start":"09:34.015 ","End":"09:39.460","Text":"and it seems to me more natural to take y as a function of x."},{"Start":"09:39.460 ","End":"09:40.930","Text":"We have 2 simple functions,"},{"Start":"09:40.930 ","End":"09:45.760","Text":"the semicircle and a straight line at the x-axis,"},{"Start":"09:45.760 ","End":"09:49.165","Text":"so let\u0027s take vertical slices, so to speak."},{"Start":"09:49.165 ","End":"09:52.030","Text":"We have this area D,"},{"Start":"09:52.030 ","End":"09:55.570","Text":"maybe I will shade it vertically,"},{"Start":"09:55.570 ","End":"09:58.460","Text":"some lines like this."},{"Start":"10:01.260 ","End":"10:09.385","Text":"We\u0027re going here from minus 1-1, and then from here,"},{"Start":"10:09.385 ","End":"10:15.700","Text":"this is the curve y equals 0, that\u0027s the x-axis,"},{"Start":"10:15.700 ","End":"10:26.005","Text":"and this is the curve y equals the square root of 1 minus x squared."},{"Start":"10:26.005 ","End":"10:33.805","Text":"Remember, if the equation of the circle is x squared plus y squared equals 1,"},{"Start":"10:33.805 ","End":"10:37.810","Text":"that we bring x squared to the other side and get 1 minus x squared,"},{"Start":"10:37.810 ","End":"10:42.760","Text":"and we take the positive square root because we\u0027ve got the upper semicircle."},{"Start":"10:42.760 ","End":"10:48.355","Text":"What we get now is that dA becomes now dydx,"},{"Start":"10:48.355 ","End":"10:54.340","Text":"refers to the integral with respect to y from this to this."},{"Start":"10:54.340 ","End":"10:58.150","Text":"Then we\u0027ll take the integral with respect to x from minus 1-1."},{"Start":"10:58.150 ","End":"11:03.820","Text":"I\u0027ll show you. We get the double integral."},{"Start":"11:03.820 ","End":"11:06.040","Text":"What we\u0027re going to take is,"},{"Start":"11:06.040 ","End":"11:13.795","Text":"on the outer integral is from minus 1-1 dx."},{"Start":"11:13.795 ","End":"11:19.705","Text":"The inner integral is from 0 to"},{"Start":"11:19.705 ","End":"11:26.290","Text":"square root of 1 minus x squared dy,"},{"Start":"11:26.290 ","End":"11:32.995","Text":"and all I have to do now is figure out what is dQ by dx minus dP by dy."},{"Start":"11:32.995 ","End":"11:40.639","Text":"Let\u0027s see if we can see what P and Q. I\u0027ll do that at the side."},{"Start":"11:41.700 ","End":"11:48.280","Text":"dQ by dx is, this is Q,"},{"Start":"11:48.280 ","End":"11:52.240","Text":"so with respect to x gives me 3y,"},{"Start":"11:52.240 ","End":"12:02.695","Text":"whereas dP by dy is equal to this thing with respect to y, which is 2y."},{"Start":"12:02.695 ","End":"12:06.880","Text":"What we get here is basically,"},{"Start":"12:06.880 ","End":"12:12.115","Text":"the difference we need is 3y minus 2y."},{"Start":"12:12.115 ","End":"12:13.615","Text":"All we get here,"},{"Start":"12:13.615 ","End":"12:18.260","Text":"and I left a lot of space for it, is y dydx."},{"Start":"12:20.190 ","End":"12:24.400","Text":"Let\u0027s do this integral first, the inner one,"},{"Start":"12:24.400 ","End":"12:30.010","Text":"the integral of y is 1/2 y"},{"Start":"12:30.010 ","End":"12:34.840","Text":"squared and this is taken"},{"Start":"12:34.840 ","End":"12:42.160","Text":"from 0 to square root of 1 minus x squared."},{"Start":"12:42.160 ","End":"12:45.025","Text":"After we\u0027ve gotten the answer to that,"},{"Start":"12:45.025 ","End":"12:47.080","Text":"it will be a function of x,"},{"Start":"12:47.080 ","End":"12:52.990","Text":"then we\u0027ll need to take the integral from minus 1-1 of this thing, dx."},{"Start":"12:52.990 ","End":"12:56.215","Text":"Let\u0027s see what happens with 1/2 y squared."},{"Start":"12:56.215 ","End":"13:00.010","Text":"If I plug in y equals this,"},{"Start":"13:00.010 ","End":"13:02.155","Text":"without the square root,"},{"Start":"13:02.155 ","End":"13:07.630","Text":"I get 1/2 of 1 minus x squared."},{"Start":"13:07.630 ","End":"13:12.265","Text":"I plug in 0 to 1/2 y squared,"},{"Start":"13:12.265 ","End":"13:14.605","Text":"then I just get 0."},{"Start":"13:14.605 ","End":"13:17.080","Text":"I don\u0027t need to say minus 0,"},{"Start":"13:17.080 ","End":"13:18.745","Text":"I\u0027ll just leave it as it is."},{"Start":"13:18.745 ","End":"13:27.970","Text":"Then we have the integral from minus 1-1 of this thing, dx."},{"Start":"13:27.970 ","End":"13:30.490","Text":"Well, this is equal to, well,"},{"Start":"13:30.490 ","End":"13:33.610","Text":"if I expand the brackets, I get 1/2 minus 1/2 x squared."},{"Start":"13:33.610 ","End":"13:37.690","Text":"The integral of 1/2 is just 1/2x,"},{"Start":"13:37.690 ","End":"13:45.760","Text":"and the integral of 1/2 x squared is 1/2 x cubed over 3."},{"Start":"13:45.760 ","End":"13:48.295","Text":"The 3 with the 2 will make 1/6,"},{"Start":"13:48.295 ","End":"13:50.440","Text":"and I can say x cubed,"},{"Start":"13:50.440 ","End":"13:55.220","Text":"and then I can take this from minus 1-1."},{"Start":"13:55.590 ","End":"13:57.895","Text":"What this will give,"},{"Start":"13:57.895 ","End":"14:00.100","Text":"if I put in 1,"},{"Start":"14:00.100 ","End":"14:06.460","Text":"I\u0027ll get, 1/2 minus 1/6 is 1/3."},{"Start":"14:06.460 ","End":"14:09.070","Text":"I put in minus 1,"},{"Start":"14:09.070 ","End":"14:10.090","Text":"I\u0027ll just get the opposite."},{"Start":"14:10.090 ","End":"14:11.365","Text":"It\u0027s an odd function."},{"Start":"14:11.365 ","End":"14:13.255","Text":"We get minus 1/2,"},{"Start":"14:13.255 ","End":"14:14.740","Text":"and then we\u0027ll get plus 1/6."},{"Start":"14:14.740 ","End":"14:19.975","Text":"We\u0027ll just get minus minus 1/3,"},{"Start":"14:19.975 ","End":"14:26.440","Text":"and the answer is 2/3."},{"Start":"14:26.440 ","End":"14:30.955","Text":"2/3 is what we got earlier on. I remember."},{"Start":"14:30.955 ","End":"14:37.315","Text":"We got the same answer with the line integral and with the double integral."},{"Start":"14:37.315 ","End":"14:39.640","Text":"That\u0027s our first example done."},{"Start":"14:39.640 ","End":"14:41.840","Text":"Let\u0027s take a break."}],"ID":8768},{"Watched":false,"Name":"Green\u0027s Theorem and Vector Fields","Duration":"23m 26s","ChapterTopicVideoID":8643,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"Continuing with Green\u0027s theorem,"},{"Start":"00:02.880 ","End":"00:05.940","Text":"I just like to spend the theoretical moment showing"},{"Start":"00:05.940 ","End":"00:09.405","Text":"you how this is related to conservative vector fields."},{"Start":"00:09.405 ","End":"00:13.425","Text":"Suppose I have a vector field F,"},{"Start":"00:13.425 ","End":"00:17.990","Text":"which is equal to components P and Q."},{"Start":"00:17.990 ","End":"00:21.160","Text":"Each of these is a function of x and y."},{"Start":"00:21.200 ","End":"00:24.525","Text":"Suppose I have a path,"},{"Start":"00:24.525 ","End":"00:28.905","Text":"this path C is written in the form of r of t,"},{"Start":"00:28.905 ","End":"00:31.635","Text":"which is x of t,"},{"Start":"00:31.635 ","End":"00:38.445","Text":"y of t, where t goes from sum a to b."},{"Start":"00:38.445 ","End":"00:45.935","Text":"Then what we get is that this line integral."},{"Start":"00:45.935 ","End":"00:47.765","Text":"Well, let me start from the other end."},{"Start":"00:47.765 ","End":"00:53.880","Text":"Let\u0027s see what is the integral of F.dr."},{"Start":"00:56.390 ","End":"01:02.970","Text":"Let\u0027s suppose this is traces of a closed curve C. I want"},{"Start":"01:02.970 ","End":"01:09.750","Text":"to compute F.dr over this curve C,"},{"Start":"01:09.750 ","End":"01:11.840","Text":"let\u0027s assume it\u0027s a counterclockwise,"},{"Start":"01:11.840 ","End":"01:13.845","Text":"doesn\u0027t really matter, actually."},{"Start":"01:13.845 ","End":"01:21.395","Text":"Closed curve. What we would get is that if we did it component-wise,"},{"Start":"01:21.395 ","End":"01:22.820","Text":"we\u0027d get F is P,"},{"Start":"01:22.820 ","End":"01:33.590","Text":"Q and dr is just dx, dy."},{"Start":"01:33.590 ","End":"01:39.465","Text":"F.dr, this integral is just the integral,"},{"Start":"01:39.465 ","End":"01:50.400","Text":"this thing becomes the integral over the curve of this dot with this is Pdx plus Qdy."},{"Start":"01:51.250 ","End":"01:54.740","Text":"Now I said something about conservative."},{"Start":"01:54.740 ","End":"01:58.715","Text":"If the vector field is conservative."},{"Start":"01:58.715 ","End":"02:07.340","Text":"Remember that conservative was characterized by the following formula."},{"Start":"02:07.340 ","End":"02:14.405","Text":"I think we wrote it that P_y equals Q_x."},{"Start":"02:14.405 ","End":"02:19.625","Text":"But if we write it in this notation,"},{"Start":"02:19.625 ","End":"02:29.150","Text":"it\u0027s dP by dy equals dQ by dx."},{"Start":"02:29.150 ","End":"02:31.115","Text":"Possibly I wrote it the other way around."},{"Start":"02:31.115 ","End":"02:34.074","Text":"Either way, if these 2 are equal,"},{"Start":"02:34.074 ","End":"02:35.380","Text":"dP, dy and dQ,"},{"Start":"02:35.380 ","End":"02:37.724","Text":"dx, this thing is 0."},{"Start":"02:37.724 ","End":"02:40.790","Text":"If we are in a conservative,"},{"Start":"02:40.790 ","End":"02:44.950","Text":"this is equal to 0."},{"Start":"02:44.950 ","End":"02:48.920","Text":"Again I\u0027m emphasizing if we\u0027re conservative."},{"Start":"02:48.920 ","End":"02:55.460","Text":"Actually what we\u0027ve shown here is that we\u0027ve just proven with Green\u0027s theorem,"},{"Start":"02:55.460 ","End":"03:04.160","Text":"the theorem that we said that the vector field is conservative, it\u0027s line independent."},{"Start":"03:04.160 ","End":"03:09.200","Text":"But we proved in particular that the integral over a closed curve is 0."},{"Start":"03:09.200 ","End":"03:13.460","Text":"Of course, it doesn\u0027t matter if it\u0027s clockwise or counterclockwise,"},{"Start":"03:13.460 ","End":"03:20.030","Text":"because if our curve happen to be in the negative direction, which is clockwise,"},{"Start":"03:20.030 ","End":"03:24.485","Text":"we would just take minus the curve and our answer would be minus 0,"},{"Start":"03:24.485 ","End":"03:27.185","Text":"but minus 0 and 0 are the same."},{"Start":"03:27.185 ","End":"03:29.540","Text":"It works actually over closed curves,"},{"Start":"03:29.540 ","End":"03:31.670","Text":"whichever orientation they have."},{"Start":"03:31.670 ","End":"03:34.550","Text":"That was theoretical remark."},{"Start":"03:34.550 ","End":"03:36.945","Text":"Green\u0027s theorem shows that"},{"Start":"03:36.945 ","End":"03:42.905","Text":"conservative fields have a line integral of 0 over a closed curve."},{"Start":"03:42.905 ","End":"03:47.930","Text":"Now let\u0027s go to an example. Let\u0027s continue."},{"Start":"03:47.930 ","End":"03:53.520","Text":"We\u0027ll take examples of domains with holes in them."},{"Start":"03:53.620 ","End":"03:59.420","Text":"But before that, I want to just review the concept of positively oriented."},{"Start":"03:59.420 ","End":"04:07.070","Text":"Let me have a domain like this with a simple closed curve around it."},{"Start":"04:07.070 ","End":"04:11.360","Text":"Then we say, positively oriented means counterclockwise."},{"Start":"04:11.360 ","End":"04:13.940","Text":"But the counterclockwise is not a great definition,"},{"Start":"04:13.940 ","End":"04:17.000","Text":"as you\u0027ll see later when we have domains with holes in them,"},{"Start":"04:17.000 ","End":"04:21.590","Text":"the better way to say it is that when I travel along the curve,"},{"Start":"04:21.590 ","End":"04:23.885","Text":"the domain is on the left."},{"Start":"04:23.885 ","End":"04:29.360","Text":"I\u0027ll just write that alternative instead of counterclockwise domain on"},{"Start":"04:29.360 ","End":"04:36.120","Text":"left as we travel the curve with increasing parameter."},{"Start":"04:36.640 ","End":"04:39.185","Text":"Now let\u0027s take an example."},{"Start":"04:39.185 ","End":"04:43.080","Text":"Let me clear what we don\u0027t need."},{"Start":"04:43.340 ","End":"04:46.655","Text":"I think you\u0027ve had enough of this history."},{"Start":"04:46.655 ","End":"04:53.389","Text":"That\u0027s better. Now tell you where I\u0027m heading next."},{"Start":"04:53.389 ","End":"04:58.100","Text":"So far we\u0027ve discussed a very simple situation where we have"},{"Start":"04:58.100 ","End":"05:04.505","Text":"some domain whose boundary is a simple curve,"},{"Start":"05:04.505 ","End":"05:07.535","Text":"it\u0027s all the other things as well, smooth,"},{"Start":"05:07.535 ","End":"05:13.445","Text":"and piecewise, at least in closed and positively oriented."},{"Start":"05:13.445 ","End":"05:18.410","Text":"But the restriction that it should be simple is too restrictive."},{"Start":"05:18.410 ","End":"05:20.675","Text":"For example, if we had a hole in it,"},{"Start":"05:20.675 ","End":"05:25.799","Text":"we couldn\u0027t bound it by a simple curve,"},{"Start":"05:25.799 ","End":"05:29.175","Text":"and many regions do have holes in them."},{"Start":"05:29.175 ","End":"05:31.550","Text":"We want to work towards that."},{"Start":"05:31.550 ","End":"05:34.265","Text":"But before I jump into that,"},{"Start":"05:34.265 ","End":"05:39.020","Text":"I need just 1 tool and that will be of help and that is to"},{"Start":"05:39.020 ","End":"05:45.870","Text":"break a region up into 2 or more sub-regions, let\u0027s say 2."},{"Start":"05:46.810 ","End":"05:49.950","Text":"I\u0027ll bring in a picture."},{"Start":"05:50.810 ","End":"05:58.295","Text":"Here we have an example of a domain or region that\u0027s split up into 2 bits."},{"Start":"05:58.295 ","End":"06:08.730","Text":"Suppose that the whole interior is D. This is broken up into D_1 and D_2."},{"Start":"06:08.730 ","End":"06:12.770","Text":"I\u0027ll use the mathematical notation of union."},{"Start":"06:12.770 ","End":"06:15.145","Text":"You must have seen this before."},{"Start":"06:15.145 ","End":"06:20.590","Text":"If not, it\u0027s a plus or it\u0027s combined with."},{"Start":"06:21.740 ","End":"06:26.000","Text":"Likewise, the curve C,"},{"Start":"06:26.000 ","End":"06:29.730","Text":"I\u0027ll call the whole curve C. It is labeled here."},{"Start":"06:31.610 ","End":"06:42.675","Text":"But our curve C, which is the counterclockwise whole thing will be C_1 union C_2."},{"Start":"06:42.675 ","End":"06:44.370","Text":"I\u0027ll just change this color."},{"Start":"06:44.370 ","End":"06:47.420","Text":"I use the term counterclockwise,"},{"Start":"06:47.420 ","End":"06:49.309","Text":"which I should stop using."},{"Start":"06:49.309 ","End":"06:53.900","Text":"I should start saying positively oriented and in fact,"},{"Start":"06:53.900 ","End":"07:00.800","Text":"more accurate is to not define it by clockwise or counterclockwise,"},{"Start":"07:00.800 ","End":"07:02.810","Text":"but that the domain is on the left."},{"Start":"07:02.810 ","End":"07:10.040","Text":"We\u0027ll soon see this very clearly when we start talking about regions with holes,"},{"Start":"07:10.040 ","End":"07:15.900","Text":"I\u0027m just introducing this tool which is going to be helpful for regions with holes."},{"Start":"07:16.390 ","End":"07:21.130","Text":"Now it makes sense that I could take"},{"Start":"07:21.130 ","End":"07:30.120","Text":"this Green\u0027s theorem separately on D_1 and D_2 and do an addition."},{"Start":"07:30.120 ","End":"07:32.490","Text":"At the moment, it\u0027s just a tool,"},{"Start":"07:32.490 ","End":"07:36.400","Text":"I\u0027m not exactly going to prove anything profound,"},{"Start":"07:36.400 ","End":"07:42.830","Text":"but I\u0027m just showing you this concept of splitting up a region into 2 or could be more."},{"Start":"07:42.830 ","End":"07:46.520","Text":"What I would say is that the line"},{"Start":"07:46.520 ","End":"07:53.435","Text":"integral and the double integral can both be split up and I\u0027ll show you what I mean."},{"Start":"07:53.435 ","End":"08:01.280","Text":"I could say that the double integral over D of this expression,"},{"Start":"08:01.280 ","End":"08:02.570","Text":"and I\u0027m not going to copy it at each time."},{"Start":"08:02.570 ","End":"08:04.070","Text":"It\u0027s always going to be the same expression."},{"Start":"08:04.070 ","End":"08:06.095","Text":"I\u0027ll just write dot, dot, dot,"},{"Start":"08:06.095 ","End":"08:10.220","Text":"will equal the double integral over"},{"Start":"08:10.220 ","End":"08:15.140","Text":"D1 plus the double"},{"Start":"08:15.140 ","End":"08:21.520","Text":"integral over D2 of the same expression."},{"Start":"08:23.120 ","End":"08:29.390","Text":"If I want to use Green\u0027s theorem separately here and here,"},{"Start":"08:29.390 ","End":"08:34.020","Text":"I have to find a curve that goes around D_1."},{"Start":"08:37.210 ","End":"08:41.960","Text":"This part, by the Green\u0027s theorem,"},{"Start":"08:41.960 ","End":"08:49.250","Text":"would be equal to the line integral over,"},{"Start":"08:49.250 ","End":"08:58.680","Text":"we can just call it C_1 union with C_3 of dot,"},{"Start":"08:58.680 ","End":"08:59.940","Text":"dot, dot, in this case,"},{"Start":"08:59.940 ","End":"09:02.759","Text":"it means this expression here."},{"Start":"09:02.759 ","End":"09:09.315","Text":"Then plus the line integral over,"},{"Start":"09:09.315 ","End":"09:11.340","Text":"I have a name for this one,"},{"Start":"09:11.340 ","End":"09:13.034","Text":"but made up of 2 pieces."},{"Start":"09:13.034 ","End":"09:17.200","Text":"It\u0027s C_2, and then here,"},{"Start":"09:17.200 ","End":"09:22.400","Text":"I don\u0027t need a new letter for this because this is the opposite direction of C_3,"},{"Start":"09:22.400 ","End":"09:25.350","Text":"so it\u0027s minus C_3."},{"Start":"09:25.470 ","End":"09:32.440","Text":"Here I write C_2 union with minus C_3,"},{"Start":"09:32.440 ","End":"09:36.860","Text":"blah, blah, blah, which is this line integral."},{"Start":"09:36.860 ","End":"09:43.040","Text":"Now, remember with the line integral just like with the double integral,"},{"Start":"09:43.040 ","End":"09:44.750","Text":"when you break it up into 2 pieces,"},{"Start":"09:44.750 ","End":"09:46.040","Text":"it\u0027s additive. You get a plus."},{"Start":"09:46.040 ","End":"09:48.470","Text":"When I break the curve up into 2 pieces,"},{"Start":"09:48.470 ","End":"09:52.760","Text":"the integral just breaks up into separate integrals."},{"Start":"09:52.760 ","End":"09:55.550","Text":"The first one is the integral,"},{"Start":"09:55.550 ","End":"09:58.550","Text":"but I can\u0027t put the circle here anymore."},{"Start":"09:58.550 ","End":"10:00.680","Text":"It\u0027s no longer a closed curve,"},{"Start":"10:00.680 ","End":"10:09.930","Text":"but it\u0027s the integral over C_1 plus the integral over C_3 and then plus the"},{"Start":"10:09.930 ","End":"10:20.505","Text":"integral over C_2 plus integral over minus C_3."},{"Start":"10:20.505 ","End":"10:25.220","Text":"Now we\u0027ve already learned that when you"},{"Start":"10:25.220 ","End":"10:30.110","Text":"take the integral over a path and of the opposite path,"},{"Start":"10:30.110 ","End":"10:34.525","Text":"the line integral, these 2 cancel each other out."},{"Start":"10:34.525 ","End":"10:38.960","Text":"This bit, with this bit cancel each other out,"},{"Start":"10:38.960 ","End":"10:45.575","Text":"and we\u0027re actually almost back full circle,"},{"Start":"10:45.575 ","End":"10:53.310","Text":"so to speak because the integral over C_1 plus the integral over C_2 is"},{"Start":"10:53.310 ","End":"11:01.850","Text":"just the integral over C. We didn\u0027t get anything new,"},{"Start":"11:01.850 ","End":"11:06.310","Text":"we started off with this double integral being equal to this line integral,"},{"Start":"11:06.310 ","End":"11:12.265","Text":"but the process of breaking up is just a tool that will help us,"},{"Start":"11:12.265 ","End":"11:18.810","Text":"very shortly I\u0027m about to get to regions with holes in them."},{"Start":"11:18.810 ","End":"11:23.700","Text":"In other words, they\u0027re not simply connected regions."},{"Start":"11:23.700 ","End":"11:28.479","Text":"I\u0027ll start on a new page."},{"Start":"11:29.090 ","End":"11:34.755","Text":"Here we see a region with a hole in it."},{"Start":"11:34.755 ","End":"11:36.960","Text":"Remember, we sometimes use R,"},{"Start":"11:36.960 ","End":"11:40.290","Text":"sometimes we use D for regions both."},{"Start":"11:40.290 ","End":"11:42.910","Text":"Any letter will do."},{"Start":"11:43.700 ","End":"11:49.380","Text":"It\u0027s not bounded by 1 simple curve, C_1 here."},{"Start":"11:49.380 ","End":"11:50.535","Text":"Because of the hole,"},{"Start":"11:50.535 ","End":"11:52.200","Text":"I need 2 curves."},{"Start":"11:52.200 ","End":"11:55.785","Text":"Look carefully at the directions of the arrows."},{"Start":"11:55.785 ","End":"12:02.910","Text":"Both of them, C_1 and C_2, are positively oriented."},{"Start":"12:02.910 ","End":"12:06.810","Text":"In the case of C_1, it\u0027s clear and you\u0027re thinking, \"Yeah, counterclockwise.\""},{"Start":"12:06.810 ","End":"12:10.080","Text":"But remember, I said counterclockwise is not the thing."},{"Start":"12:10.080 ","End":"12:12.930","Text":"It\u0027s as you traverse the curve,"},{"Start":"12:12.930 ","End":"12:15.465","Text":"the domain is on your left."},{"Start":"12:15.465 ","End":"12:19.980","Text":"If you think about it, C_2 is actually clockwise,"},{"Start":"12:19.980 ","End":"12:22.890","Text":"but it\u0027s also positive in"},{"Start":"12:22.890 ","End":"12:28.110","Text":"orientation in this context because when I\u0027m going along C_2 clockwise,"},{"Start":"12:28.110 ","End":"12:30.915","Text":"then the curve is on my left."},{"Start":"12:30.915 ","End":"12:35.250","Text":"We actually like to think of R, this region,"},{"Start":"12:35.250 ","End":"12:39.630","Text":"as being bounded by the curve at C,"},{"Start":"12:39.630 ","End":"12:45.480","Text":"which is the union; C_1 union C_2."},{"Start":"12:45.480 ","End":"12:51.330","Text":"Together they form a curve even though the two disjointed, disconnected bits"},{"Start":"12:51.330 ","End":"12:59.610","Text":"but the boundary of the region R is the curve C_1 together with C_2."},{"Start":"12:59.610 ","End":"13:08.415","Text":"What happens, and I\u0027m here jumping to the result part,"},{"Start":"13:08.415 ","End":"13:11.685","Text":"is that Green\u0027s theorem still holds in a way."},{"Start":"13:11.685 ","End":"13:13.080","Text":"D, in this case,"},{"Start":"13:13.080 ","End":"13:21.300","Text":"is R. Green\u0027s theorem still holds if we take C as C_1 union C_2,"},{"Start":"13:21.300 ","End":"13:29.219","Text":"which means I replace this integral by the integral of the same thing"},{"Start":"13:29.219 ","End":"13:39.560","Text":"over C_1 plus the integral of the same thing over C_1 and C_2,"},{"Start":"13:39.560 ","End":"13:42.890","Text":"where they\u0027re oriented as in the diagram."},{"Start":"13:42.890 ","End":"13:45.410","Text":"The reason is like we saw before,"},{"Start":"13:45.410 ","End":"13:48.665","Text":"we can break a region up into 2 regions."},{"Start":"13:48.665 ","End":"13:58.230","Text":"If you think about it, the double integral over R is the double integral over R2 plus R1,"},{"Start":"13:58.230 ","End":"14:00.090","Text":"just like we saw before."},{"Start":"14:00.090 ","End":"14:09.730","Text":"The border of this is from here going around here,"},{"Start":"14:09.950 ","End":"14:13.215","Text":"I\u0027m doing R2 first but never mind,"},{"Start":"14:13.215 ","End":"14:15.210","Text":"then I go in here,"},{"Start":"14:15.210 ","End":"14:17.100","Text":"I\u0027ll just put it dotted,"},{"Start":"14:17.100 ","End":"14:25.650","Text":"and then I\u0027m going along this part here and put the arrow,"},{"Start":"14:25.650 ","End":"14:28.920","Text":"put the arrow, and then along here."},{"Start":"14:28.920 ","End":"14:32.400","Text":"This is the curve for R2."},{"Start":"14:32.400 ","End":"14:36.480","Text":"For R1, this is also a closed curve."},{"Start":"14:36.480 ","End":"14:39.480","Text":"If I add these 2 line integrals together,"},{"Start":"14:39.480 ","End":"14:46.679","Text":"the bits with the opposite arrows cancel and I actually get the integral over"},{"Start":"14:46.679 ","End":"14:50.925","Text":"the outside in the positive direction plus"},{"Start":"14:50.925 ","End":"14:57.760","Text":"the inside in the positive direction so that really explains why this works."},{"Start":"14:58.280 ","End":"15:04.350","Text":"An example will really help to explain it and the example"},{"Start":"15:04.350 ","End":"15:09.675","Text":"we\u0027ll use will be to find the double integral over an annulus."},{"Start":"15:09.675 ","End":"15:12.660","Text":"Here\u0027s an illustration of the shape I meant."},{"Start":"15:12.660 ","End":"15:20.640","Text":"Now that would be my D and this would be bounded by the curve C,"},{"Start":"15:20.640 ","End":"15:23.430","Text":"which is C_1 union C_2."},{"Start":"15:23.430 ","End":"15:29.970","Text":"If I choose the arrows in the positive orientation,"},{"Start":"15:29.970 ","End":"15:36.765","Text":"then C_1 goes in this direction and C_2 in this direction."},{"Start":"15:36.765 ","End":"15:40.320","Text":"This is counterclockwise and this is clockwise,"},{"Start":"15:40.320 ","End":"15:41.730","Text":"but in each case,"},{"Start":"15:41.730 ","End":"15:46.605","Text":"the domain are the regions on the left."},{"Start":"15:46.605 ","End":"15:51.060","Text":"If I want to use Green\u0027s theorem and I do the arrows this way,"},{"Start":"15:51.060 ","End":"15:55.440","Text":"then I could take a double integral over the annulus as"},{"Start":"15:55.440 ","End":"16:01.210","Text":"the line integral over C_1 plus C_2 in the orientation shown."},{"Start":"16:01.210 ","End":"16:06.195","Text":"I think I\u0027m finally getting to an actual worked example."},{"Start":"16:06.195 ","End":"16:14.730","Text":"Let\u0027s say this outer circle is radius 2 and the inner circle is radius"},{"Start":"16:14.730 ","End":"16:23.130","Text":"1 and that our curve C is C_1, union C_2."},{"Start":"16:23.130 ","End":"16:26.475","Text":"We consider it as 1 curve just in 2 bits."},{"Start":"16:26.475 ","End":"16:29.700","Text":"I want to compute"},{"Start":"16:29.700 ","End":"16:37.785","Text":"the line integral over curve C,"},{"Start":"16:37.785 ","End":"16:45.315","Text":"which as I say is really 2 separate bits of y cubed,"},{"Start":"16:45.315 ","End":"16:50.985","Text":"dx minus x cubed, dy."},{"Start":"16:50.985 ","End":"16:56.380","Text":"I want to compute it using Green\u0027s theorem."},{"Start":"16:56.510 ","End":"17:01.650","Text":"We\u0027re back to D. Well anyway,"},{"Start":"17:01.650 ","End":"17:03.675","Text":"I\u0027ll leave that to help."},{"Start":"17:03.675 ","End":"17:07.605","Text":"Let\u0027s see what this is going to equal."},{"Start":"17:07.605 ","End":"17:11.070","Text":"It\u0027s going to equal the double"},{"Start":"17:11.070 ","End":"17:20.625","Text":"integral over the region D. This part is P,"},{"Start":"17:20.625 ","End":"17:23.715","Text":"this part is Q."},{"Start":"17:23.715 ","End":"17:32.380","Text":"Derivative of Q partially with respect to X would just be 3X squared."},{"Start":"17:35.060 ","End":"17:40.140","Text":"I forgot the minus here, not too late."},{"Start":"17:40.140 ","End":"17:45.330","Text":"Then minus dP by dy is 3y"},{"Start":"17:45.330 ","End":"17:51.640","Text":"squared and this is dA."},{"Start":"17:53.240 ","End":"18:00.495","Text":"Up till now, we\u0027ve been doing dA in general."},{"Start":"18:00.495 ","End":"18:05.560","Text":"We\u0027ve had either dx, dy or dy,"},{"Start":"18:06.080 ","End":"18:16.660","Text":"dx but there is a third possibility is that we use polar coordinates."},{"Start":"18:16.760 ","End":"18:25.210","Text":"In polar coordinates, I hope you remember polar coordinates with r and Theta,"},{"Start":"18:25.340 ","End":"18:33.195","Text":"the element of area there turned out to be r dr d Theta."},{"Start":"18:33.195 ","End":"18:35.985","Text":"Just in case you\u0027ve forgotten,"},{"Start":"18:35.985 ","End":"18:42.705","Text":"this is dA and we get that x is r"},{"Start":"18:42.705 ","End":"18:50.490","Text":"cosine Theta in polar and y equals r sine Theta."},{"Start":"18:50.490 ","End":"18:53.130","Text":"Condensed here but okay."},{"Start":"18:53.130 ","End":"18:56.595","Text":"Let\u0027s do this in polar coordinates."},{"Start":"18:56.595 ","End":"18:58.575","Text":"The reason why I\u0027m using polar,"},{"Start":"18:58.575 ","End":"19:02.940","Text":"because this thing has circular symmetry that\u0027s natural way to go."},{"Start":"19:02.940 ","End":"19:07.020","Text":"Also, neither of these curves I can express"},{"Start":"19:07.020 ","End":"19:12.060","Text":"as x as a function of y or y as a function of x because they\u0027re not really function,"},{"Start":"19:12.060 ","End":"19:15.720","Text":"so polar seems to be the natural choice here."},{"Start":"19:15.720 ","End":"19:19.230","Text":"This thing becomes; look,"},{"Start":"19:19.230 ","End":"19:23.835","Text":"I can take the minus 3 outside the brackets,"},{"Start":"19:23.835 ","End":"19:32.610","Text":"the double integral over D of the x squared plus y squared."},{"Start":"19:32.610 ","End":"19:37.710","Text":"In polar x squared plus y squared equals r squared,"},{"Start":"19:37.710 ","End":"19:39.525","Text":"it\u0027s one of the other formulas."},{"Start":"19:39.525 ","End":"19:45.820","Text":"What I have here now is x squared plus y squared is r squared."},{"Start":"19:46.100 ","End":"19:48.675","Text":"I write the r squared."},{"Start":"19:48.675 ","End":"19:55.660","Text":"Then the dA is r dr d Theta."},{"Start":"19:58.670 ","End":"20:01.770","Text":"Scroll a bit here."},{"Start":"20:01.770 ","End":"20:06.430","Text":"Minus 3;"},{"Start":"20:09.620 ","End":"20:15.375","Text":"the full circle will be Theta from 0-2Pi."},{"Start":"20:15.375 ","End":"20:18.175","Text":"For each particular Theta,"},{"Start":"20:18.175 ","End":"20:22.970","Text":"r goes from 1-2."},{"Start":"20:22.970 ","End":"20:26.305","Text":"That\u0027s what this is."},{"Start":"20:26.305 ","End":"20:35.390","Text":"Any radius, r goes from 1 to 2 and then I get r cubed dr,"},{"Start":"20:35.390 ","End":"20:39.105","Text":"and this thing is the inner integral,"},{"Start":"20:39.105 ","End":"20:44.130","Text":"and the result is a function of Theta, well,"},{"Start":"20:44.130 ","End":"20:46.515","Text":"it comes out to be a constant in this case,"},{"Start":"20:46.515 ","End":"20:50.260","Text":"and then d Theta."},{"Start":"20:51.140 ","End":"20:53.835","Text":"What does that give us?"},{"Start":"20:53.835 ","End":"21:02.400","Text":"Minus 3 and then the integral of r cubed is r^4 over 4."},{"Start":"21:02.400 ","End":"21:12.490","Text":"Let me take the over 4 and put it right here in front and then 0 to 2Pi."},{"Start":"21:12.490 ","End":"21:15.180","Text":"This already is r^4,"},{"Start":"21:15.180 ","End":"21:16.920","Text":"that\u0027s already been integrated,"},{"Start":"21:16.920 ","End":"21:19.020","Text":"the r^4 over 4 is here."},{"Start":"21:19.020 ","End":"21:26.340","Text":"But this thing has to be between 1 and 2,"},{"Start":"21:26.340 ","End":"21:29.320","Text":"and then the d Theta."},{"Start":"21:30.170 ","End":"21:37.130","Text":"We get r^4. If I plug in 2,"},{"Start":"21:37.130 ","End":"21:39.890","Text":"it\u0027s 16, if I plug in 1,"},{"Start":"21:39.890 ","End":"21:45.320","Text":"it\u0027s 1, 16 minus 1 is 15."},{"Start":"21:45.320 ","End":"21:51.075","Text":"I get, and the 15 I can bring out here,"},{"Start":"21:51.075 ","End":"21:56.130","Text":"minus 45 over 4 times"},{"Start":"21:56.130 ","End":"22:04.150","Text":"the integral from 0-2Pi."},{"Start":"22:04.490 ","End":"22:10.680","Text":"After I took the 15 and I\u0027m just left with 1d Theta,"},{"Start":"22:10.680 ","End":"22:12.850","Text":"I\u0027ll put the 1 in."},{"Start":"22:14.000 ","End":"22:18.429","Text":"This is now equal to;"},{"Start":"22:18.429 ","End":"22:21.710","Text":"the integral of 1 is just Theta,"},{"Start":"22:21.710 ","End":"22:26.220","Text":"from 0-2Pi, it\u0027s 2 Pi minus 0, is 2Pi."},{"Start":"22:29.330 ","End":"22:33.570","Text":"This bit comes out to be 2Pi, like I said."},{"Start":"22:33.570 ","End":"22:37.005","Text":"The 2 will cancel with the 4 partially."},{"Start":"22:37.005 ","End":"22:45.075","Text":"I\u0027m going to get minus 45Pi over 2,"},{"Start":"22:45.075 ","End":"22:48.465","Text":"and that\u0027s the answer."},{"Start":"22:48.465 ","End":"22:54.780","Text":"You could try on your own to see if you could do the line integral directly."},{"Start":"22:54.780 ","End":"23:01.890","Text":"But remember, you\u0027ll have to do two-line integrals because C is C_1 and C_2."},{"Start":"23:01.890 ","End":"23:06.540","Text":"You\u0027ll have to parameterize this one to go in this direction."},{"Start":"23:06.540 ","End":"23:11.075","Text":"It will be quite a bit of work, not enormous amount,"},{"Start":"23:11.075 ","End":"23:15.170","Text":"but it\u0027s definitely easier to do the double integral over the annulus,"},{"Start":"23:15.170 ","End":"23:17.555","Text":"especially in polar coordinates."},{"Start":"23:17.555 ","End":"23:19.100","Text":"We\u0027ll take a break now."},{"Start":"23:19.100 ","End":"23:27.090","Text":"After that, I\u0027ll show you how to use Green\u0027s theorem to compute certain areas."}],"ID":8769},{"Watched":false,"Name":"Area of a Region","Duration":"10m 49s","ChapterTopicVideoID":8644,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"1 more topic in Green\u0027s theorem,"},{"Start":"00:03.330 ","End":"00:10.180","Text":"and how it can help us to find the area of a region sometimes."},{"Start":"00:10.460 ","End":"00:18.075","Text":"Here\u0027s the setup. Supposing that this expression here in the double integral,"},{"Start":"00:18.075 ","End":"00:21.330","Text":"suppose that this was equal to 1,"},{"Start":"00:21.330 ","End":"00:23.955","Text":"or in our shorthand notation,"},{"Start":"00:23.955 ","End":"00:29.850","Text":"Q with respect to x minus P with respect to"},{"Start":"00:29.850 ","End":"00:36.540","Text":"y was equal to the constant function 1."},{"Start":"00:36.540 ","End":"00:41.135","Text":"In that case, what we would get is that"},{"Start":"00:41.135 ","End":"00:46.070","Text":"this double integral would just be the double integral"},{"Start":"00:46.070 ","End":"00:52.230","Text":"over the region D of dA and that is just equal to"},{"Start":"00:52.230 ","End":"01:01.510","Text":"the area of D. Integral of 1 over region is its area."},{"Start":"01:02.990 ","End":"01:07.634","Text":"That\u0027s 1 way of finding an area."},{"Start":"01:07.634 ","End":"01:12.735","Text":"Now, there\u0027s lots of combinations of Q and P that will give us that,"},{"Start":"01:12.735 ","End":"01:18.680","Text":"but in practice, there are 3 main combinations that are used."},{"Start":"01:18.680 ","End":"01:24.800","Text":"1 example would be if P was equal"},{"Start":"01:24.800 ","End":"01:33.685","Text":"to the constant 0 function and Q was equal to x."},{"Start":"01:33.685 ","End":"01:35.985","Text":"Of course, I mean Q of xy,"},{"Start":"01:35.985 ","End":"01:38.430","Text":"the function and P of xy,"},{"Start":"01:38.430 ","End":"01:40.935","Text":"I\u0027m just writing this in a condensed form."},{"Start":"01:40.935 ","End":"01:49.680","Text":"Then we have that Px here would equal 0, sorry, Py."},{"Start":"01:49.680 ","End":"01:53.130","Text":"Px is also 0, but it\u0027s not what I want."},{"Start":"01:53.130 ","End":"01:57.180","Text":"Then Qx is equal to 1,"},{"Start":"01:57.180 ","End":"02:04.605","Text":"so we can see that Qx minus Py is equal to 1, so that\u0027s okay."},{"Start":"02:04.605 ","End":"02:08.625","Text":"Combination 2, that\u0027s popular,"},{"Start":"02:08.625 ","End":"02:14.640","Text":"is to take P equals minus y,"},{"Start":"02:14.640 ","End":"02:22.809","Text":"Q equals 0, and then Py is minus 1,"},{"Start":"02:22.809 ","End":"02:28.850","Text":"partial derivative of Q with respect to x is 0."},{"Start":"02:28.850 ","End":"02:35.720","Text":"This difference is equal to 1 because 0 less minus 1 is 1,"},{"Start":"02:35.720 ","End":"02:38.075","Text":"so that\u0027s a good combination."},{"Start":"02:38.075 ","End":"02:46.515","Text":"The third commonly used combination is to take the compromise between the 2,"},{"Start":"02:46.515 ","End":"02:50.950","Text":"where P equals minus"},{"Start":"02:51.830 ","End":"02:59.880","Text":"1/2y and Q is equal to 1/2x or x over 2."},{"Start":"02:59.880 ","End":"03:05.320","Text":"In this case, derivative of P with respect to y is minus 1/2,"},{"Start":"03:05.320 ","End":"03:11.570","Text":"the derivative of Q with respect to x is equal to plus 1/2,"},{"Start":"03:11.570 ","End":"03:15.440","Text":"and then this expression is 1/2 minus minus 1/2 is 1,"},{"Start":"03:15.440 ","End":"03:17.790","Text":"this is also good."},{"Start":"03:18.110 ","End":"03:23.610","Text":"What this in practice means,"},{"Start":"03:23.610 ","End":"03:26.210","Text":"is that if we have the region D,"},{"Start":"03:26.210 ","End":"03:29.100","Text":"I\u0027ll just do a sketch here,"},{"Start":"03:29.180 ","End":"03:35.080","Text":"region D. Suppose the region D"},{"Start":"03:35.150 ","End":"03:42.600","Text":"is bounded by a closed,"},{"Start":"03:42.600 ","End":"03:51.080","Text":"simple, piecewise, smooth curve C,"},{"Start":"03:51.080 ","End":"03:59.840","Text":"then we can compute the area of D as a line integral so in 3 possible ways."},{"Start":"03:59.840 ","End":"04:09.555","Text":"I could say that the area of D equals,"},{"Start":"04:09.555 ","End":"04:13.180","Text":"and then 3 possibilities."},{"Start":"04:14.770 ","End":"04:20.990","Text":"I could take the line integral over"},{"Start":"04:20.990 ","End":"04:27.105","Text":"C. If I use case 1,"},{"Start":"04:27.105 ","End":"04:34.270","Text":"P dx plus Q dy is just x dy."},{"Start":"04:37.310 ","End":"04:41.810","Text":"The second possibility would be from here,"},{"Start":"04:41.810 ","End":"04:46.170","Text":"I could take the integral over C,"},{"Start":"04:46.170 ","End":"04:48.450","Text":"of minus y dx,"},{"Start":"04:48.450 ","End":"04:53.800","Text":"put the minus in front, y dx."},{"Start":"04:54.170 ","End":"05:05.070","Text":"The third possibility, using number 3 and taking 1/2 out in front,"},{"Start":"05:05.070 ","End":"05:07.395","Text":"would be 1/2 the integral,"},{"Start":"05:07.395 ","End":"05:13.785","Text":"these are all closed curves, from here,"},{"Start":"05:13.785 ","End":"05:21.540","Text":"I would get 1/2 of the 1/2x dy,"},{"Start":"05:21.540 ","End":"05:24.580","Text":"so it\u0027s just x dy,"},{"Start":"05:24.740 ","End":"05:33.310","Text":"and from the other 1, minus y dx."},{"Start":"05:33.320 ","End":"05:36.420","Text":"These are 3 formulas,"},{"Start":"05:36.420 ","End":"05:40.630","Text":"and let\u0027s use these to derive a well known formula"},{"Start":"05:40.630 ","End":"05:45.115","Text":"for the area of a circle or rather of a disk,"},{"Start":"05:45.115 ","End":"05:47.965","Text":"so let\u0027s say D,"},{"Start":"05:47.965 ","End":"05:51.460","Text":"I call it disk because a circle is"},{"Start":"05:51.460 ","End":"05:55.255","Text":"usually just the boundary and the disk includes the interior,"},{"Start":"05:55.255 ","End":"06:02.490","Text":"disk of radius r. Everyone knows to say Pi r squared,"},{"Start":"06:02.490 ","End":"06:05.640","Text":"but let\u0027s say I didn\u0027t know Pi r squared."},{"Start":"06:05.640 ","End":"06:09.230","Text":"We\u0027ll do it using the last 1,"},{"Start":"06:09.230 ","End":"06:11.125","Text":"seems to be the easiest,"},{"Start":"06:11.125 ","End":"06:13.900","Text":"I need a parameterization."},{"Start":"06:13.900 ","End":"06:19.380","Text":"Here\u0027s our disk and its boundary is the circle,"},{"Start":"06:19.380 ","End":"06:25.220","Text":"and it\u0027s going to go in the positive direction around the disk C,"},{"Start":"06:25.220 ","End":"06:29.285","Text":"and if it\u0027s centered at the origin with radius r,"},{"Start":"06:29.285 ","End":"06:34.415","Text":"then the parameterization of C,"},{"Start":"06:34.415 ","End":"06:39.200","Text":"write it in parameter form which is almost the same as vector form,"},{"Start":"06:39.200 ","End":"06:48.500","Text":"which is that x equals r cosine t,"},{"Start":"06:48.500 ","End":"06:53.640","Text":"y equals r sine t,"},{"Start":"06:53.640 ","End":"06:55.575","Text":"I\u0027ll put some brackets,"},{"Start":"06:55.575 ","End":"07:00.990","Text":"and t goes from 0-2Pi."},{"Start":"07:00.990 ","End":"07:03.450","Text":"What I get is that the area,"},{"Start":"07:03.450 ","End":"07:06.639","Text":"I\u0027ll just call that A,"},{"Start":"07:06.740 ","End":"07:12.400","Text":"will equal the integral"},{"Start":"07:13.370 ","End":"07:18.210","Text":"over this curve C of,"},{"Start":"07:18.210 ","End":"07:19.800","Text":"I don\u0027t have dy and dx,"},{"Start":"07:19.800 ","End":"07:27.165","Text":"so let\u0027s just write those in dx will be minus"},{"Start":"07:27.165 ","End":"07:29.920","Text":"r sine t"},{"Start":"07:37.400 ","End":"07:40.185","Text":"dt of course,"},{"Start":"07:40.185 ","End":"07:49.815","Text":"and dy will equal r cosine of t, also dt,"},{"Start":"07:49.815 ","End":"07:52.874","Text":"and so we get from this formula,"},{"Start":"07:52.874 ","End":"08:01.980","Text":"the 1/2 in front of x dy r cosine t,"},{"Start":"08:01.980 ","End":"08:11.340","Text":"dy is r cosine t dt,"},{"Start":"08:11.340 ","End":"08:13.440","Text":"I\u0027ll write the dt at the end,"},{"Start":"08:13.440 ","End":"08:16.215","Text":"then minus y dx,"},{"Start":"08:16.215 ","End":"08:20.920","Text":"y is r sine t,"},{"Start":"08:21.890 ","End":"08:27.765","Text":"then dx is also minus r"},{"Start":"08:27.765 ","End":"08:34.705","Text":"sine t. This minus can become a plus rather than write an extra minus here,"},{"Start":"08:34.705 ","End":"08:40.390","Text":"so again, r sine t. This is looking a mess,"},{"Start":"08:40.390 ","End":"08:48.270","Text":"we better put the brackets around here so we can see what\u0027s going on."},{"Start":"08:49.380 ","End":"08:55.825","Text":"Then finally, all this dt,"},{"Start":"08:55.825 ","End":"08:59.820","Text":"r is no longer over the curve C,"},{"Start":"08:59.820 ","End":"09:03.745","Text":"it\u0027s an integral in t from 0-2Pi."},{"Start":"09:03.745 ","End":"09:12.765","Text":"Now look, this r with this r is r squared in the first term."},{"Start":"09:12.765 ","End":"09:15.830","Text":"We also have an r with an r in the second term."},{"Start":"09:15.830 ","End":"09:18.055","Text":"An r is a constant,"},{"Start":"09:18.055 ","End":"09:21.600","Text":"so we can take r squared outside,"},{"Start":"09:21.600 ","End":"09:27.810","Text":"so this is equal to 1/2r squared,"},{"Start":"09:27.810 ","End":"09:32.010","Text":"and then we have the integral from 0-2Pi."},{"Start":"09:32.010 ","End":"09:33.815","Text":"But what are we left with?"},{"Start":"09:33.815 ","End":"09:36.005","Text":"What we\u0027re left with,"},{"Start":"09:36.005 ","End":"09:43.140","Text":"from here and here we would have cosine squared of t,"},{"Start":"09:43.140 ","End":"09:47.440","Text":"and from this and this we would get sine squared of t,"},{"Start":"09:47.440 ","End":"09:50.825","Text":"and cosine squared plus sine squared is 1,"},{"Start":"09:50.825 ","End":"09:55.875","Text":"so it\u0027s the integral of 1dt from 0-2Pi,"},{"Start":"09:55.875 ","End":"09:59.760","Text":"and the integral of 1 from a-b is just b minus a."},{"Start":"09:59.760 ","End":"10:02.090","Text":"You could integrate it and say that\u0027s t,"},{"Start":"10:02.090 ","End":"10:04.970","Text":"when I substitute 2Pi and 0,"},{"Start":"10:04.970 ","End":"10:07.600","Text":"I get 2Pi minus 0,"},{"Start":"10:07.600 ","End":"10:13.875","Text":"this is equal to 1/2r squared,"},{"Start":"10:13.875 ","End":"10:16.845","Text":"2Pi minus 0 is 2Pi,"},{"Start":"10:16.845 ","End":"10:19.575","Text":"the 2 with the 1/2 cancels."},{"Start":"10:19.575 ","End":"10:25.710","Text":"What we\u0027re left with is Pi r squared,"},{"Start":"10:25.710 ","End":"10:30.750","Text":"and this is the formula we were expecting for a disk of radius r,"},{"Start":"10:30.750 ","End":"10:32.360","Text":"this is its area."},{"Start":"10:32.360 ","End":"10:38.240","Text":"But now we\u0027ve proved it using Green\u0027s theorem."},{"Start":"10:39.650 ","End":"10:45.935","Text":"That\u0027s it. We\u0027re done with the Green\u0027s theorem now,"},{"Start":"10:45.935 ","End":"10:49.560","Text":"but we still haven\u0027t finished with line integrals."}],"ID":8770},{"Watched":false,"Name":"Exercise 1","Duration":"15m 37s","ChapterTopicVideoID":8645,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.685","Text":"In this exercise, we\u0027re going to verify Green\u0027s theorem for a particular example."},{"Start":"00:05.685 ","End":"00:08.220","Text":"This here is Green\u0027s theorem."},{"Start":"00:08.220 ","End":"00:10.800","Text":"R is some region which is closed,"},{"Start":"00:10.800 ","End":"00:14.640","Text":"simply connected, bounded as it is here,"},{"Start":"00:14.640 ","End":"00:18.990","Text":"and the boundary is the curve C,"},{"Start":"00:18.990 ","End":"00:24.645","Text":"which has taken counterclockwise here I\u0027ve marked C and we equate"},{"Start":"00:24.645 ","End":"00:30.780","Text":"a line integral over a closed curve with the double integral over such a region."},{"Start":"00:30.780 ","End":"00:33.240","Text":"We\u0027re going to evaluate both the left-hand side and"},{"Start":"00:33.240 ","End":"00:36.420","Text":"the right-hand side and show that they\u0027re equal,"},{"Start":"00:36.420 ","End":"00:39.710","Text":"but it turns out that this double integral is much easier to"},{"Start":"00:39.710 ","End":"00:43.745","Text":"compute than the line integral and I\u0027m not sure which to start with."},{"Start":"00:43.745 ","End":"00:48.010","Text":"Let\u0027s start with the line integral and end with the easy one."},{"Start":"00:48.010 ","End":"00:53.310","Text":"For the line integral, I\u0027m going to break this curve C up into 3 pieces."},{"Start":"00:54.770 ","End":"00:57.540","Text":"Now let\u0027s make this C_1,"},{"Start":"00:57.540 ","End":"00:59.670","Text":"this will be C_2,"},{"Start":"00:59.670 ","End":"01:01.500","Text":"and the last one will be"},{"Start":"01:01.500 ","End":"01:09.380","Text":"C_3 and let me just mark some of the points in this point is the origin, so it\u0027s 00."},{"Start":"01:09.380 ","End":"01:12.845","Text":"Here obviously it\u0027s the 1,"},{"Start":"01:12.845 ","End":"01:15.305","Text":"0 and here 1,"},{"Start":"01:15.305 ","End":"01:22.715","Text":"2 and I want to parameterize each of the pieces separately and at the end,"},{"Start":"01:22.715 ","End":"01:28.510","Text":"we\u0027ll take the integral of this over C as the integral"},{"Start":"01:28.510 ","End":"01:34.745","Text":"over C_1 plus the integral over C_2 plus the integral over C_3,"},{"Start":"01:34.745 ","End":"01:36.370","Text":"so we\u0027ll have 3 integrals to do,"},{"Start":"01:36.370 ","End":"01:38.500","Text":"and then we\u0027ll add the answers."},{"Start":"01:38.500 ","End":"01:44.740","Text":"Let\u0027s parameterize C_1, change your mind let\u0027s start with a difficult one, C_3."},{"Start":"01:44.740 ","End":"01:50.520","Text":"One way of parametrizing the line segment between 2 points is to do as follows,"},{"Start":"01:50.520 ","End":"01:53.005","Text":"just take x and y equal."},{"Start":"01:53.005 ","End":"01:55.390","Text":"Now x will be the x of the start point."},{"Start":"01:55.390 ","End":"01:56.770","Text":"We\u0027re going from here to here,"},{"Start":"01:56.770 ","End":"01:59.180","Text":"so we put here a 1,"},{"Start":"01:59.180 ","End":"02:01.200","Text":"here we start from 2,"},{"Start":"02:01.200 ","End":"02:08.290","Text":"then t and then t and then in each case we\u0027ll do the end minus the start,"},{"Start":"02:08.290 ","End":"02:11.645","Text":"so for x its 0 minus 1,"},{"Start":"02:11.645 ","End":"02:17.955","Text":"and for y it\u0027s 0 minus 2,"},{"Start":"02:17.955 ","End":"02:20.135","Text":"t will go from 0-1."},{"Start":"02:20.135 ","End":"02:21.190","Text":"I\u0027ll write that in a moment,"},{"Start":"02:21.190 ","End":"02:23.020","Text":"but I just want to simplify this."},{"Start":"02:23.020 ","End":"02:27.520","Text":"That we could just say that x is equal to"},{"Start":"02:27.520 ","End":"02:35.680","Text":"1 minus t and y equals 2 minus 2t."},{"Start":"02:35.680 ","End":"02:38.650","Text":"As I said, when you do it this way,"},{"Start":"02:38.650 ","End":"02:42.265","Text":"t goes from 0-1."},{"Start":"02:42.265 ","End":"02:46.000","Text":"Now I can take some shortcuts for C_1 and C_2 because they\u0027re"},{"Start":"02:46.000 ","End":"02:52.715","Text":"horizontal to the right and vertically upwards."},{"Start":"02:52.715 ","End":"02:56.330","Text":"C_1, what you can do is as follows."},{"Start":"02:56.330 ","End":"03:00.190","Text":"We see that y is 0 all along the x-axis,"},{"Start":"03:00.190 ","End":"03:04.970","Text":"so I\u0027ll write y equals 0 and x goes from 0-1,"},{"Start":"03:04.970 ","End":"03:10.540","Text":"so I\u0027ll write it as a parameter t and make t go from 0-1."},{"Start":"03:10.540 ","End":"03:14.755","Text":"Similarly with C_2. With C_2,"},{"Start":"03:14.755 ","End":"03:21.730","Text":"what we have is x stays 1 and y is the one that moves from 0-2,"},{"Start":"03:21.730 ","End":"03:28.130","Text":"so we\u0027ll take y equals t and we\u0027ll let t go from 0-2."},{"Start":"03:28.130 ","End":"03:30.990","Text":"I\u0027ve got all 3 curves parameterized."},{"Start":"03:30.990 ","End":"03:34.880","Text":"This one is been replaced by this one,"},{"Start":"03:34.880 ","End":"03:38.430","Text":"which is simpler, just a simplification."},{"Start":"03:38.800 ","End":"03:42.340","Text":"Now before we substitute,"},{"Start":"03:42.340 ","End":"03:48.600","Text":"I see we\u0027ll also need dx and dy for each of the 3 pieces."},{"Start":"03:48.600 ","End":"03:56.270","Text":"For the curve C_3, I have from here that dx is equal to"},{"Start":"03:56.270 ","End":"04:04.070","Text":"minus dt and dy is equal to minus 2dt."},{"Start":"04:04.070 ","End":"04:09.485","Text":"For C_1, let\u0027s see,"},{"Start":"04:09.485 ","End":"04:12.140","Text":"dx is equal to dt,"},{"Start":"04:12.140 ","End":"04:19.820","Text":"but dy is equal to 0 and for C_2,"},{"Start":"04:19.820 ","End":"04:28.200","Text":"dx will equal 0 because x is a constant and dy will equal dt."},{"Start":"04:28.870 ","End":"04:32.360","Text":"Now we have 3 substitutions to make,"},{"Start":"04:32.360 ","End":"04:38.780","Text":"let\u0027s start with C_1 and what we\u0027ll get is,"},{"Start":"04:38.780 ","End":"04:42.320","Text":"yeah, for C_1, C_1 is here."},{"Start":"04:42.320 ","End":"04:44.525","Text":"t goes from 0-1,"},{"Start":"04:44.525 ","End":"04:51.059","Text":"so the integral from 0-1 of x squared y,"},{"Start":"04:51.830 ","End":"04:56.220","Text":"that\u0027s 0 because y is 0,"},{"Start":"04:56.220 ","End":"05:00.345","Text":"so the first bit is 0 and let\u0027s see the x dy,"},{"Start":"05:00.345 ","End":"05:06.945","Text":"dy is 0, so it\u0027s another 0."},{"Start":"05:06.945 ","End":"05:10.110","Text":"Altogether C_1 just comes out to be 0."},{"Start":"05:10.110 ","End":"05:11.625","Text":"Nice, easy one."},{"Start":"05:11.625 ","End":"05:13.510","Text":"Let\u0027s go for C_2."},{"Start":"05:13.510 ","End":"05:18.930","Text":"Here we have the integral from"},{"Start":"05:18.930 ","End":"05:27.340","Text":"0-2 of x squared y dx,"},{"Start":"05:27.350 ","End":"05:32.739","Text":"but dx is 0, so we start off with a 0."},{"Start":"05:32.739 ","End":"05:34.450","Text":"Let\u0027s see what\u0027s the second bid,"},{"Start":"05:34.450 ","End":"05:37.495","Text":"x dy not zero,"},{"Start":"05:37.495 ","End":"05:46.215","Text":"x dy is 1dt and this goes from 0-2,"},{"Start":"05:46.215 ","End":"05:49.120","Text":"because the 0 as if it\u0027s not there,"},{"Start":"05:49.120 ","End":"05:51.140","Text":"when we have the integral of 1,"},{"Start":"05:51.140 ","End":"05:55.155","Text":"we just take the upper limit minus the lower limit,"},{"Start":"05:55.155 ","End":"05:59.445","Text":"which happens to be 2."},{"Start":"05:59.445 ","End":"06:03.300","Text":"That\u0027s that. As for C_3,"},{"Start":"06:03.300 ","End":"06:05.685","Text":"a little bit more difficult,"},{"Start":"06:05.685 ","End":"06:07.385","Text":"not too bad though."},{"Start":"06:07.385 ","End":"06:09.035","Text":"What we have is the integral,"},{"Start":"06:09.035 ","End":"06:13.505","Text":"the parameter is from 0-1, that\u0027s the dt."},{"Start":"06:13.505 ","End":"06:18.125","Text":"Now I need x squared y dx."},{"Start":"06:18.125 ","End":"06:20.480","Text":"I\u0027m looking at x squared y,"},{"Start":"06:20.480 ","End":"06:23.660","Text":"and I\u0027m noticing there\u0027s a 1 minus t here and"},{"Start":"06:23.660 ","End":"06:27.790","Text":"this begs to be written as twice 1 minus t,"},{"Start":"06:27.790 ","End":"06:30.780","Text":"so allow me to write 2 minus 2t,"},{"Start":"06:30.780 ","End":"06:33.765","Text":"that\u0027s twice 1 minus t,"},{"Start":"06:33.765 ","End":"06:36.240","Text":"and that\u0027ll make it easier."},{"Start":"06:36.240 ","End":"06:45.770","Text":"x squared y is just 1 minus t squared times twice 1 minus t,"},{"Start":"06:45.770 ","End":"06:50.315","Text":"so it\u0027s twice 1 minus t cubed,"},{"Start":"06:50.315 ","End":"06:52.690","Text":"and then I need the dx,"},{"Start":"06:52.690 ","End":"06:55.350","Text":"and dx is minus dt,"},{"Start":"06:55.350 ","End":"06:59.465","Text":"so I\u0027ll put the minus here and the dt here."},{"Start":"06:59.465 ","End":"07:00.860","Text":"That\u0027s just the first bit,"},{"Start":"07:00.860 ","End":"07:02.880","Text":"then I need x dy."},{"Start":"07:04.760 ","End":"07:07.635","Text":"Oh yeah, I mean, C_3,"},{"Start":"07:07.635 ","End":"07:10.275","Text":"x is 1 minus t,"},{"Start":"07:10.275 ","End":"07:17.290","Text":"and dy is minus 2 dt."},{"Start":"07:18.260 ","End":"07:24.885","Text":"I have minus 2, 1 minus t,"},{"Start":"07:24.885 ","End":"07:34.730","Text":"dt and let\u0027s see there\u0027s several ways I could do this You know what,"},{"Start":"07:34.730 ","End":"07:37.620","Text":"I\u0027d like to do a substitution."},{"Start":"07:37.860 ","End":"07:41.395","Text":"I see I have 1 minus t everywhere. Let\u0027s do that."},{"Start":"07:41.395 ","End":"07:42.730","Text":"You don\u0027t have to do it this way."},{"Start":"07:42.730 ","End":"07:45.700","Text":"You can expand or many other ways."},{"Start":"07:45.700 ","End":"07:54.415","Text":"I\u0027ll let u equal 1 minus t and then I have that du is minus dt."},{"Start":"07:54.415 ","End":"08:01.705","Text":"Then also when, let\u0027s substitute the limits, t equals 0,"},{"Start":"08:01.705 ","End":"08:08.440","Text":"t equals 1, 1 minus 0 is 1,"},{"Start":"08:08.440 ","End":"08:10.750","Text":"1 minus 1 is 0."},{"Start":"08:10.750 ","End":"08:13.390","Text":"If I make all these substitutions,"},{"Start":"08:13.390 ","End":"08:17.165","Text":"then what I\u0027ll get, I\u0027ll write it over here."},{"Start":"08:17.165 ","End":"08:20.790","Text":"I\u0027ll also take the 2 in front of the brackets."},{"Start":"08:20.790 ","End":"08:25.180","Text":"I\u0027ve got twice the integral."},{"Start":"08:25.180 ","End":"08:29.260","Text":"Now, not from 0-1, but from 1-0."},{"Start":"08:29.260 ","End":"08:32.350","Text":"It\u0027s going to be du."},{"Start":"08:32.350 ","End":"08:35.845","Text":"Now, here I have"},{"Start":"08:35.845 ","End":"08:44.905","Text":"u cubed and the minus dt is du."},{"Start":"08:44.905 ","End":"08:48.830","Text":"This is u cubed du."},{"Start":"08:49.230 ","End":"08:53.650","Text":"The 2 I\u0027ve just taken out front and here also."},{"Start":"08:53.650 ","End":"08:59.605","Text":"This part, again, minus dt is du,"},{"Start":"08:59.605 ","End":"09:08.860","Text":"and 1 minus t is u. I have u cubed plus u du."},{"Start":"09:08.860 ","End":"09:13.850","Text":"This is equal to, let\u0027s see, twice."},{"Start":"09:14.910 ","End":"09:22.030","Text":"What I have is"},{"Start":"09:22.030 ","End":"09:29.920","Text":"u^4 over 4 plus u squared over 2 from 1-0."},{"Start":"09:29.920 ","End":"09:32.815","Text":"At 0, I don\u0027t get anything."},{"Start":"09:32.815 ","End":"09:36.460","Text":"At 1, I get, 1/4 plus 1/2 is 3/4,"},{"Start":"09:36.460 ","End":"09:41.680","Text":"3/4 times 2 is 1 and 1/2."},{"Start":"09:41.680 ","End":"09:45.820","Text":"Altogether, this is 1 and 1/2."},{"Start":"09:45.820 ","End":"09:50.500","Text":"Now I can substitute the 3 pieces here."},{"Start":"09:50.500 ","End":"09:53.560","Text":"The first integral came out 0,"},{"Start":"09:53.560 ","End":"09:58.135","Text":"the second integral came out to be,"},{"Start":"09:58.135 ","End":"10:02.600","Text":"sorry, this is minus 1 and 1/2."},{"Start":"10:03.150 ","End":"10:06.160","Text":"I\u0027m subtracting what I get when I substitute 1."},{"Start":"10:06.160 ","End":"10:07.210","Text":"This is minus 3/4,"},{"Start":"10:07.210 ","End":"10:09.260","Text":"this is minus 1 and 1/2."},{"Start":"10:09.260 ","End":"10:13.240","Text":"Just caught that in time, it\u0027s a minus."},{"Start":"10:14.590 ","End":"10:21.730","Text":"C_2 is 2 and C_3 is minus 1 and 1/2."},{"Start":"10:21.730 ","End":"10:32.245","Text":"Altogether, I get that this is equal to 1/2 because it\u0027s 0 plus 2 minus 1 and 1/2."},{"Start":"10:32.245 ","End":"10:35.380","Text":"This is what I hope to get when we do"},{"Start":"10:35.380 ","End":"10:41.050","Text":"the right-hand side method using the double integral."},{"Start":"10:41.050 ","End":"10:43.210","Text":"Just before I clear the board,"},{"Start":"10:43.210 ","End":"10:45.610","Text":"let me just write down that the left-hand side,"},{"Start":"10:45.610 ","End":"10:47.200","Text":"we got the answer of 1/2,"},{"Start":"10:47.200 ","End":"10:50.530","Text":"so we\u0027re expecting to get the same on the right-hand side."},{"Start":"10:50.530 ","End":"10:53.470","Text":"Now let\u0027s clear the board."},{"Start":"10:53.470 ","End":"10:56.860","Text":"First thing you want to do is some technical stuff."},{"Start":"10:56.860 ","End":"11:00.310","Text":"That\u0027s to find out what is g_x and f_y."},{"Start":"11:00.310 ","End":"11:04.390","Text":"Well, this part is our f,"},{"Start":"11:04.390 ","End":"11:07.510","Text":"it\u0027s the part that goes with dx,"},{"Start":"11:07.510 ","End":"11:11.800","Text":"and g is the part that goes with dy,"},{"Start":"11:11.800 ","End":"11:15.085","Text":"so we have that."},{"Start":"11:15.085 ","End":"11:19.195","Text":"I need to know what g_x minus f_y is, basically."},{"Start":"11:19.195 ","End":"11:20.755","Text":"This is equal to,"},{"Start":"11:20.755 ","End":"11:24.460","Text":"g with respect to x is 1,"},{"Start":"11:24.460 ","End":"11:26.290","Text":"and I have a minus,"},{"Start":"11:26.290 ","End":"11:31.195","Text":"and f with respect to y is x squared."},{"Start":"11:31.195 ","End":"11:35.935","Text":"The integral we\u0027re going to take this bit is going to be 1 minus x squared."},{"Start":"11:35.935 ","End":"11:39.460","Text":"Now, we want to do this as an iterated integral."},{"Start":"11:39.460 ","End":"11:42.550","Text":"Let\u0027s think, we want type 1 or type 2?"},{"Start":"11:42.550 ","End":"11:49.135","Text":"Both would work, but I think it\u0027s easier if we take vertical slices, type 1 region."},{"Start":"11:49.135 ","End":"11:55.434","Text":"Like a typical vertical slice at a typical x,"},{"Start":"11:55.434 ","End":"12:01.180","Text":"would enter the region here and exit here."},{"Start":"12:01.180 ","End":"12:04.090","Text":"What I need at this point and this point, in other words,"},{"Start":"12:04.090 ","End":"12:08.545","Text":"I want the y as a function of x for here and here."},{"Start":"12:08.545 ","End":"12:13.090","Text":"I think it\u0027s fairly clear that this line from 0,"},{"Start":"12:13.090 ","End":"12:14.770","Text":"0 to 1, 2,"},{"Start":"12:14.770 ","End":"12:19.180","Text":"this is the line y equals 2x."},{"Start":"12:19.180 ","End":"12:20.905","Text":"I could explain."},{"Start":"12:20.905 ","End":"12:22.210","Text":"It goes through the origin,"},{"Start":"12:22.210 ","End":"12:25.495","Text":"so y equals some number times x,"},{"Start":"12:25.495 ","End":"12:28.255","Text":"and that number times 1 has to equal 2,"},{"Start":"12:28.255 ","End":"12:30.130","Text":"so y equals 2x."},{"Start":"12:30.130 ","End":"12:32.719","Text":"As for this line here,"},{"Start":"12:32.719 ","End":"12:35.040","Text":"this is just the x-axis,"},{"Start":"12:35.040 ","End":"12:37.725","Text":"so this is y equals 0."},{"Start":"12:37.725 ","End":"12:43.260","Text":"We\u0027ll be going from 0-2x as x travels from here to here,"},{"Start":"12:43.260 ","End":"12:45.210","Text":"which means from 0-1."},{"Start":"12:45.210 ","End":"12:51.120","Text":"This double integral will become the integral,"},{"Start":"12:51.120 ","End":"12:57.250","Text":"as I said, x goes from 0-1,"},{"Start":"12:57.250 ","End":"13:00.340","Text":"and that will be dx."},{"Start":"13:00.340 ","End":"13:04.790","Text":"Then for each x, y goes from 0-2x."},{"Start":"13:04.830 ","End":"13:12.100","Text":"That will be y goes from 0-2x, and that\u0027s dy."},{"Start":"13:12.100 ","End":"13:15.070","Text":"The function of x,"},{"Start":"13:15.070 ","End":"13:17.140","Text":"y to be integrated is this,"},{"Start":"13:17.140 ","End":"13:22.490","Text":"which is 1 minus x squared."},{"Start":"13:22.740 ","End":"13:30.025","Text":"Now, 1 minus x squared is a constant as far as y goes."},{"Start":"13:30.025 ","End":"13:34.990","Text":"I meant to highlight there because I\u0027m just going to work on the inner integral."},{"Start":"13:34.990 ","End":"13:38.065","Text":"As I was saying, 1 minus x squared is a constant."},{"Start":"13:38.065 ","End":"13:47.545","Text":"The integral of a constant times y is just equal to, I\u0027ll write this,"},{"Start":"13:47.545 ","End":"13:50.590","Text":"x goes from 0-1,"},{"Start":"13:50.590 ","End":"13:56.600","Text":"this part is just 1 minus x squared times y,"},{"Start":"14:00.660 ","End":"14:11.035","Text":"but taken between 0 and 2x, and then dx."},{"Start":"14:11.035 ","End":"14:13.720","Text":"I could have just computed this at the side,"},{"Start":"14:13.720 ","End":"14:15.790","Text":"but I\u0027m leaving it in here."},{"Start":"14:15.790 ","End":"14:21.460","Text":"This is easy to do in our heads because this thing is a constant."},{"Start":"14:21.460 ","End":"14:24.984","Text":"When I take y from 0-2x,"},{"Start":"14:24.984 ","End":"14:28.825","Text":"it\u0027s just 2x minus 0, it\u0027s just 2x."},{"Start":"14:28.825 ","End":"14:35.905","Text":"What we get is the integral from 0-1 of 1 minus x squared."},{"Start":"14:35.905 ","End":"14:42.670","Text":"As I said, the y in substitution just gives us 2x dx,"},{"Start":"14:42.670 ","End":"14:46.435","Text":"that\u0027s equal, I\u0027ll continue over here,"},{"Start":"14:46.435 ","End":"14:49.090","Text":"let\u0027s just open brackets,"},{"Start":"14:49.090 ","End":"14:52.795","Text":"from 0-1 of 2x"},{"Start":"14:52.795 ","End":"14:59.425","Text":"minus 2x cubed dx."},{"Start":"14:59.425 ","End":"15:03.220","Text":"This is equal to, integral of 2x is x squared,"},{"Start":"15:03.220 ","End":"15:06.595","Text":"the integral of this is 2x^4 over 4,"},{"Start":"15:06.595 ","End":"15:09.760","Text":"2 over 4, I can write as 1/2."},{"Start":"15:09.760 ","End":"15:14.650","Text":"This I want from 0-1."},{"Start":"15:14.650 ","End":"15:17.920","Text":"At 0, I get nothing, at 1,"},{"Start":"15:17.920 ","End":"15:21.625","Text":"I get 1 minus 1/2."},{"Start":"15:21.625 ","End":"15:24.445","Text":"This is equal to 1/2."},{"Start":"15:24.445 ","End":"15:28.870","Text":"The right-hand side is 1/2."},{"Start":"15:28.870 ","End":"15:30.930","Text":"1/2 does equal to 1/2."},{"Start":"15:30.930 ","End":"15:37.110","Text":"At least we verified Green\u0027s theorem for this example. We\u0027re done."}],"ID":8771},{"Watched":false,"Name":"Exercise 2","Duration":"15m 4s","ChapterTopicVideoID":8646,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this exercise, we\u0027re going to verify Green\u0027s theorem for 1 particular example."},{"Start":"00:04.770 ","End":"00:08.700","Text":"Just briefly, Green\u0027s theorem says that if we have a region"},{"Start":"00:08.700 ","End":"00:15.015","Text":"that\u0027s simply connected and bounded like R would be the inside,"},{"Start":"00:15.015 ","End":"00:22.650","Text":"what\u0027s between these curves and the area region,"},{"Start":"00:22.650 ","End":"00:26.670","Text":"and C would be all the way around."},{"Start":"00:26.670 ","End":"00:28.970","Text":"Like made up of 3 pieces."},{"Start":"00:28.970 ","End":"00:30.995","Text":"In fact, let\u0027s give them names."},{"Start":"00:30.995 ","End":"00:36.515","Text":"This piece is C_1, this will be C_2,"},{"Start":"00:36.515 ","End":"00:38.735","Text":"and this is C_3,"},{"Start":"00:38.735 ","End":"00:42.815","Text":"and we\u0027re going to first of all compute"},{"Start":"00:42.815 ","End":"00:49.234","Text":"the line integral along C in 3 pieces and get an answer,"},{"Start":"00:49.234 ","End":"00:52.040","Text":"and then we\u0027ll do the double integral on"},{"Start":"00:52.040 ","End":"00:56.940","Text":"the region of this and hopefully we\u0027ll get the same answer."},{"Start":"00:57.680 ","End":"01:01.790","Text":"Let\u0027s begin with the more difficult one,"},{"Start":"01:01.790 ","End":"01:03.905","Text":"which is the line integral."},{"Start":"01:03.905 ","End":"01:08.210","Text":"Basically we\u0027re going to say that the integral of the closed curve"},{"Start":"01:08.210 ","End":"01:14.285","Text":"C is equal to the integral of the separate pieces added together,"},{"Start":"01:14.285 ","End":"01:20.125","Text":"integral of a C_1 plus the integral of a C_2 plus the integral of a C_3,"},{"Start":"01:20.125 ","End":"01:22.380","Text":"and to do this,"},{"Start":"01:22.380 ","End":"01:24.560","Text":"you want to parameterize these,"},{"Start":"01:24.560 ","End":"01:26.630","Text":"then it\u0027s easier to compute it."},{"Start":"01:26.630 ","End":"01:33.785","Text":"Let\u0027s parameterize each of the 3 pieces. We\u0027ll start with C_1."},{"Start":"01:33.785 ","End":"01:42.260","Text":"C_1, whenever you have 1 variable as a function of the other here y is a function of x."},{"Start":"01:42.260 ","End":"01:46.100","Text":"The independent variable will just be t,"},{"Start":"01:46.100 ","End":"01:52.130","Text":"and the dependent variable is just according to the formula here it\u0027s x squared,"},{"Start":"01:52.130 ","End":"01:53.615","Text":"so it\u0027s t squared,"},{"Start":"01:53.615 ","End":"01:57.480","Text":"and let\u0027s see where does t go from and to."},{"Start":"01:57.590 ","End":"01:59.840","Text":"We\u0027ll be doing this in our heads."},{"Start":"01:59.840 ","End":"02:02.645","Text":"If y is 1, then x squared is 1,"},{"Start":"02:02.645 ","End":"02:05.960","Text":"and if x squared is 1 and x is positive,"},{"Start":"02:05.960 ","End":"02:08.210","Text":"then x is 1."},{"Start":"02:08.210 ","End":"02:15.200","Text":"What we\u0027re going to get is t goes between 0 and 1."},{"Start":"02:15.200 ","End":"02:18.885","Text":"That\u0027s it for C_1. Now, C_2."},{"Start":"02:18.885 ","End":"02:21.045","Text":"C_2 Is this bit."},{"Start":"02:21.045 ","End":"02:24.650","Text":"There\u0027s a standard way of doing this for line segments."},{"Start":"02:24.650 ","End":"02:31.220","Text":"If we have the start point is 1, 1."},{"Start":"02:31.220 ","End":"02:38.415","Text":"The end point here is 0, 1,"},{"Start":"02:38.415 ","End":"02:41.585","Text":"and it might as well write the origin also"},{"Start":"02:41.585 ","End":"02:45.420","Text":"because we\u0027re going to need it for the next bit that\u0027s 0,"},{"Start":"02:45.420 ","End":"02:47.540","Text":"0 is the origin, of course."},{"Start":"02:47.540 ","End":"02:51.035","Text":"Anyway, in this situation,"},{"Start":"02:51.035 ","End":"02:53.930","Text":"I like to work on both bits simultaneously."},{"Start":"02:53.930 ","End":"02:59.740","Text":"What we do is we take the start of x and y respectively."},{"Start":"02:59.740 ","End":"03:01.460","Text":"The start for x is 1,"},{"Start":"03:01.460 ","End":"03:03.205","Text":"the start for y is 1."},{"Start":"03:03.205 ","End":"03:07.115","Text":"Then we add t times,"},{"Start":"03:07.115 ","End":"03:11.180","Text":"in each case, the end minus the start,"},{"Start":"03:11.180 ","End":"03:18.985","Text":"so for x, the end is 0 and the start is 1, and for y,"},{"Start":"03:18.985 ","End":"03:23.220","Text":"the end is 1 and the start is 1,"},{"Start":"03:23.220 ","End":"03:33.180","Text":"and it\u0027s always t goes between 0 and 1 when we do it using this method."},{"Start":"03:33.180 ","End":"03:36.590","Text":"Just to simplify this a bit."},{"Start":"03:36.590 ","End":"03:38.930","Text":"I\u0027ll write this part again."},{"Start":"03:38.930 ","End":"03:44.595","Text":"That x is equal to 1 minus t,"},{"Start":"03:44.595 ","End":"03:48.620","Text":"and y is equal to 1 plus t times 0,"},{"Start":"03:48.620 ","End":"03:55.655","Text":"y equals 1, and that\u0027s pretty clear because y is on the curve, y equals 1."},{"Start":"03:55.655 ","End":"04:01.755","Text":"That\u0027s C_2, and now the third bit C_3,"},{"Start":"04:01.755 ","End":"04:04.990","Text":"using the same method."},{"Start":"04:04.990 ","End":"04:07.955","Text":"We take the start."},{"Start":"04:07.955 ","End":"04:11.585","Text":"In this case the start is 0, 1,"},{"Start":"04:11.585 ","End":"04:16.535","Text":"and then I add t times something minus something,"},{"Start":"04:16.535 ","End":"04:20.734","Text":"t times something minus something and minus start."},{"Start":"04:20.734 ","End":"04:27.520","Text":"For x, the end and the start are both 0,"},{"Start":"04:27.710 ","End":"04:33.224","Text":"and for y the end is 0 the start is 1,"},{"Start":"04:33.224 ","End":"04:38.535","Text":"and t goes from 0 to 1,"},{"Start":"04:38.535 ","End":"04:41.840","Text":"again rewrite this more neatly,"},{"Start":"04:41.840 ","End":"04:45.095","Text":"that x is equal to, well, it\u0027s just 0."},{"Start":"04:45.095 ","End":"04:48.695","Text":"Of course it\u0027s 0 because we\u0027re traveling along the y axis,"},{"Start":"04:48.695 ","End":"04:53.510","Text":"and y is equal to 1 minus t,"},{"Start":"04:53.510 ","End":"04:57.850","Text":"so each of these 3 is parameterized."},{"Start":"04:57.850 ","End":"04:59.850","Text":"But when we substitute,"},{"Start":"04:59.850 ","End":"05:01.560","Text":"we\u0027ll also need dx and dy,"},{"Start":"05:01.560 ","End":"05:05.265","Text":"so let\u0027s get that out of the way now."},{"Start":"05:05.265 ","End":"05:12.345","Text":"For C_1, I will get that dx is"},{"Start":"05:12.345 ","End":"05:19.275","Text":"equal to dt and dy equals 2t,"},{"Start":"05:19.275 ","End":"05:22.405","Text":"a derivative of that, dt."},{"Start":"05:22.405 ","End":"05:30.500","Text":"Over here we\u0027ll get that dx is equal to the derivative, this is minus 1,"},{"Start":"05:30.500 ","End":"05:32.740","Text":"so it\u0027s just minus dt,"},{"Start":"05:32.740 ","End":"05:34.829","Text":"y is a constant,"},{"Start":"05:34.829 ","End":"05:36.360","Text":"so dy is 0,"},{"Start":"05:36.360 ","End":"05:38.609","Text":"dt, or just plain 0."},{"Start":"05:38.609 ","End":"05:41.340","Text":"Here, dx is just 0,"},{"Start":"05:41.340 ","End":"05:45.480","Text":"and here dy is equal to derivative,"},{"Start":"05:45.480 ","End":"05:49.290","Text":"this is minus 1 times dt minus dt."},{"Start":"05:49.290 ","End":"05:52.175","Text":"Okay. That\u0027s all this stuff out of the way."},{"Start":"05:52.175 ","End":"05:56.190","Text":"Now let\u0027s take them piece by piece."},{"Start":"05:59.450 ","End":"06:03.855","Text":"I meant to point out we\u0027re working on C_1."},{"Start":"06:03.855 ","End":"06:11.700","Text":"Yeah. Look at the range from 0 to 1 for the parameter,"},{"Start":"06:11.700 ","End":"06:14.730","Text":"and then I look at the formula."},{"Start":"06:14.730 ","End":"06:16.680","Text":"This part, by the way,"},{"Start":"06:16.680 ","End":"06:22.155","Text":"is the f from the formula"},{"Start":"06:22.155 ","End":"06:28.500","Text":"and I can write it as 1dy and that 1 is the g from the formula,"},{"Start":"06:28.500 ","End":"06:30.555","Text":"I\u0027m just mentioning, we\u0027ll use that later."},{"Start":"06:30.555 ","End":"06:34.769","Text":"Anyway, let\u0027s see what f is."},{"Start":"06:34.769 ","End":"06:40.715","Text":"For C_1, we have x minus y squared."},{"Start":"06:40.715 ","End":"06:46.025","Text":"It\u0027s t minus t squared squared."},{"Start":"06:46.025 ","End":"06:52.130","Text":"That will be t minus t squared squared"},{"Start":"06:52.130 ","End":"06:58.920","Text":"is t^4 and dx equals dt."},{"Start":"07:00.170 ","End":"07:03.045","Text":"That\u0027s just the first bit,"},{"Start":"07:03.045 ","End":"07:06.390","Text":"and now we need the dy bit,"},{"Start":"07:06.390 ","End":"07:11.830","Text":"so plus dy is 2tdt."},{"Start":"07:14.130 ","End":"07:18.250","Text":"This is the integral we have to evaluate."},{"Start":"07:18.250 ","End":"07:21.745","Text":"This is equal to,"},{"Start":"07:21.745 ","End":"07:26.365","Text":"let\u0027s see if I just collect all the terms together."},{"Start":"07:26.365 ","End":"07:32.155","Text":"I\u0027ve got t plus 2t is 3t,"},{"Start":"07:32.155 ","End":"07:38.800","Text":"minus t to 4th dt from 0-1."},{"Start":"07:38.800 ","End":"07:45.565","Text":"This is equal to 3t squared over"},{"Start":"07:45.565 ","End":"07:53.440","Text":"2 minus t to the 5th/5, from 0-1."},{"Start":"07:53.440 ","End":"07:57.940","Text":"I just need to plug in the 1 because 0 gives nothing."},{"Start":"07:57.940 ","End":"08:00.970","Text":"What I get is"},{"Start":"08:00.970 ","End":"08:08.920","Text":"3/2 minus 1/5."},{"Start":"08:08.920 ","End":"08:10.975","Text":"Let\u0027s see, I\u0027ll think of it in decimals."},{"Start":"08:10.975 ","End":"08:17.960","Text":"1.5 minus 0.2 is 1.3."},{"Start":"08:18.540 ","End":"08:23.200","Text":"I\u0027ll write it as 13/10,"},{"Start":"08:23.200 ","End":"08:25.930","Text":"and that\u0027s the first part for C_1."},{"Start":"08:25.930 ","End":"08:28.450","Text":"Maybe I\u0027ll highlight that,"},{"Start":"08:28.450 ","End":"08:30.430","Text":"and I can find it later."},{"Start":"08:30.430 ","End":"08:33.740","Text":"Now let\u0027s go on to C_2,"},{"Start":"08:33.750 ","End":"08:39.445","Text":"and here also it\u0027s from 0-1 for the parameter."},{"Start":"08:39.445 ","End":"08:41.545","Text":"What I want is,"},{"Start":"08:41.545 ","End":"08:44.900","Text":"x minus y squared."},{"Start":"08:45.570 ","End":"08:49.270","Text":"Well, x is 1 minus t,"},{"Start":"08:49.270 ","End":"08:51.100","Text":"and y squared is 1,"},{"Start":"08:51.100 ","End":"08:59.289","Text":"so x minus y squared is going to be minus t. Then I need dx,"},{"Start":"08:59.289 ","End":"09:01.460","Text":"which is minus dt."},{"Start":"09:03.480 ","End":"09:08.990","Text":"The second part is plus dy, which is 0."},{"Start":"09:09.420 ","End":"09:18.080","Text":"This is just the integral from 0-1 of t dt."},{"Start":"09:18.330 ","End":"09:22.900","Text":"Let\u0027s see, we can do this in our head, well, maybe not,"},{"Start":"09:22.900 ","End":"09:29.240","Text":"t squared over 2 from 0-1."},{"Start":"09:29.240 ","End":"09:31.260","Text":"That\u0027s 0, we get 0,"},{"Start":"09:31.260 ","End":"09:34.575","Text":"1 we get 1/2, this is just 1.5."},{"Start":"09:34.575 ","End":"09:39.365","Text":"I\u0027ll highlight that, 2 down. 1 to go."},{"Start":"09:39.365 ","End":"09:43.930","Text":"C_3 also from 0-1,"},{"Start":"09:43.930 ","End":"09:51.220","Text":"we need x minus y squared, and then dx."},{"Start":"09:51.220 ","End":"09:53.679","Text":"But dx is 0,"},{"Start":"09:53.679 ","End":"09:55.465","Text":"so that saves us a bit."},{"Start":"09:55.465 ","End":"09:57.850","Text":"This whole first bit is 0."},{"Start":"09:57.850 ","End":"09:59.410","Text":"Let\u0027s see, plus dy,"},{"Start":"09:59.410 ","End":"10:03.280","Text":"dy is minus dt,"},{"Start":"10:03.280 ","End":"10:05.860","Text":"so it\u0027s just minus dt."},{"Start":"10:05.860 ","End":"10:11.750","Text":"The integral of minus 1,"},{"Start":"10:12.210 ","End":"10:19.390","Text":"is just minus t. I need that from 0-1,"},{"Start":"10:19.390 ","End":"10:24.235","Text":"and so I get minus 1."},{"Start":"10:24.235 ","End":"10:26.140","Text":"I\u0027ll highlight that."},{"Start":"10:26.140 ","End":"10:28.870","Text":"Now I\u0027ve got the 3 pieces,"},{"Start":"10:28.870 ","End":"10:31.600","Text":"so I\u0027m looking here, I have to add them."},{"Start":"10:31.600 ","End":"10:36.655","Text":"I\u0027ve got the integral along the closed curve C,"},{"Start":"10:36.655 ","End":"10:39.040","Text":"made up of the 3 bits."},{"Start":"10:39.040 ","End":"10:40.870","Text":"I didn\u0027t write what\u0027s here,"},{"Start":"10:40.870 ","End":"10:43.250","Text":"it\u0027s just whatever\u0027s here."},{"Start":"10:46.860 ","End":"10:54.430","Text":"C_1, 13/10 plus 1/2,"},{"Start":"10:54.430 ","End":"10:57.850","Text":"and then C_3 is minus 1."},{"Start":"10:57.850 ","End":"11:05.785","Text":"What do I get? 1/3/10 minus 1/3/10 plus 1/2 is 5/10."},{"Start":"11:05.785 ","End":"11:08.560","Text":"3 times 5, 8/10,"},{"Start":"11:08.560 ","End":"11:11.240","Text":"4/5 is what I make it."},{"Start":"11:12.000 ","End":"11:14.845","Text":"This is the left-hand side,"},{"Start":"11:14.845 ","End":"11:16.360","Text":"in fact I\u0027ll write it in here,"},{"Start":"11:16.360 ","End":"11:20.140","Text":"that the left-hand side gave me 4/5."},{"Start":"11:20.140 ","End":"11:22.690","Text":"That\u0027s quite a bit of work."},{"Start":"11:22.690 ","End":"11:28.600","Text":"Let\u0027s now use Green\u0027s theorem and evaluate the right-hand side,"},{"Start":"11:28.600 ","End":"11:32.000","Text":"and hope that we also get 4/5."},{"Start":"11:32.070 ","End":"11:35.845","Text":"I\u0027m going to erase everything I don\u0027t need."},{"Start":"11:35.845 ","End":"11:41.229","Text":"Here we are, and I think it\u0027ll be nice to shade the region."},{"Start":"11:41.229 ","End":"11:44.020","Text":"We want to compute this double integral."},{"Start":"11:44.020 ","End":"11:45.880","Text":"Let me first compute the integral,"},{"Start":"11:45.880 ","End":"11:49.495","Text":"and that\u0027s this g_x minus f _y,"},{"Start":"11:49.495 ","End":"11:50.845","Text":"and see what we get."},{"Start":"11:50.845 ","End":"11:56.155","Text":"G with respect to x minus f with respect to y is equal to,"},{"Start":"11:56.155 ","End":"12:03.040","Text":"let\u0027s see, g with respect to x is 1, derivative is 0."},{"Start":"12:03.040 ","End":"12:11.815","Text":"F with respect to y is minus 2y,"},{"Start":"12:11.815 ","End":"12:15.430","Text":"and so we just get 2y,"},{"Start":"12:15.430 ","End":"12:18.080","Text":"so that\u0027s this bit here."},{"Start":"12:18.080 ","End":"12:20.820","Text":"Now, the double integral,"},{"Start":"12:20.820 ","End":"12:22.979","Text":"I want to do is an iterated integral."},{"Start":"12:22.979 ","End":"12:26.100","Text":"Since we have already y extracted in terms of x,"},{"Start":"12:26.100 ","End":"12:27.510","Text":"we have y equals this,"},{"Start":"12:27.510 ","End":"12:31.175","Text":"y equals this, we\u0027ll do it as a Type 1 region."},{"Start":"12:31.175 ","End":"12:34.480","Text":"I see that x goes from 0-1,"},{"Start":"12:34.480 ","End":"12:36.100","Text":"and for each particular x,"},{"Start":"12:36.100 ","End":"12:41.350","Text":"if I take a vertical slice through the region,"},{"Start":"12:41.350 ","End":"12:44.665","Text":"it enters here and exit here."},{"Start":"12:44.665 ","End":"12:47.530","Text":"I know what these are, if this is x,"},{"Start":"12:47.530 ","End":"12:49.810","Text":"I know that this point is x squared,"},{"Start":"12:49.810 ","End":"12:52.570","Text":"and this point is just 1."},{"Start":"12:52.570 ","End":"12:58.450","Text":"Putting this together, what we get is the integral,"},{"Start":"12:58.450 ","End":"13:00.970","Text":"x goes from 0-1,"},{"Start":"13:00.970 ","End":"13:03.565","Text":"that\u0027s the outer integral."},{"Start":"13:03.565 ","End":"13:07.330","Text":"For each let\u0027s say it would be dx."},{"Start":"13:07.330 ","End":"13:12.370","Text":"Then for each x, we have that y goes from"},{"Start":"13:12.370 ","End":"13:17.620","Text":"x squared to 1, that\u0027s dy."},{"Start":"13:17.620 ","End":"13:22.280","Text":"Then this bit we have computed here is 2y."},{"Start":"13:22.860 ","End":"13:27.290","Text":"That\u0027s straightforward enough."},{"Start":"13:33.300 ","End":"13:38.290","Text":"I should have said I\u0027m doing the inner integral first."},{"Start":"13:38.290 ","End":"13:44.785","Text":"For 2y, dy, I could do this bit at the side and just say,"},{"Start":"13:44.785 ","End":"13:50.650","Text":"integral from x squared to 1 of 2y,"},{"Start":"13:50.650 ","End":"13:55.645","Text":"dy, is just, 2y gives me y squared."},{"Start":"13:55.645 ","End":"14:00.445","Text":"I need to do this from x squared to 1."},{"Start":"14:00.445 ","End":"14:03.490","Text":"You plug in 1, we get 1 squared,"},{"Start":"14:03.490 ","End":"14:05.065","Text":"and that\u0027s just 1."},{"Start":"14:05.065 ","End":"14:08.065","Text":"Plug in x squared to y,"},{"Start":"14:08.065 ","End":"14:11.964","Text":"and I get x squared squared is x to 4th,"},{"Start":"14:11.964 ","End":"14:13.735","Text":"so that\u0027s what this is."},{"Start":"14:13.735 ","End":"14:16.210","Text":"Then I put it back here,"},{"Start":"14:16.210 ","End":"14:20.470","Text":"I get the integral from 0-1 of this,"},{"Start":"14:20.470 ","End":"14:25.405","Text":"1 minus x to the 4th, dx."},{"Start":"14:25.405 ","End":"14:27.580","Text":"This is equal to,"},{"Start":"14:27.580 ","End":"14:29.860","Text":"the integral of 1 is x,"},{"Start":"14:29.860 ","End":"14:32.050","Text":"integral of x to the 4th,"},{"Start":"14:32.050 ","End":"14:35.965","Text":"is x to the 5th/5."},{"Start":"14:35.965 ","End":"14:38.380","Text":"I want this from 0-1."},{"Start":"14:38.380 ","End":"14:39.940","Text":"0 doesn\u0027t give me anything,"},{"Start":"14:39.940 ","End":"14:41.095","Text":"is 0 and 0."},{"Start":"14:41.095 ","End":"14:44.320","Text":"I put in 1, I get 1 minus 1/5,"},{"Start":"14:44.320 ","End":"14:48.460","Text":"and this is equal to 4/5."},{"Start":"14:48.460 ","End":"14:50.455","Text":"I write here 4/5,"},{"Start":"14:50.455 ","End":"14:53.900","Text":"and then I immediately see that this is equal to this,"},{"Start":"14:53.900 ","End":"14:56.930","Text":"so this is equal to this, yes."},{"Start":"14:56.930 ","End":"15:00.230","Text":"That\u0027s what I\u0027d call verifying."},{"Start":"15:00.230 ","End":"15:02.180","Text":"Yeah, it\u0027s indeed equal,"},{"Start":"15:02.180 ","End":"15:04.800","Text":"and so we are done."}],"ID":8772},{"Watched":false,"Name":"Exercise 3","Duration":"23m 10s","ChapterTopicVideoID":8647,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.135","Text":"In this exercise, we\u0027re going to verify Green\u0027s theorem for a particular example."},{"Start":"00:06.135 ","End":"00:11.430","Text":"Green\u0027s theorem equates a line integral over"},{"Start":"00:11.430 ","End":"00:14.280","Text":"a closed curve with a double integral over"},{"Start":"00:14.280 ","End":"00:18.360","Text":"a region where that region has certain conditions."},{"Start":"00:18.360 ","End":"00:27.960","Text":"It\u0027s bounded and simply connected and basically that\u0027s what it is,"},{"Start":"00:27.960 ","End":"00:34.850","Text":"and C goes around the border of it in a counterclockwise fashion."},{"Start":"00:34.850 ","End":"00:40.370","Text":"We\u0027re going to verify it for the particular example of this line integral."},{"Start":"00:40.370 ","End":"00:42.200","Text":"This will be our curve."},{"Start":"00:42.200 ","End":"00:45.780","Text":"It\u0027s probably best I give a sketch."},{"Start":"00:45.910 ","End":"00:52.865","Text":"The axes. Let\u0027s take the point in order 0,0 2,0."},{"Start":"00:52.865 ","End":"00:56.030","Text":"Let\u0027s say this is 2 here."},{"Start":"00:56.030 ","End":"01:01.740","Text":"Then we go up to 2,2 could be here,"},{"Start":"01:01.740 ","End":"01:03.615","Text":"that would be 2 here."},{"Start":"01:03.615 ","End":"01:07.290","Text":"Then to 0,2 here,"},{"Start":"01:07.290 ","End":"01:09.805","Text":"and then back again."},{"Start":"01:09.805 ","End":"01:11.990","Text":"I traced it out,"},{"Start":"01:11.990 ","End":"01:16.190","Text":"and let\u0027s put some arrows because we\u0027re going in the positive direction,"},{"Start":"01:16.190 ","End":"01:23.390","Text":"and that\u0027s our curve C. I\u0027d like to start out by doing the left-hand side,"},{"Start":"01:23.390 ","End":"01:26.270","Text":"which is to compute the integral of,"},{"Start":"01:26.270 ","End":"01:28.520","Text":"in our case, it\u0027s what?"},{"Start":"01:28.520 ","End":"01:34.160","Text":"X squared minus xy cubed, dx."},{"Start":"01:34.160 ","End":"01:39.425","Text":"This is like the part which is f of xy,"},{"Start":"01:39.425 ","End":"01:46.430","Text":"and the other bit is y squared minus 2xy dy."},{"Start":"01:46.430 ","End":"01:51.000","Text":"In a specific case and in general,"},{"Start":"01:51.000 ","End":"01:54.790","Text":"this is g of xy."},{"Start":"01:57.470 ","End":"02:02.250","Text":"To do this is break it up into 4 separate pieces."},{"Start":"02:02.250 ","End":"02:06.015","Text":"Let\u0027s say that this is C_1,"},{"Start":"02:06.015 ","End":"02:10.380","Text":"this bit is C_2,"},{"Start":"02:10.380 ","End":"02:14.085","Text":"this bit will be C_3,"},{"Start":"02:14.085 ","End":"02:17.620","Text":"and this bit will be C_4."},{"Start":"02:18.170 ","End":"02:20.210","Text":"After we\u0027ve done this,"},{"Start":"02:20.210 ","End":"02:26.660","Text":"we\u0027ll record the answer and then we\u0027ll go and do the double integral over the region R,"},{"Start":"02:26.660 ","End":"02:30.420","Text":"which will be the square with its interior."},{"Start":"02:30.890 ","End":"02:33.305","Text":"First of all, this,"},{"Start":"02:33.305 ","End":"02:34.640","Text":"so this is going to,"},{"Start":"02:34.640 ","End":"02:36.380","Text":"as I said, equal 4 parts,"},{"Start":"02:36.380 ","End":"02:42.710","Text":"the integral over C_1 of the same plus the integral over C_2 plus the integral over"},{"Start":"02:42.710 ","End":"02:51.130","Text":"C_3 plus the integral over C_4 for calculations and an addition at the end."},{"Start":"02:51.130 ","End":"02:54.330","Text":"Only later we\u0027ll do the right-hand side."},{"Start":"02:54.330 ","End":"02:58.080","Text":"Let\u0027s parametrize each bit and see."},{"Start":"02:58.080 ","End":"03:03.245","Text":"C_1 can be parametrized as follows."},{"Start":"03:03.245 ","End":"03:07.940","Text":"There is a formula for the parameterized line segment between 2 points,"},{"Start":"03:07.940 ","End":"03:09.185","Text":"but in this case,"},{"Start":"03:09.185 ","End":"03:13.200","Text":"we can take a shortcut, do it directly."},{"Start":"03:13.240 ","End":"03:21.255","Text":"First of all, y stays 0 and x goes from 0 to 2."},{"Start":"03:21.255 ","End":"03:24.340","Text":"This here is 0."},{"Start":"03:24.530 ","End":"03:31.530","Text":"I could say that x equals t,"},{"Start":"03:31.530 ","End":"03:39.840","Text":"and then I\u0027ll write in a moment that t goes from 0 to 2 and that y is equal to just 0."},{"Start":"03:40.270 ","End":"03:45.680","Text":"Like I said, 0 to 2 is where t goes."},{"Start":"03:45.680 ","End":"03:48.560","Text":"Similarly, the curve C_2,"},{"Start":"03:48.560 ","End":"03:52.175","Text":"there\u0027s a very easy way to express that."},{"Start":"03:52.175 ","End":"04:00.315","Text":"Here x remains constant at 2 and y goes from 0 to 2,"},{"Start":"04:00.315 ","End":"04:05.465","Text":"so y is the parameter t and t goes from 0 to 2."},{"Start":"04:05.465 ","End":"04:07.795","Text":"That\u0027s one way of doing it."},{"Start":"04:07.795 ","End":"04:11.825","Text":"Here because we\u0027re going in the other direction."},{"Start":"04:11.825 ","End":"04:17.780","Text":"I could use the formula for the line segment between 2 points."},{"Start":"04:17.780 ","End":"04:27.140","Text":"This is the point 2,2 and this is the point 0,2."},{"Start":"04:27.140 ","End":"04:32.165","Text":"What we can say about C_3,"},{"Start":"04:32.165 ","End":"04:38.975","Text":"we can use the formula that the x,"},{"Start":"04:38.975 ","End":"04:45.005","Text":"I\u0027ll do it with the curly braces that x and y are,"},{"Start":"04:45.005 ","End":"04:48.835","Text":"first of all, the starting point which is 2,2."},{"Start":"04:48.835 ","End":"04:53.990","Text":"Then I put t times the end minus the start."},{"Start":"04:53.990 ","End":"04:57.395","Text":"In each case I\u0027ll have end minus start."},{"Start":"04:57.395 ","End":"05:02.090","Text":"For x, the end"},{"Start":"05:02.090 ","End":"05:08.340","Text":"is 0 and the start is 2,"},{"Start":"05:08.340 ","End":"05:11.100","Text":"so it\u0027s 0 minus 2."},{"Start":"05:11.100 ","End":"05:14.774","Text":"For y it remains at 2,"},{"Start":"05:14.774 ","End":"05:18.065","Text":"and so if I simplify it,"},{"Start":"05:18.065 ","End":"05:20.255","Text":"it just comes out to be that x"},{"Start":"05:20.255 ","End":"05:30.180","Text":"is 2 minus 2t."},{"Start":"05:30.180 ","End":"05:33.010","Text":"y is just equal to 2,"},{"Start":"05:33.010 ","End":"05:36.785","Text":"which is what we expected because y is a constant."},{"Start":"05:36.785 ","End":"05:42.230","Text":"For C_4, using the same idea,"},{"Start":"05:42.230 ","End":"05:49.065","Text":"I\u0027ll get that, let\u0027s see what x is equal and what is y equal."},{"Start":"05:49.065 ","End":"05:55.670","Text":"The start point, which is 0,2,"},{"Start":"05:55.670 ","End":"06:00.190","Text":"and then t times,"},{"Start":"06:00.190 ","End":"06:05.780","Text":"end minus start, t times end minus start."},{"Start":"06:05.780 ","End":"06:12.255","Text":"For x, I start at 0 and end at 0,"},{"Start":"06:12.255 ","End":"06:16.890","Text":"but y goes from 2 to 0,"},{"Start":"06:16.890 ","End":"06:20.070","Text":"so it\u0027s 0 minus 2,"},{"Start":"06:20.070 ","End":"06:22.390","Text":"the end minus the start."},{"Start":"06:22.390 ","End":"06:31.530","Text":"So x is just equal to 0 and y is equal to 2 minus 2t."},{"Start":"06:31.970 ","End":"06:34.445","Text":"When we use this formula,"},{"Start":"06:34.445 ","End":"06:38.195","Text":"the parameter is always from 0 to 1."},{"Start":"06:38.195 ","End":"06:42.650","Text":"So for here it goes from 0 to 1."},{"Start":"06:42.650 ","End":"06:44.240","Text":"Now in each of these,"},{"Start":"06:44.240 ","End":"06:48.020","Text":"I\u0027m also going to be substituting besides x and y,"},{"Start":"06:48.020 ","End":"06:50.975","Text":"which I\u0027ll take from the parametrized form,"},{"Start":"06:50.975 ","End":"06:53.195","Text":"I\u0027ll also need dx and dy,"},{"Start":"06:53.195 ","End":"06:56.555","Text":"so it might as well get that technical stuff out of the way."},{"Start":"06:56.555 ","End":"06:59.540","Text":"Let\u0027s see what dx is equal to here."},{"Start":"06:59.540 ","End":"07:05.610","Text":"I\u0027ll just write this 4 times,"},{"Start":"07:05.610 ","End":"07:09.210","Text":"and then we\u0027ll see what dy equals also."},{"Start":"07:09.210 ","End":"07:13.275","Text":"dy equals, dy equals,"},{"Start":"07:13.275 ","End":"07:17.995","Text":"dy equals, sometimes I do it like production line."},{"Start":"07:17.995 ","End":"07:21.770","Text":"Let\u0027s see, x equals t,"},{"Start":"07:21.770 ","End":"07:23.915","Text":"so dx equals dt,"},{"Start":"07:23.915 ","End":"07:27.560","Text":"y is 0, so dy is 0,"},{"Start":"07:27.560 ","End":"07:29.585","Text":"dt or just plain 0."},{"Start":"07:29.585 ","End":"07:31.429","Text":"Here, x is constant,"},{"Start":"07:31.429 ","End":"07:35.890","Text":"so dx is 0, dy is dt."},{"Start":"07:35.890 ","End":"07:39.840","Text":"From here, I\u0027ve got that dx,"},{"Start":"07:39.840 ","End":"07:44.530","Text":"is this bit minus 2dt,"},{"Start":"07:45.500 ","End":"07:48.270","Text":"but y is a constant,"},{"Start":"07:48.270 ","End":"07:50.670","Text":"so dy is 0."},{"Start":"07:50.670 ","End":"07:59.380","Text":"Here dx is 0 and dy is minus 2dt."},{"Start":"07:59.390 ","End":"08:04.170","Text":"That\u0027s about everything we need for the substitution."},{"Start":"08:04.170 ","End":"08:07.910","Text":"Let\u0027s get some space here."},{"Start":"08:10.470 ","End":"08:16.090","Text":"For C_1, I get the integral."},{"Start":"08:16.090 ","End":"08:18.160","Text":"This is just a regular integral,"},{"Start":"08:18.160 ","End":"08:23.320","Text":"not a closed curve integral of whatever it says here,"},{"Start":"08:23.320 ","End":"08:29.310","Text":"so need x squared minus xy cubed from C_1,"},{"Start":"08:29.310 ","End":"08:34.990","Text":"x squared minus xy cube."},{"Start":"08:34.990 ","End":"08:41.675","Text":"Now y is 0, so this is just x squared which is t squared,"},{"Start":"08:41.675 ","End":"08:45.900","Text":"and dx is dt."},{"Start":"08:45.900 ","End":"08:46.935","Text":"That\u0027s the first bit."},{"Start":"08:46.935 ","End":"08:50.070","Text":"The second bit, as we said,"},{"Start":"08:50.070 ","End":"08:56.240","Text":"dy is 0, so this whole second part is 0."},{"Start":"08:56.240 ","End":"09:01.450","Text":"This doesn\u0027t appear also the parameter goes from 0-2."},{"Start":"09:01.450 ","End":"09:04.060","Text":"This is what C_1 is,"},{"Start":"09:04.060 ","End":"09:12.955","Text":"and this is equal to 1/3 of t cubed from 0-2,"},{"Start":"09:12.955 ","End":"09:18.790","Text":"2 cubed is 8, so it\u0027s just 8 over 3 for the first one."},{"Start":"09:18.790 ","End":"09:22.780","Text":"Now let\u0027s do C_2."},{"Start":"09:22.780 ","End":"09:31.490","Text":"C_2 is also from 0-2 and this time,"},{"Start":"09:31.530 ","End":"09:34.090","Text":"what do we get?"},{"Start":"09:34.090 ","End":"09:37.704","Text":"We need x squared minus xy cubed dx"},{"Start":"09:37.704 ","End":"09:39.625","Text":"but here dx is 0,"},{"Start":"09:39.625 ","End":"09:42.534","Text":"so the whole first part is not needed."},{"Start":"09:42.534 ","End":"09:48.610","Text":"We just need the second part where dy is equal to dt,"},{"Start":"09:48.610 ","End":"09:50.170","Text":"so we\u0027ve got something dt."},{"Start":"09:50.170 ","End":"09:57.010","Text":"Now y squared is t squared,"},{"Start":"09:57.010 ","End":"10:02.275","Text":"and 2xy is 2,"},{"Start":"10:02.275 ","End":"10:04.645","Text":"2t, which is 4t,"},{"Start":"10:04.645 ","End":"10:06.340","Text":"so it\u0027s minus 4t."},{"Start":"10:06.340 ","End":"10:10.580","Text":"As I said dy is dt."},{"Start":"10:10.620 ","End":"10:17.755","Text":"This is equal to integral is 1/3 of t cubed."},{"Start":"10:17.755 ","End":"10:21.520","Text":"Here we\u0027ll get minus 4t squared over 2,"},{"Start":"10:21.520 ","End":"10:28.675","Text":"2t squared, from 0-2, 0 gives nothing."},{"Start":"10:28.675 ","End":"10:34.930","Text":"2 gives us 8 over 3 minus 2,"},{"Start":"10:34.930 ","End":"10:39.500","Text":"2 squared is 8, minus 8."},{"Start":"10:39.720 ","End":"10:44.380","Text":"I\u0027ll leave the computation for later when I add them all up."},{"Start":"10:44.380 ","End":"10:50.755","Text":"Next, maybe I can fit everything into this page."},{"Start":"10:50.755 ","End":"10:54.110","Text":"Let\u0027s put a separator here."},{"Start":"10:54.210 ","End":"11:01.060","Text":"For C_3, this integral is,"},{"Start":"11:01.060 ","End":"11:05.210","Text":"first of all, look at what the range is from 0-1."},{"Start":"11:07.170 ","End":"11:13.360","Text":"Then I need to compute x squared minus xy cubed. Where am I?"},{"Start":"11:13.360 ","End":"11:19.570","Text":"Here. X squared is"},{"Start":"11:19.570 ","End":"11:27.640","Text":"2 minus 2t squared minus xy cubed,"},{"Start":"11:27.640 ","End":"11:34.660","Text":"x is 2 minus 2t,"},{"Start":"11:34.660 ","End":"11:40.300","Text":"and y cubed is 8,"},{"Start":"11:40.300 ","End":"11:44.335","Text":"because y is 2 times 8,"},{"Start":"11:44.335 ","End":"11:50.740","Text":"and dx here is minus 2dt."},{"Start":"11:50.740 ","End":"11:57.220","Text":"All this times minus 2dt,"},{"Start":"11:57.220 ","End":"12:03.190","Text":"but luckily, the second part dy is 0,"},{"Start":"12:03.190 ","End":"12:05.575","Text":"so that\u0027s all we need."},{"Start":"12:05.575 ","End":"12:09.310","Text":"What does this come out to be?"},{"Start":"12:09.310 ","End":"12:15.445","Text":"Need a bit of computation here. You know what?"},{"Start":"12:15.445 ","End":"12:19.180","Text":"May be an overkill but I\u0027d like to do it with a substitution to let"},{"Start":"12:19.180 ","End":"12:23.890","Text":"2 minus 2t equal u. I think it\u0027ll come out nicely."},{"Start":"12:23.890 ","End":"12:30.415","Text":"For here, I\u0027ll just say I\u0027m going to substitute u equals 2 minus 2t."},{"Start":"12:30.415 ","End":"12:33.715","Text":"I need to know what also du equals,"},{"Start":"12:33.715 ","End":"12:39.475","Text":"du is just the derivative of this minus 2 times dt."},{"Start":"12:39.475 ","End":"12:42.190","Text":"That looks nice already because this last bit is"},{"Start":"12:42.190 ","End":"12:48.700","Text":"du and might as well substitute the limits of integration as well,"},{"Start":"12:48.700 ","End":"12:56.280","Text":"so that\u0027s when t is equal to 0,"},{"Start":"12:56.280 ","End":"13:01.280","Text":"I get, maybe I\u0027ll write this a bit higher."},{"Start":"13:01.280 ","End":"13:08.200","Text":"Then u is 2 minus twice 0 is 2,"},{"Start":"13:08.200 ","End":"13:12.745","Text":"and when t is equal to 1,"},{"Start":"13:12.745 ","End":"13:18.370","Text":"then I get that u equals 0."},{"Start":"13:18.370 ","End":"13:23.545","Text":"This whole thing transforms to be the integral."},{"Start":"13:23.545 ","End":"13:28.870","Text":"This time from 2-0,"},{"Start":"13:28.870 ","End":"13:37.135","Text":"2 minus 2t is u squared minus 8u,"},{"Start":"13:37.135 ","End":"13:42.620","Text":"and the minus 2dt is du."},{"Start":"13:43.530 ","End":"13:47.050","Text":"This is equal to,"},{"Start":"13:47.050 ","End":"13:49.879","Text":"I\u0027ll continue over here."},{"Start":"13:51.330 ","End":"14:00.805","Text":"Let\u0027s see, u squared gives me u cubed over 3."},{"Start":"14:00.805 ","End":"14:05.920","Text":"This gives me 8u squared over 2,"},{"Start":"14:05.920 ","End":"14:14.350","Text":"just 4u squared altogether from 2-0,"},{"Start":"14:14.350 ","End":"14:18.970","Text":"and this is equal to if I put in u equals 0,"},{"Start":"14:18.970 ","End":"14:22.795","Text":"I get nothing, so I just need to subtract."},{"Start":"14:22.795 ","End":"14:27.895","Text":"What happens when I put u equal 2,"},{"Start":"14:27.895 ","End":"14:34.930","Text":"2 cubed over 8 is 8 over 3,"},{"Start":"14:34.930 ","End":"14:44.900","Text":"and 4 times 2 squared is 4 times 4 is 16."},{"Start":"14:46.470 ","End":"14:49.015","Text":"I\u0027ll just write it as,"},{"Start":"14:49.015 ","End":"14:50.395","Text":"just reverse the order,"},{"Start":"14:50.395 ","End":"14:53.710","Text":"16 minus 8 over 3."},{"Start":"14:53.710 ","End":"14:56.470","Text":"I won\u0027t actually compute this just like here,"},{"Start":"14:56.470 ","End":"14:59.695","Text":"I\u0027ll do it at the end when I do all the additions,"},{"Start":"14:59.695 ","End":"15:05.395","Text":"all I\u0027m missing now is the integral over C_4."},{"Start":"15:05.395 ","End":"15:10.450","Text":"For C_4, I have the integral. Here we are."},{"Start":"15:10.450 ","End":"15:13.390","Text":"The parameter is from 0-1,"},{"Start":"15:13.390 ","End":"15:18.265","Text":"that\u0027s t. Then let\u0027s see where I\u0027m going to get a 0,"},{"Start":"15:18.265 ","End":"15:21.140","Text":"the dx is 0."},{"Start":"15:21.780 ","End":"15:24.310","Text":"All this first part is 0."},{"Start":"15:24.310 ","End":"15:25.780","Text":"I Just need the second part,"},{"Start":"15:25.780 ","End":"15:28.225","Text":"y squared minus 2xy,"},{"Start":"15:28.225 ","End":"15:30.910","Text":"and I\u0027m looking here,"},{"Start":"15:30.910 ","End":"15:37.390","Text":"y squared is 2 minus 2t"},{"Start":"15:37.390 ","End":"15:46.915","Text":"squared minus 2xy is,"},{"Start":"15:46.915 ","End":"15:50.930","Text":"where am I now? Here."},{"Start":"15:51.630 ","End":"15:58.300","Text":"x is 0, so I don\u0027t get anything else."},{"Start":"15:58.300 ","End":"16:05.095","Text":"That\u0027s just times the dy,"},{"Start":"16:05.095 ","End":"16:10.340","Text":"which is minus 2 dt."},{"Start":"16:12.870 ","End":"16:17.380","Text":"You know what? I think I could use the same substitutions just to remind myself."},{"Start":"16:17.380 ","End":"16:20.515","Text":"This is t goes from 0-1."},{"Start":"16:20.515 ","End":"16:23.570","Text":"If I make the same substitution,"},{"Start":"16:24.150 ","End":"16:33.715","Text":"then I\u0027ve got that 0-1 means from 2-0,"},{"Start":"16:33.715 ","End":"16:36.980","Text":"that\u0027s as far as u goes."},{"Start":"16:38.400 ","End":"16:45.980","Text":"I\u0027ve got this is u squared and this bit is du."},{"Start":"16:46.230 ","End":"16:55.615","Text":"This comes out to be u cubed over 3 from 2-0."},{"Start":"16:55.615 ","End":"16:58.285","Text":"At 0 I get nothing,"},{"Start":"16:58.285 ","End":"17:01.135","Text":"at 2 I\u0027ve got 8/3,"},{"Start":"17:01.135 ","End":"17:06.070","Text":"so this is minus 8/3."},{"Start":"17:06.070 ","End":"17:08.440","Text":"Let me highlight the partial sums."},{"Start":"17:08.440 ","End":"17:10.390","Text":"This is the integral over C1,"},{"Start":"17:10.390 ","End":"17:12.445","Text":"the integral over C2,"},{"Start":"17:12.445 ","End":"17:14.455","Text":"the integral over C3,"},{"Start":"17:14.455 ","End":"17:17.485","Text":"and the integral over C4."},{"Start":"17:17.485 ","End":"17:20.350","Text":"What I want to do finally,"},{"Start":"17:20.350 ","End":"17:24.740","Text":"is to add these 4 together."},{"Start":"17:25.290 ","End":"17:28.669","Text":"Let me just get some space."},{"Start":"17:29.580 ","End":"17:38.040","Text":"Our integral over C is equal to 8/3"},{"Start":"17:38.040 ","End":"17:42.565","Text":"plus 8/3 minus 8"},{"Start":"17:42.565 ","End":"17:50.650","Text":"plus 16 minus 8/3,"},{"Start":"17:50.650 ","End":"17:57.880","Text":"and then minus 8/3."},{"Start":"17:57.880 ","End":"17:59.995","Text":"Let\u0027s see."},{"Start":"17:59.995 ","End":"18:02.515","Text":"Stuff cancels,"},{"Start":"18:02.515 ","End":"18:07.690","Text":"8/3 and 8/3 cancels minus 8/3 and minus 8/3."},{"Start":"18:07.690 ","End":"18:10.945","Text":"I\u0027m left with minus 8 plus 16,"},{"Start":"18:10.945 ","End":"18:14.140","Text":"and this is equal to 8."},{"Start":"18:14.140 ","End":"18:19.700","Text":"Let me just go and record this at the top."},{"Start":"18:19.700 ","End":"18:23.325","Text":"Where am I? Here we are."},{"Start":"18:23.325 ","End":"18:26.555","Text":"We did the left-hand side,"},{"Start":"18:26.555 ","End":"18:29.995","Text":"and this comes out to be 8."},{"Start":"18:29.995 ","End":"18:32.860","Text":"Now we\u0027ll work on the right-hand side"},{"Start":"18:32.860 ","End":"18:37.030","Text":"but I\u0027ll erase all the stuff I don\u0027t need."},{"Start":"18:37.030 ","End":"18:45.370","Text":"That\u0027s better, and now let me just shade the region which is a square,"},{"Start":"18:45.370 ","End":"18:51.190","Text":"this R. I\u0027m going to need to compute g_x minus f_y."},{"Start":"18:51.190 ","End":"19:01.375","Text":"Let\u0027s see, g_x minus f_y means partial derivatives, this is f,"},{"Start":"19:01.375 ","End":"19:09.080","Text":"this is g, g with respect to x,"},{"Start":"19:09.990 ","End":"19:12.400","Text":"since y is a constant,"},{"Start":"19:12.400 ","End":"19:15.650","Text":"is just minus 2y,"},{"Start":"19:16.410 ","End":"19:20.320","Text":"and f with respect to y."},{"Start":"19:20.320 ","End":"19:25.810","Text":"Let\u0027s see, x is a constant with respect to y."},{"Start":"19:25.810 ","End":"19:33.820","Text":"This would be minus 3xy squared,"},{"Start":"19:33.820 ","End":"19:39.205","Text":"which is just, let me put the plus before the minus,"},{"Start":"19:39.205 ","End":"19:43.465","Text":"3xy squared"},{"Start":"19:43.465 ","End":"19:50.680","Text":"minus 2y."},{"Start":"19:50.680 ","End":"19:54.860","Text":"This double integral over R,"},{"Start":"19:56.910 ","End":"20:04.480","Text":"this here 3xy squared minus 2y dA."},{"Start":"20:04.480 ","End":"20:07.585","Text":"I can do it as dxdy or dydx."},{"Start":"20:07.585 ","End":"20:11.665","Text":"A rectangular region is always very straightforward."},{"Start":"20:11.665 ","End":"20:19.240","Text":"Let\u0027s do it that the outer integral will be x from 0-2."},{"Start":"20:20.640 ","End":"20:24.670","Text":"The inner 1 is also simple,"},{"Start":"20:24.670 ","End":"20:28.030","Text":"also y goes from 0-2."},{"Start":"20:28.030 ","End":"20:34.750","Text":"I just have this here, 3xy squared minus 2y,"},{"Start":"20:34.750 ","End":"20:40.190","Text":"dy closes this and the dx closes this."},{"Start":"20:40.760 ","End":"20:45.855","Text":"This way will be much easier than this way, much less work."},{"Start":"20:45.855 ","End":"20:50.290","Text":"Let\u0027s start with the inner integral as always."},{"Start":"20:51.560 ","End":"20:56.069","Text":"Let me do this 1 separately at the side here."},{"Start":"20:56.069 ","End":"21:04.520","Text":"What we get dy is this is y cube over 3,"},{"Start":"21:04.520 ","End":"21:08.470","Text":"because x is constant and the 3 will cancel with the 3,"},{"Start":"21:08.470 ","End":"21:12.490","Text":"and we\u0027ll get xy cubed for the first bit,"},{"Start":"21:12.490 ","End":"21:17.390","Text":"and for the second bit will get just y squared."},{"Start":"21:17.520 ","End":"21:23.960","Text":"All this, y goes from 0-2,"},{"Start":"21:24.600 ","End":"21:27.835","Text":"make a note, that\u0027s y."},{"Start":"21:27.835 ","End":"21:30.010","Text":"When y is 0,"},{"Start":"21:30.010 ","End":"21:34.135","Text":"I get nothing, so I just have to let y equals 2."},{"Start":"21:34.135 ","End":"21:41.540","Text":"Then this is equal to 2 cubed is 8, so it\u0027s 8x."},{"Start":"21:42.240 ","End":"21:46.195","Text":"Here 2 squared is 4,"},{"Start":"21:46.195 ","End":"21:50.030","Text":"so it\u0027s 8x minus 4."},{"Start":"21:50.880 ","End":"21:55.270","Text":"I can put this back in here, let\u0027s make a note."},{"Start":"21:55.270 ","End":"21:59.900","Text":"This is 8x minus 4."},{"Start":"22:04.350 ","End":"22:09.445","Text":"This becomes the integral from 0-2,"},{"Start":"22:09.445 ","End":"22:14.185","Text":"why don\u0027t I take 4 outside the brackets,"},{"Start":"22:14.185 ","End":"22:22.795","Text":"and then I\u0027ll just have 2x minus 1 dx."},{"Start":"22:22.795 ","End":"22:26.425","Text":"This is equal to 4 times,"},{"Start":"22:26.425 ","End":"22:37.100","Text":"now this becomes x squared minus x, taken from 0-2."},{"Start":"22:37.800 ","End":"22:41.560","Text":"When x is 0, this is just 0."},{"Start":"22:41.560 ","End":"22:43.735","Text":"All I need is the 2,"},{"Start":"22:43.735 ","End":"22:49.945","Text":"so it\u0027s 4 times 2 squared minus 2,"},{"Start":"22:49.945 ","End":"22:51.730","Text":"2 squared is 4,"},{"Start":"22:51.730 ","End":"22:53.080","Text":"4 minus 2 is 2,"},{"Start":"22:53.080 ","End":"22:57.490","Text":"2 times 4 is equal to 8."},{"Start":"22:57.490 ","End":"23:00.385","Text":"I\u0027ll record that here,"},{"Start":"23:00.385 ","End":"23:03.400","Text":"and yes, 8 equals 8,"},{"Start":"23:03.400 ","End":"23:08.425","Text":"so yes, that has been verified for this example,"},{"Start":"23:08.425 ","End":"23:11.570","Text":"and so we are done."}],"ID":8773},{"Watched":false,"Name":"Exercise 4","Duration":"8m 21s","ChapterTopicVideoID":8648,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:04.530","Text":"This is an exercise from physics involving work,"},{"Start":"00:04.530 ","End":"00:08.610","Text":"and we\u0027ve done these before and I\u0027ll provide the formula in a moment."},{"Start":"00:08.610 ","End":"00:12.870","Text":"Compute the work done by a force field F"},{"Start":"00:12.870 ","End":"00:17.040","Text":"as follows on a particle which moves anticlockwise,"},{"Start":"00:17.040 ","End":"00:18.570","Text":"that\u0027s the positive direction,"},{"Start":"00:18.570 ","End":"00:22.695","Text":"on the unit circle and completes 1 revolution."},{"Start":"00:22.695 ","End":"00:28.290","Text":"We\u0027re going once around the circle and the sketch."},{"Start":"00:28.290 ","End":"00:31.094","Text":"Now, the formula for work."},{"Start":"00:31.094 ","End":"00:34.124","Text":"Before that we have to label these."},{"Start":"00:34.124 ","End":"00:39.360","Text":"Let\u0027s say that this bit from here to here,"},{"Start":"00:39.360 ","End":"00:42.780","Text":"is f of xy,"},{"Start":"00:42.780 ","End":"00:47.950","Text":"and this bit is g of xy."},{"Start":"00:48.920 ","End":"00:57.590","Text":"The formula for the work done is just the line integral."},{"Start":"00:57.590 ","End":"01:00.440","Text":"In this case along the curve c,"},{"Start":"01:00.440 ","End":"01:09.955","Text":"let\u0027s call this c of f dx plus g dy."},{"Start":"01:09.955 ","End":"01:15.140","Text":"Sometimes I use the letters p and q here I\u0027m using f and g. In this case,"},{"Start":"01:15.140 ","End":"01:16.700","Text":"because it\u0027s a closed curve,"},{"Start":"01:16.700 ","End":"01:20.250","Text":"I would write it with a little circle here."},{"Start":"01:20.880 ","End":"01:26.590","Text":"Now it turns out that this integral is really difficult to compute."},{"Start":"01:26.590 ","End":"01:29.380","Text":"I\u0027m just going to start doing it and then I\u0027ll erase it just"},{"Start":"01:29.380 ","End":"01:32.410","Text":"to show you what a difficulty we get into."},{"Start":"01:32.410 ","End":"01:36.815","Text":"Normally what we would do would be to parametrize the circle."},{"Start":"01:36.815 ","End":"01:38.530","Text":"Let\u0027s just say we start here,"},{"Start":"01:38.530 ","End":"01:40.855","Text":"it doesn\u0027t really matter the start and endpoint."},{"Start":"01:40.855 ","End":"01:46.510","Text":"Then we could parametrize it as x equals cosine t,"},{"Start":"01:46.510 ","End":"01:57.290","Text":"y equals sine t and t goes from 0 to 2 Pi."},{"Start":"01:57.720 ","End":"02:03.445","Text":"Also, dx would be minus"},{"Start":"02:03.445 ","End":"02:13.075","Text":"sine t dt and dy would equal cosine t dt."},{"Start":"02:13.075 ","End":"02:14.750","Text":"If we did all of this,"},{"Start":"02:14.750 ","End":"02:19.850","Text":"we would get the integral from 0 to 2 Pi."},{"Start":"02:19.850 ","End":"02:21.950","Text":"Now, f is this function,"},{"Start":"02:21.950 ","End":"02:31.295","Text":"so it come out to be e^cosine t and then we\u0027d have y cubed is sine cubed"},{"Start":"02:31.295 ","End":"02:38.390","Text":"t. Then dx would be minus"},{"Start":"02:38.390 ","End":"02:47.220","Text":"sine t. Already plus the other part g dy,"},{"Start":"02:47.220 ","End":"02:48.780","Text":"so on, so on, so on,"},{"Start":"02:48.780 ","End":"02:51.360","Text":"you can see that it\u0027s a mess."},{"Start":"02:51.360 ","End":"02:55.550","Text":"Fortunately, there\u0027s another way of computing this integral,"},{"Start":"02:55.550 ","End":"02:58.850","Text":"I mean we\u0027re in the chapter on Green\u0027s theorem."},{"Start":"02:58.850 ","End":"03:02.880","Text":"Let me erase this first."},{"Start":"03:03.800 ","End":"03:06.620","Text":"Let\u0027s see what Green\u0027s theorem says."},{"Start":"03:06.620 ","End":"03:09.235","Text":"Well, it applies to a closed curve,"},{"Start":"03:09.235 ","End":"03:15.815","Text":"that\u0027s simple, and it goes anticlockwise and it\u0027s piecewise differentiable."},{"Start":"03:15.815 ","End":"03:18.875","Text":"Circle is all these good things."},{"Start":"03:18.875 ","End":"03:22.075","Text":"Notice that this is the positive direction."},{"Start":"03:22.075 ","End":"03:25.970","Text":"Green\u0027s theorem says that we can evaluate this as the"},{"Start":"03:25.970 ","End":"03:30.879","Text":"double integral over the region that\u0027s inside the curve,"},{"Start":"03:30.879 ","End":"03:34.425","Text":"and I\u0027ve shaded it and I\u0027m calling this the region,"},{"Start":"03:34.425 ","End":"03:38.230","Text":"R. A double integral,"},{"Start":"03:39.080 ","End":"03:42.845","Text":"the integral is of the derivative of"},{"Start":"03:42.845 ","End":"03:50.640","Text":"g_x minus partial derivative of f_y over the region."},{"Start":"03:50.640 ","End":"03:53.790","Text":"Let\u0027s see what we get in our case."},{"Start":"03:53.790 ","End":"03:58.550","Text":"I can assure you it\u0027s going to be a lot simpler than doing it along the curve."},{"Start":"03:58.550 ","End":"04:02.545","Text":"As we saw, we got very messy expressions."},{"Start":"04:02.545 ","End":"04:06.350","Text":"In our case for our particular g and f,"},{"Start":"04:06.350 ","End":"04:12.540","Text":"we get the double integral over this circle, the disk really,"},{"Start":"04:12.540 ","End":"04:18.530","Text":"of g_x, this derivative with respect to x"},{"Start":"04:18.530 ","End":"04:25.770","Text":"is just 3x squared minus f_y,"},{"Start":"04:26.080 ","End":"04:31.705","Text":"this derivative with respect to y is minus 3y squared."},{"Start":"04:31.705 ","End":"04:34.420","Text":"But there is a minus so minus,"},{"Start":"04:34.420 ","End":"04:41.860","Text":"minus is plus 3y squared dA."},{"Start":"04:41.960 ","End":"04:48.020","Text":"Now, this is a perfect time for using polar coordinates."},{"Start":"04:48.800 ","End":"04:53.020","Text":"Let me remind you in case you\u0027ve forgotten what the polar coordinates."},{"Start":"04:53.020 ","End":"04:58.330","Text":"Let\u0027s x be r cosine Theta in general,"},{"Start":"04:58.330 ","End":"05:04.725","Text":"y equals r sine Theta and we replace"},{"Start":"05:04.725 ","End":"05:12.560","Text":"dA by r dr dTheta and the fourth formula,"},{"Start":"05:12.560 ","End":"05:14.030","Text":"which is very useful,"},{"Start":"05:14.030 ","End":"05:18.740","Text":"x squared plus y squared equals r squared."},{"Start":"05:19.010 ","End":"05:23.870","Text":"We also have to convert the region in each particular case."},{"Start":"05:23.870 ","End":"05:27.585","Text":"In our case, the region is very simple."},{"Start":"05:27.585 ","End":"05:32.690","Text":"If we\u0027re starting from here and going around here,"},{"Start":"05:32.690 ","End":"05:35.210","Text":"let\u0027s say this is a general point,"},{"Start":"05:35.210 ","End":"05:39.075","Text":"Theta and r. Well,"},{"Start":"05:39.075 ","End":"05:43.460","Text":"Theta goes from 0 all the way around to"},{"Start":"05:43.460 ","End":"05:50.730","Text":"2 Pi and r goes from 0 to 1."},{"Start":"05:50.730 ","End":"05:55.330","Text":"What we get after we do all the conversion,"},{"Start":"05:55.640 ","End":"05:57.840","Text":"let\u0027s do it in a different color,"},{"Start":"05:57.840 ","End":"05:59.550","Text":"we\u0027re now in polar."},{"Start":"05:59.550 ","End":"06:04.620","Text":"We have Theta going from 0 to"},{"Start":"06:04.620 ","End":"06:13.840","Text":"2 Pi and we have r going from 0 to 1."},{"Start":"06:13.940 ","End":"06:19.620","Text":"The x squared plus y squared is r"},{"Start":"06:19.620 ","End":"06:26.670","Text":"squared so we have here 3 squared,"},{"Start":"06:26.670 ","End":"06:35.880","Text":"and dA is r dr dTheta and this is our integral."},{"Start":"06:35.880 ","End":"06:43.960","Text":"Now, this is just 3r cubed."},{"Start":"06:43.960 ","End":"06:49.250","Text":"Well, I\u0027ll highlight the inner integral, the dr integral."},{"Start":"06:49.250 ","End":"06:51.250","Text":"We\u0027ll do that one first,"},{"Start":"06:51.250 ","End":"06:53.820","Text":"and I\u0027ll do that one at the side."},{"Start":"06:53.820 ","End":"06:58.290","Text":"This is the integral from 0 to 1,"},{"Start":"06:58.290 ","End":"07:04.650","Text":"3r cubed dr and this is equal"},{"Start":"07:04.650 ","End":"07:09.840","Text":"to 3r^4 over 4 or"},{"Start":"07:09.840 ","End":"07:17.265","Text":"3/4r^4 between 0 and 1."},{"Start":"07:17.265 ","End":"07:19.020","Text":"At 0, we don\u0027t get anything,"},{"Start":"07:19.020 ","End":"07:23.760","Text":"and at 1 we just get 3/4, a constant."},{"Start":"07:23.760 ","End":"07:29.805","Text":"This whole inner integral is 3/4."},{"Start":"07:29.805 ","End":"07:33.780","Text":"Now I can take the 3/4 in fact,"},{"Start":"07:33.780 ","End":"07:37.680","Text":"right in front because it\u0027s a constant."},{"Start":"07:37.680 ","End":"07:42.360","Text":"I can say this is equal to 3/4 times the"},{"Start":"07:42.360 ","End":"07:49.700","Text":"integral from 0 to 2 Pi of just d Theta,"},{"Start":"07:49.700 ","End":"07:52.670","Text":"or I can write it as 1d Theta."},{"Start":"07:52.670 ","End":"07:54.410","Text":"I only have the integral of 1,"},{"Start":"07:54.410 ","End":"07:56.690","Text":"it\u0027s just the upper limit minus the lower limit,"},{"Start":"07:56.690 ","End":"08:04.440","Text":"which is 2 Pi so I get 3/4 times 2 pi or if you like,"},{"Start":"08:04.440 ","End":"08:07.740","Text":"3 Pi over 2."},{"Start":"08:07.740 ","End":"08:11.510","Text":"This was much simpler to do with Green\u0027s theorem."},{"Start":"08:11.510 ","End":"08:13.520","Text":"It would have, as I started to show you,"},{"Start":"08:13.520 ","End":"08:16.055","Text":"got in a real mess if you did it directly."},{"Start":"08:16.055 ","End":"08:20.760","Text":"Anyway, this is our answer and we are done."}],"ID":8774},{"Watched":false,"Name":"Exercise 5","Duration":"12m 28s","ChapterTopicVideoID":8649,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"In this exercise, we have a line integral of type 2 to perform on"},{"Start":"00:05.160 ","End":"00:10.780","Text":"the curve C. C is described in words,"},{"Start":"00:10.820 ","End":"00:14.190","Text":"but we really need a sketch."},{"Start":"00:14.190 ","End":"00:18.360","Text":"Here are some axes. Now we need these 2 curves,"},{"Start":"00:18.360 ","End":"00:21.570","Text":"y equals x squared and y equals 8 minus x squared."},{"Start":"00:21.570 ","End":"00:22.950","Text":"As for x squared,"},{"Start":"00:22.950 ","End":"00:24.810","Text":"we\u0027ll only need it in the first quadrant,"},{"Start":"00:24.810 ","End":"00:27.585","Text":"so it might look something like this;"},{"Start":"00:27.585 ","End":"00:32.940","Text":"and then 8 minus x squared would start at 8,"},{"Start":"00:32.940 ","End":"00:37.455","Text":"just be the reflection to this upside down parabola,"},{"Start":"00:37.455 ","End":"00:42.190","Text":"so something like maybe this."},{"Start":"00:42.200 ","End":"00:44.700","Text":"This is their intersection,"},{"Start":"00:44.700 ","End":"00:47.055","Text":"we\u0027ll compute that in a moment."},{"Start":"00:47.055 ","End":"00:53.010","Text":"Our curve C is made up of 2 bits;"},{"Start":"00:53.010 ","End":"00:58.855","Text":"this bit here, and this bit here going clockwise, I\u0027ll highlight it."},{"Start":"00:58.855 ","End":"01:01.805","Text":"This part up to here,"},{"Start":"01:01.805 ","End":"01:06.030","Text":"and then this part to here."},{"Start":"01:06.130 ","End":"01:10.490","Text":"Now, our curve C is made up of 2 bits."},{"Start":"01:10.490 ","End":"01:12.800","Text":"The curve C, I\u0027ll just write it here,"},{"Start":"01:12.800 ","End":"01:15.610","Text":"is made up from,"},{"Start":"01:15.610 ","End":"01:21.120","Text":"let\u0027s call this C_1 and we\u0027ll call this bit C_2,"},{"Start":"01:21.120 ","End":"01:26.080","Text":"so C is C_1 plus C_2."},{"Start":"01:27.410 ","End":"01:32.440","Text":"Now, I want to be able to use Green\u0027s theorem here,"},{"Start":"01:32.440 ","End":"01:40.085","Text":"because this is going to be difficult integral to compute along C_1 and C_2."},{"Start":"01:40.085 ","End":"01:42.510","Text":"When you start making the substitution, you\u0027ll see it\u0027s a mess."},{"Start":"01:42.510 ","End":"01:44.305","Text":"I want to use Green\u0027s theorem."},{"Start":"01:44.305 ","End":"01:48.610","Text":"The trouble is that Green\u0027s theorem applies to closed curves."},{"Start":"01:48.610 ","End":"01:54.130","Text":"How about if I add an extra bit from"},{"Start":"01:54.130 ","End":"02:00.620","Text":"here to here in this direction,"},{"Start":"02:00.620 ","End":"02:05.760","Text":"and we\u0027ll call this 1, C_3."},{"Start":"02:06.790 ","End":"02:09.230","Text":"The plan is this."},{"Start":"02:09.230 ","End":"02:12.680","Text":"If we compute the integral, let\u0027s call it L,"},{"Start":"02:12.680 ","End":"02:18.170","Text":"which is the closed curve C_1 plus C_2 plus C_3."},{"Start":"02:18.170 ","End":"02:23.760","Text":"If I can compute with the help of Green\u0027s theorem the line"},{"Start":"02:23.760 ","End":"02:32.190","Text":"integral over L which is a closed curve of whatever it is here,"},{"Start":"02:32.190 ","End":"02:38.135","Text":"and then I subtract the integral over C_3,"},{"Start":"02:38.135 ","End":"02:45.590","Text":"then I\u0027ll get just the integral of C_1 plus C_2 which is the integral of the original C,"},{"Start":"02:45.590 ","End":"02:47.150","Text":"which is this plus this."},{"Start":"02:47.150 ","End":"02:48.980","Text":"That\u0027s the idea."},{"Start":"02:48.980 ","End":"02:53.360","Text":"Also, because this C_3 is a vertical line,"},{"Start":"02:53.360 ","End":"03:00.450","Text":"it should be fairly easy to compute as opposed to computing this over pieces of parabola."},{"Start":"03:00.450 ","End":"03:06.870","Text":"That\u0027s the strategy, so let\u0027s just do some labeling first."},{"Start":"03:06.870 ","End":"03:11.050","Text":"This first piece of the dx,"},{"Start":"03:11.050 ","End":"03:12.955","Text":"I\u0027ll call this piece f,"},{"Start":"03:12.955 ","End":"03:16.930","Text":"and this function of xy that goes with the dy,"},{"Start":"03:16.930 ","End":"03:25.710","Text":"I\u0027ll call it g. Green\u0027s theorem says that the integral of"},{"Start":"03:25.710 ","End":"03:28.690","Text":"a closed path"},{"Start":"03:34.520 ","End":"03:35.730","Text":"made"},{"Start":"03:35.730 ","End":"03:39.145","Text":"up piecewise of differentiable curves,"},{"Start":"03:39.145 ","End":"03:42.870","Text":"but it also has to go counterclockwise."},{"Start":"03:42.870 ","End":"03:45.435","Text":"That\u0027s another snag we\u0027re hitting."},{"Start":"03:45.435 ","End":"03:51.825","Text":"Actually, the integral of minus L would"},{"Start":"03:51.825 ","End":"03:58.280","Text":"be this thing in a counterclockwise direction,"},{"Start":"03:58.280 ","End":"04:03.060","Text":"the positive direction, so the integral of minus L of"},{"Start":"04:03.060 ","End":"04:11.175","Text":"fdx plus gdy would equal the double integral."},{"Start":"04:11.175 ","End":"04:13.725","Text":"I\u0027ll call this region R,"},{"Start":"04:13.725 ","End":"04:20.510","Text":"the region that the curve L goes around,"},{"Start":"04:20.510 ","End":"04:22.680","Text":"it\u0027s this bit here,"},{"Start":"04:22.680 ","End":"04:32.730","Text":"of g with respect to x minus f with respect to ydA."},{"Start":"04:33.370 ","End":"04:36.590","Text":"This is Green\u0027s theorem."},{"Start":"04:36.590 ","End":"04:42.755","Text":"Now, because of the minus here,"},{"Start":"04:42.755 ","End":"04:47.040","Text":"the integral of the clockwise which is"},{"Start":"04:47.040 ","End":"04:51.440","Text":"a negative direction is just minus the integral in the other direction."},{"Start":"04:51.440 ","End":"04:54.725","Text":"What I can get is that the integral"},{"Start":"04:54.725 ","End":"05:02.844","Text":"of over L of fdx plus gdy,"},{"Start":"05:02.844 ","End":"05:04.370","Text":"which is what we want,"},{"Start":"05:04.370 ","End":"05:07.940","Text":"is equal to minus this."},{"Start":"05:07.940 ","End":"05:14.090","Text":"What I can do is instead of putting a minus in front,"},{"Start":"05:14.090 ","End":"05:17.875","Text":"I can reverse the order of these 2."},{"Start":"05:17.875 ","End":"05:21.800","Text":"So putting a minus is the same as changing the order of the subtraction,"},{"Start":"05:21.800 ","End":"05:25.880","Text":"that\u0027s just 1 extra little snag that we had to pay attention to."},{"Start":"05:25.880 ","End":"05:30.510","Text":"Positive direction is the counterclockwise, so let\u0027s see."},{"Start":"05:31.120 ","End":"05:37.260","Text":"I want here to put f_y minus g_x."},{"Start":"05:38.420 ","End":"05:49.340","Text":"Let\u0027s see, f with respect to y is just e to the power of y because x is a constant,"},{"Start":"05:49.340 ","End":"05:54.270","Text":"minus g with respect to x."},{"Start":"05:54.270 ","End":"05:58.590","Text":"Let\u0027s see. The second term has no x at all, that\u0027s 0,"},{"Start":"05:58.590 ","End":"06:00.855","Text":"and this with respect to x,"},{"Start":"06:00.855 ","End":"06:03.410","Text":"just if the constant times x,"},{"Start":"06:03.410 ","End":"06:05.605","Text":"so it\u0027s just e to the y."},{"Start":"06:05.605 ","End":"06:09.260","Text":"Look at that, this is very lucky."},{"Start":"06:09.260 ","End":"06:11.195","Text":"This minus this is 0,"},{"Start":"06:11.195 ","End":"06:15.840","Text":"so this whole integral is just 0."},{"Start":"06:16.000 ","End":"06:19.470","Text":"It\u0027s so lucky that we didn\u0027t even have to compute,"},{"Start":"06:19.470 ","End":"06:24.994","Text":"we didn\u0027t have to convert the region to an iterated integral."},{"Start":"06:24.994 ","End":"06:27.050","Text":"But it would have been,"},{"Start":"06:27.050 ","End":"06:29.365","Text":"I\u0027ll just mention it."},{"Start":"06:29.365 ","End":"06:35.105","Text":"If we had to do the integral we\u0027d compute the intersection point,"},{"Start":"06:35.105 ","End":"06:42.660","Text":"and that would be done by letting 8 minus x squared equals x squared,"},{"Start":"06:42.660 ","End":"06:49.660","Text":"but this point comes out to be the point 2, 4."},{"Start":"06:49.660 ","End":"06:56.125","Text":"We would have gotten the integral as,"},{"Start":"06:56.125 ","End":"06:58.289","Text":"this is the origin,"},{"Start":"06:58.289 ","End":"07:03.780","Text":"as x goes from 0 to"},{"Start":"07:03.780 ","End":"07:11.524","Text":"2 and then we would have got each vertical slice."},{"Start":"07:11.524 ","End":"07:15.720","Text":"We would have entered the region here,"},{"Start":"07:15.720 ","End":"07:17.625","Text":"and exited the region here."},{"Start":"07:17.625 ","End":"07:21.180","Text":"We would have gone from the lower parabola,"},{"Start":"07:21.200 ","End":"07:23.969","Text":"which would be x squared,"},{"Start":"07:23.969 ","End":"07:28.560","Text":"and then 8 minus x squared is the upper parabola."},{"Start":"07:28.560 ","End":"07:31.720","Text":"Then we would have had,"},{"Start":"07:32.350 ","End":"07:35.060","Text":"well, it happened to be 0 here,"},{"Start":"07:35.060 ","End":"07:37.070","Text":"but it might not have been 0,"},{"Start":"07:37.070 ","End":"07:42.350","Text":"and it would have been then dy and dx."},{"Start":"07:42.350 ","End":"07:46.160","Text":"But I just wanted to show you how you would have written it as an iterated integral,"},{"Start":"07:46.160 ","End":"07:48.830","Text":"if it hadn\u0027t come out 0."},{"Start":"07:48.830 ","End":"07:52.260","Text":"Fortunately, we were spared all that."},{"Start":"07:52.910 ","End":"08:00.800","Text":"Now, what we have left to do is to compute the integral over C_3,"},{"Start":"08:00.800 ","End":"08:08.670","Text":"and then we can from there get to C_1 plus C_2 which is C. At this point,"},{"Start":"08:08.670 ","End":"08:11.895","Text":"x is 0 and y is 8,"},{"Start":"08:11.895 ","End":"08:16.240","Text":"and so I need to parametrize the curve C_3."},{"Start":"08:16.240 ","End":"08:20.410","Text":"C_3, I can do it quite simply,"},{"Start":"08:20.410 ","End":"08:24.335","Text":"because it\u0027s just along the y-axis and going upward."},{"Start":"08:24.335 ","End":"08:31.815","Text":"I can just say that x equals 0 and y goes from 0 to 8,"},{"Start":"08:31.815 ","End":"08:40.020","Text":"so I\u0027d say, y equals t and t goes from 0 to 8."},{"Start":"08:40.020 ","End":"08:44.340","Text":"For the formula, we\u0027ll need dx and dy."},{"Start":"08:44.340 ","End":"08:49.760","Text":"Well, dx is just 0dt or plain 0,"},{"Start":"08:49.760 ","End":"08:55.105","Text":"and dy is equal to just dt."},{"Start":"08:55.105 ","End":"08:59.530","Text":"This is what the parameterization of C_3,"},{"Start":"08:59.530 ","End":"09:06.805","Text":"and so I\u0027ve got that the integral along C_3"},{"Start":"09:06.805 ","End":"09:16.265","Text":"of fdx plus gdy is equal to the integral."},{"Start":"09:16.265 ","End":"09:21.415","Text":"The parameter goes from 0 to 8."},{"Start":"09:21.415 ","End":"09:27.085","Text":"We know that dx is 0,"},{"Start":"09:27.085 ","End":"09:30.605","Text":"and because of this, I don\u0027t need all the first bit."},{"Start":"09:30.605 ","End":"09:35.690","Text":"Just cross this first piece out because dx is 0,"},{"Start":"09:35.690 ","End":"09:38.245","Text":"and so I\u0027ve just got gdy."},{"Start":"09:38.245 ","End":"09:44.160","Text":"Notice that in g we have an x here and x is 0,"},{"Start":"09:44.160 ","End":"09:48.120","Text":"so we just need y cosine y squared dy."},{"Start":"09:48.120 ","End":"09:55.600","Text":"y is t so we get t cosine t squared,"},{"Start":"09:56.150 ","End":"10:00.435","Text":"and ty is, here it is dt,"},{"Start":"10:00.435 ","End":"10:04.900","Text":"and this will give us the integral over C_3 and then we\u0027ll be able to"},{"Start":"10:04.900 ","End":"10:11.350","Text":"find the integral along C. What I can do here,"},{"Start":"10:11.350 ","End":"10:14.215","Text":"I could do it by substitution."},{"Start":"10:14.215 ","End":"10:17.624","Text":"I could substitute the t squared,"},{"Start":"10:17.624 ","End":"10:20.075","Text":"but that may be a bit of an overkill."},{"Start":"10:20.075 ","End":"10:24.085","Text":"I think we just use the trick of putting a 2 here,"},{"Start":"10:24.085 ","End":"10:27.490","Text":"and compensating by putting a 1.5 here,"},{"Start":"10:27.490 ","End":"10:32.450","Text":"because then I have the derivative of t squared here as 2t."},{"Start":"10:32.450 ","End":"10:37.055","Text":"Then because the integral of cosine is sine,"},{"Start":"10:37.055 ","End":"10:44.295","Text":"what I will get is just sine of t squared,"},{"Start":"10:44.295 ","End":"10:49.430","Text":"because the derivative of this is cosine of t squared times 2t,"},{"Start":"10:49.430 ","End":"10:52.625","Text":"but I also need this 1.5,"},{"Start":"10:52.625 ","End":"10:55.355","Text":"and then I need to evaluate this,"},{"Start":"10:55.355 ","End":"11:00.840","Text":"where t goes from 0 to 8."},{"Start":"11:00.840 ","End":"11:06.140","Text":"What we get squared of sine 0 is 0,"},{"Start":"11:06.140 ","End":"11:08.420","Text":"so we just need to substitute the 8,"},{"Start":"11:08.420 ","End":"11:11.270","Text":"8 squared is 64, it\u0027s just a number,"},{"Start":"11:11.270 ","End":"11:18.190","Text":"what we get is 1.5 sine of 64 radians."},{"Start":"11:18.230 ","End":"11:23.370","Text":"We\u0027ve got the integral of C_3,"},{"Start":"11:23.370 ","End":"11:32.215","Text":"this bit is 1.5 sine 64."},{"Start":"11:32.215 ","End":"11:36.875","Text":"The integral over L we got was 0,"},{"Start":"11:36.875 ","End":"11:39.865","Text":"so we need 0 minus this,"},{"Start":"11:39.865 ","End":"11:44.015","Text":"and this will equal what we want which is the integral on C,"},{"Start":"11:44.015 ","End":"11:51.115","Text":"and it will just be minus a 1/2 of sine 64."},{"Start":"11:51.115 ","End":"11:55.080","Text":"I\u0027ll highlight that, and that\u0027s the answer."},{"Start":"11:55.080 ","End":"11:59.825","Text":"We are done. But I would like again to stress this trick."},{"Start":"11:59.825 ","End":"12:02.270","Text":"It\u0027s worth learning this trick,"},{"Start":"12:02.270 ","End":"12:05.750","Text":"because it\u0027s frequently useful."},{"Start":"12:05.750 ","End":"12:10.040","Text":"You don\u0027t have the closed path and you can close it with"},{"Start":"12:10.040 ","End":"12:14.854","Text":"a very simple piece of a vertical or horizontal line,"},{"Start":"12:14.854 ","End":"12:16.310","Text":"then you can do that,"},{"Start":"12:16.310 ","End":"12:18.295","Text":"and then you can use Green\u0027s theorem,"},{"Start":"12:18.295 ","End":"12:27.660","Text":"and then just add or subtract this extra piece of path. I\u0027m done."}],"ID":8775},{"Watched":false,"Name":"Exercise 6","Duration":"19m 22s","ChapterTopicVideoID":8650,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:06.150","Text":"In this exercise, we have to compute the following type 2 line"},{"Start":"00:06.150 ","End":"00:11.880","Text":"integral along the curve C. I\u0027ve drawn a picture of it here."},{"Start":"00:11.880 ","End":"00:16.575","Text":"But what it is is the upper semi ellipse."},{"Start":"00:16.575 ","End":"00:20.110","Text":"This is the equation of ellipse."},{"Start":"00:21.320 ","End":"00:24.660","Text":"If we slightly rewrite it in this form,"},{"Start":"00:24.660 ","End":"00:29.295","Text":"x squared over a squared plus y squared over b squared equals 1."},{"Start":"00:29.295 ","End":"00:32.190","Text":"Then x goes from plus or minus a,"},{"Start":"00:32.190 ","End":"00:34.185","Text":"in this case plus or minus 2."},{"Start":"00:34.185 ","End":"00:37.050","Text":"Y goes up to 1,"},{"Start":"00:37.050 ","End":"00:41.070","Text":"but y here has given as bigger or equal to 0."},{"Start":"00:41.070 ","End":"00:45.420","Text":"It\u0027s just the top half that didn\u0027t have this completed."},{"Start":"00:45.420 ","End":"00:48.790","Text":"From this point to this point."},{"Start":"00:50.270 ","End":"00:53.175","Text":"This is the curve C,"},{"Start":"00:53.175 ","End":"00:54.770","Text":"and the question is,"},{"Start":"00:54.770 ","End":"00:56.375","Text":"how do we do this?"},{"Start":"00:56.375 ","End":"00:58.730","Text":"This looks already complicated enough."},{"Start":"00:58.730 ","End":"01:03.020","Text":"If I parametrize the ellipse and we can parametrize it."},{"Start":"01:03.020 ","End":"01:05.645","Text":"There is a parameterization,"},{"Start":"01:05.645 ","End":"01:13.880","Text":"a cosine t, y is b sine t. It gets very involved."},{"Start":"01:13.880 ","End":"01:16.100","Text":"The idea is, as you\u0027ve probably guessed,"},{"Start":"01:16.100 ","End":"01:20.180","Text":"is to use Green\u0027s Theorem because we\u0027re in the chapter on Green\u0027s Theorem."},{"Start":"01:20.180 ","End":"01:26.365","Text":"The trouble is that Green\u0027s Theorem applies to closed curves."},{"Start":"01:26.365 ","End":"01:33.790","Text":"They also have to be piece wise smooth differentiable."},{"Start":"01:33.790 ","End":"01:36.669","Text":"Anyway, this is not closed,"},{"Start":"01:36.669 ","End":"01:39.070","Text":"so there\u0027s a trick I\u0027ve used before,"},{"Start":"01:39.070 ","End":"01:41.710","Text":"but in case you missed it,"},{"Start":"01:41.710 ","End":"01:43.600","Text":"I\u0027m not going to assume that you know it."},{"Start":"01:43.600 ","End":"01:51.250","Text":"The trick is to complete this to a closed path by adding some very simple piece of curve."},{"Start":"01:51.250 ","End":"01:55.975","Text":"That very simple piece of curve will just be the straight line"},{"Start":"01:55.975 ","End":"02:03.105","Text":"from minus 2 to 2 along the x-axis,"},{"Start":"02:03.105 ","End":"02:05.590","Text":"and I\u0027ll give this piece a name."},{"Start":"02:05.590 ","End":"02:09.190","Text":"Let\u0027s call this one l for line c for curve."},{"Start":"02:09.190 ","End":"02:17.020","Text":"The combined curve c plus l satisfies the conditions of Green\u0027s Theorem."},{"Start":"02:17.020 ","End":"02:26.575","Text":"That is, it\u0027s a simple curve, doesn\u0027t cross itself."},{"Start":"02:26.575 ","End":"02:31.015","Text":"It\u0027s closed, and it\u0027s piece-wise smooth."},{"Start":"02:31.015 ","End":"02:33.805","Text":"This bit is smooth and this bit is smooth."},{"Start":"02:33.805 ","End":"02:41.880","Text":"Then we could use Green\u0027s Theorem to convert what could be messy integral,"},{"Start":"02:41.880 ","End":"02:45.610","Text":"the type two line integral to a much nicer,"},{"Start":"02:45.610 ","End":"02:49.360","Text":"double integral over the semi ellipse."},{"Start":"02:49.360 ","End":"02:53.370","Text":"Let\u0027s call this region R,"},{"Start":"02:53.370 ","End":"02:55.605","Text":"and I shaded it."},{"Start":"02:55.605 ","End":"03:00.210","Text":"Let me tell you a general strategy before we get into the details."},{"Start":"03:00.410 ","End":"03:09.680","Text":"The c plus l together is a closed path and in the correct orientation counterclockwise."},{"Start":"03:09.680 ","End":"03:15.650","Text":"I\u0027ll write it integral over c plus l of we\u0027ll take the same thing here,"},{"Start":"03:15.650 ","End":"03:19.759","Text":"but I\u0027ll just put dot meanwhile because it\u0027s just talking in general,"},{"Start":"03:19.759 ","End":"03:23.060","Text":"will equal according to Green\u0027s Theorem\u0027s double"},{"Start":"03:23.060 ","End":"03:26.405","Text":"integral of something else over the region"},{"Start":"03:26.405 ","End":"03:35.360","Text":"R. The sum of"},{"Start":"03:35.360 ","End":"03:39.950","Text":"two paths is the integral of one path plus the other."},{"Start":"03:39.950 ","End":"03:46.505","Text":"What I\u0027ll be able to say in the end is that the integral along just c,"},{"Start":"03:46.505 ","End":"03:55.680","Text":"will be equal to this double integral of along on the region R of,"},{"Start":"03:55.680 ","End":"03:56.960","Text":"well, here\u0027s one thing."},{"Start":"03:56.960 ","End":"04:02.285","Text":"Here\u0027s something else, minus the integral of"},{"Start":"04:02.285 ","End":"04:09.740","Text":"this function along l. This will turn out to be an easy double integral."},{"Start":"04:09.740 ","End":"04:17.470","Text":"This will turn out to be an easy line integral because it\u0027s just on a horizontal segment."},{"Start":"04:17.470 ","End":"04:21.440","Text":"Doing these two will be much easier even though there\u0027s"},{"Start":"04:21.440 ","End":"04:24.580","Text":"two things to compute than computing this."},{"Start":"04:24.580 ","End":"04:29.070","Text":"Which will come at a real mess if we parameterize the ellipse."},{"Start":"04:29.150 ","End":"04:31.490","Text":"Let\u0027s get to it."},{"Start":"04:31.490 ","End":"04:36.005","Text":"I\u0027ll remind you what Green\u0027s Theorem says."},{"Start":"04:36.005 ","End":"04:42.530","Text":"But I\u0027ll introduce some notation which helps this piece that goes with the dx."},{"Start":"04:42.530 ","End":"04:46.475","Text":"I\u0027ll call this function f it\u0027s f of x and y."},{"Start":"04:46.475 ","End":"04:50.750","Text":"This piece that\u0027s in the brackets that goes with dy,"},{"Start":"04:50.750 ","End":"04:54.970","Text":"I\u0027ll call this g, also a function of x and y,"},{"Start":"04:54.970 ","End":"04:58.735","Text":"and Green\u0027s Theorem says,"},{"Start":"04:58.735 ","End":"05:04.925","Text":"that the integral over a closed curve,"},{"Start":"05:04.925 ","End":"05:06.365","Text":"but in our case,"},{"Start":"05:06.365 ","End":"05:12.210","Text":"it\u0027s going to be c plus l of this,"},{"Start":"05:12.210 ","End":"05:20.500","Text":"which is fdx plus gdy is equal to the double integral over"},{"Start":"05:20.500 ","End":"05:26.410","Text":"the region R^ g with respect to"},{"Start":"05:26.410 ","End":"05:33.590","Text":"x minus f with respect to y partial derivatives, I mean dA."},{"Start":"05:35.370 ","End":"05:38.110","Text":"That will do this,"},{"Start":"05:38.110 ","End":"05:39.800","Text":"and that will give us this part,"},{"Start":"05:39.800 ","End":"05:42.970","Text":"then we just have to also compute the line integral over"},{"Start":"05:42.970 ","End":"05:49.300","Text":"l. Doing this part first we get the"},{"Start":"05:49.300 ","End":"05:55.600","Text":"double integral all over"},{"Start":"05:55.600 ","End":"06:03.460","Text":"the upper semi ellipse of,"},{"Start":"06:03.460 ","End":"06:06.575","Text":"lets see g with respect to x."},{"Start":"06:06.575 ","End":"06:09.320","Text":"If I differentiate this,"},{"Start":"06:09.320 ","End":"06:12.715","Text":"y is a constant."},{"Start":"06:12.715 ","End":"06:21.270","Text":"With respect to x, I\u0027ll have to take the derivative"},{"Start":"06:21.270 ","End":"06:30.560","Text":"of e^2x minus y is just 2e^2x minus y it\u0027s e to the something,"},{"Start":"06:30.560 ","End":"06:33.365","Text":"so it\u0027s e to the something in a derivative is 2."},{"Start":"06:33.365 ","End":"06:43.470","Text":"This bit is the constant sine y plus cosine y."},{"Start":"06:43.470 ","End":"06:49.320","Text":"The derivative of this with respect to x is 2y."},{"Start":"06:50.320 ","End":"06:57.034","Text":"All this is just the g with respect to x."},{"Start":"06:57.034 ","End":"07:03.120","Text":"Then I have to subtract f with respect to y."},{"Start":"07:05.480 ","End":"07:08.735","Text":"Because there\u0027s a minus here,"},{"Start":"07:08.735 ","End":"07:12.950","Text":"I\u0027ll make that a plus and then we\u0027ll be okay."},{"Start":"07:12.950 ","End":"07:15.760","Text":"With respect to y,"},{"Start":"07:15.760 ","End":"07:18.370","Text":"this will give us,"},{"Start":"07:22.550 ","End":"07:26.240","Text":"and I guess I\u0027ll have to use the product rule in general,"},{"Start":"07:26.240 ","End":"07:37.090","Text":"this will be say u and this will be v. I need u prime v plus v prime u."},{"Start":"07:37.180 ","End":"07:40.685","Text":"The minus I already took care of here,"},{"Start":"07:40.685 ","End":"07:44.910","Text":"and so u prime will be"},{"Start":"07:47.950 ","End":"07:58.300","Text":"2e^2x minus y but times the derivative of this with respect to y is minus 1."},{"Start":"07:58.300 ","End":"08:01.625","Text":"Better put curly braces here,"},{"Start":"08:01.625 ","End":"08:04.940","Text":"times that\u0027s u prime times v,"},{"Start":"08:04.940 ","End":"08:08.690","Text":"which is cosine y."},{"Start":"08:08.690 ","End":"08:12.185","Text":"Then plus u v prime."},{"Start":"08:12.185 ","End":"08:19.145","Text":"It\u0027s 2e^2x minus y v prime"},{"Start":"08:19.145 ","End":"08:23.730","Text":"with respect to y minus cosine y."},{"Start":"08:24.880 ","End":"08:34.950","Text":"Quite a mess really, and that\u0027s dA."},{"Start":"08:36.810 ","End":"08:42.625","Text":"But I think I need more brackets because the whole thing"},{"Start":"08:42.625 ","End":"08:49.540","Text":"here is dA looks complicated,"},{"Start":"08:49.540 ","End":"08:51.985","Text":"I hope it simplifies,"},{"Start":"08:51.985 ","End":"08:56.000","Text":"and , I meant to put a sine y here."},{"Start":"08:56.000 ","End":"08:58.500","Text":"Let\u0027s see what we get,"},{"Start":"08:58.500 ","End":"09:01.700","Text":"we get the integral,"},{"Start":"09:01.700 ","End":"09:09.760","Text":"now most everything contains this 2e^ 2 x minus y except for the bit with the 2y,"},{"Start":"09:09.760 ","End":"09:12.550","Text":"so let me write it as 2y,"},{"Start":"09:12.550 ","End":"09:21.520","Text":"and then I\u0027ll take out 2e^2 x minus y from whatever\u0027s left."},{"Start":"09:21.520 ","End":"09:24.505","Text":"Let\u0027s see what is left here,"},{"Start":"09:24.505 ","End":"09:35.110","Text":"I\u0027ve got sine y plus cosine y from here,"},{"Start":"09:35.110 ","End":"09:44.470","Text":"and from here minus cosine y, from here,"},{"Start":"09:44.470 ","End":"09:47.350","Text":"a minus sine y,"},{"Start":"09:47.350 ","End":"09:53.560","Text":"so this is a minus for the cosine y and a minus sine y look,"},{"Start":"09:53.560 ","End":"09:57.170","Text":"everything\u0027s going to cancel, isn\u0027t that great?"},{"Start":"09:58.500 ","End":"10:05.725","Text":"I just should have put extra brackets because there\u0027s a plus here."},{"Start":"10:05.725 ","End":"10:10.195","Text":"As I was saying, sine y, sine y,"},{"Start":"10:10.195 ","End":"10:12.010","Text":"cosine y cosine y,"},{"Start":"10:12.010 ","End":"10:17.110","Text":"all this disappears, I\u0027m just left with 2y."},{"Start":"10:17.110 ","End":"10:22.360","Text":"Next I want to write the double integral over the region as an iterated integral,"},{"Start":"10:22.360 ","End":"10:29.095","Text":"it seems like the best thing to do is to take it as a type 1 region with vertical slices,"},{"Start":"10:29.095 ","End":"10:32.005","Text":"which means I take x,"},{"Start":"10:32.005 ","End":"10:37.405","Text":"the outer loop from minus 2- 2,"},{"Start":"10:37.405 ","End":"10:45.040","Text":"and we take vertical slices for a given x from minus 2-2, I mean,"},{"Start":"10:45.040 ","End":"10:46.225","Text":"we cut the region,"},{"Start":"10:46.225 ","End":"10:50.050","Text":"we enter this point and we exit at this point,"},{"Start":"10:50.050 ","End":"10:52.525","Text":"which means I need to know both these functions."},{"Start":"10:52.525 ","End":"10:55.810","Text":"Well, this function is the x-axis,"},{"Start":"10:55.810 ","End":"11:00.220","Text":"that\u0027s where y equals 0,"},{"Start":"11:00.220 ","End":"11:03.880","Text":"and this function is the upper half of the ellipse."},{"Start":"11:03.880 ","End":"11:06.000","Text":"That\u0027s y equals, well,"},{"Start":"11:06.000 ","End":"11:07.620","Text":"what is it equal to?"},{"Start":"11:07.620 ","End":"11:13.650","Text":"If I look here and I bring the x squared over 4 to the other side,"},{"Start":"11:13.650 ","End":"11:19.360","Text":"you can see that y is plus or minus the square root,"},{"Start":"11:19.360 ","End":"11:22.240","Text":"I need the plus square root because I\u0027m positive,"},{"Start":"11:22.240 ","End":"11:28.160","Text":"so it\u0027s just the square root of 1 minus x squared over 4,"},{"Start":"11:29.250 ","End":"11:35.590","Text":"and that\u0027s okay, as I said,"},{"Start":"11:35.590 ","End":"11:43.750","Text":"x goes minus 2-2 so we have the integral from minus 2-2,"},{"Start":"11:43.750 ","End":"11:45.850","Text":"and that is the dx."},{"Start":"11:45.850 ","End":"11:52.720","Text":"Inside that I have the integral from 0 to square root of"},{"Start":"11:52.720 ","End":"12:01.270","Text":"1 minus x squared over 4 dy,"},{"Start":"12:01.270 ","End":"12:09.040","Text":"and all that\u0027s left of it is the 2y."},{"Start":"12:09.040 ","End":"12:12.730","Text":"As always, we start from the inside,"},{"Start":"12:12.730 ","End":"12:18.655","Text":"which is this, and I think I\u0027ll do this as a side exercise."},{"Start":"12:18.655 ","End":"12:22.735","Text":"From the 2y, I get y squared,"},{"Start":"12:22.735 ","End":"12:27.265","Text":"and I have to evaluate this"},{"Start":"12:27.265 ","End":"12:34.825","Text":"between 0 and the square root of 1 minus x squared over 4,"},{"Start":"12:34.825 ","End":"12:39.385","Text":"which means that this is equal to,"},{"Start":"12:39.385 ","End":"12:45.895","Text":"I just have to plug in the top and the bottom and then subtract,"},{"Start":"12:45.895 ","End":"12:49.450","Text":"if I plug in y equals this,"},{"Start":"12:49.450 ","End":"12:56.639","Text":"then y squared is just 1 minus x squared over 4,"},{"Start":"12:56.639 ","End":"13:00.525","Text":"I plug-in 0, I get 0 so there\u0027s nothing to subtract."},{"Start":"13:00.525 ","End":"13:03.990","Text":"This is the answer for this better to write it over here,"},{"Start":"13:03.990 ","End":"13:10.000","Text":"1 minus x squared over 4, and continuing,"},{"Start":"13:10.000 ","End":"13:13.975","Text":"I get the integral from"},{"Start":"13:13.975 ","End":"13:23.420","Text":"minus 2-2 of 1 minus x squared over 4 dx."},{"Start":"13:23.640 ","End":"13:26.800","Text":"Let\u0027s see what that gives us,"},{"Start":"13:26.800 ","End":"13:33.685","Text":"that gives us x minus,"},{"Start":"13:33.685 ","End":"13:41.165","Text":"this will be x cubed over 3 with the 4 goes makes it 12,"},{"Start":"13:41.165 ","End":"13:46.740","Text":"and this has to be taken from minus 2-2."},{"Start":"13:46.740 ","End":"13:48.105","Text":"Let\u0027s see what we get,"},{"Start":"13:48.105 ","End":"13:50.310","Text":"if I plug in the 2,"},{"Start":"13:50.310 ","End":"13:57.460","Text":"I\u0027ve got 2 minus 2 cubed is 8,"},{"Start":"13:57.460 ","End":"14:07.225","Text":"8 over 12 is 2/3,"},{"Start":"14:07.225 ","End":"14:08.440","Text":"when I put in minus 2,"},{"Start":"14:08.440 ","End":"14:11.860","Text":"I\u0027m just going to get the same thing but with opposite sign,"},{"Start":"14:11.860 ","End":"14:18.800","Text":"so it will just be the minus of this,"},{"Start":"14:19.410 ","End":"14:21.445","Text":"okay [inaudible] right anyway,"},{"Start":"14:21.445 ","End":"14:23.470","Text":"minus 2 plus 2/3."},{"Start":"14:23.470 ","End":"14:25.870","Text":"I could have just computed this and then doubled it,"},{"Start":"14:25.870 ","End":"14:27.805","Text":"because if I\u0027m subtracting the negative,"},{"Start":"14:27.805 ","End":"14:33.550","Text":"like that\u0027s what I\u0027ll do, 2 minus 2/3 is 1-and-a-third ,"},{"Start":"14:33.550 ","End":"14:40.780","Text":"1-and-a-third minus, minus 1-and-a -third will make it 2 and 2-thirds,"},{"Start":"14:40.780 ","End":"14:44.110","Text":"or if you like, 8 over 3,"},{"Start":"14:44.110 ","End":"14:46.810","Text":"we\u0027ll use whichever is more convenient for us,"},{"Start":"14:46.810 ","End":"14:51.190","Text":"and that is the double integral."},{"Start":"14:51.190 ","End":"14:58.810","Text":"We\u0027ve got that this part here is 8/3,"},{"Start":"14:58.810 ","End":"15:01.630","Text":"and now we\u0027ll go and compute this bit here,"},{"Start":"15:01.630 ","End":"15:09.040","Text":"the line integral over L. I cleared the board a bit let\u0027s see,"},{"Start":"15:09.040 ","End":"15:11.650","Text":"I need the beginning,"},{"Start":"15:11.650 ","End":"15:13.780","Text":"and what we need is this,"},{"Start":"15:13.780 ","End":"15:20.890","Text":"which means the line integral over L of fdx plus gdy."},{"Start":"15:20.890 ","End":"15:26.110","Text":"For this, I need to parametrize L, let\u0027s see,"},{"Start":"15:26.110 ","End":"15:32.485","Text":"L is a straight line from minus 2-2 along the x-axis,"},{"Start":"15:32.485 ","End":"15:36.130","Text":"and so let\u0027s see how I parametrize it."},{"Start":"15:36.130 ","End":"15:40.390","Text":"There is of course, a formula for a line segment from one point to the other,"},{"Start":"15:40.390 ","End":"15:43.390","Text":"but we won\u0027t need it because we have a simple case."},{"Start":"15:43.390 ","End":"15:45.955","Text":"Because we\u0027re along the x-axis,"},{"Start":"15:45.955 ","End":"15:50.665","Text":"we know that y is constantly equal to 0 on this segment."},{"Start":"15:50.665 ","End":"15:54.175","Text":"I\u0027ll just have to say that x goes from minus 2-2,"},{"Start":"15:54.175 ","End":"15:57.640","Text":"and I do that by saying that x equals my parameter t,"},{"Start":"15:57.640 ","End":"16:01.255","Text":"but that t goes from minus 2-2."},{"Start":"16:01.255 ","End":"16:05.530","Text":"That\u0027s most straightforward way of doing it rather than cranking out"},{"Start":"16:05.530 ","End":"16:10.960","Text":"the formula for the general segment between 2 given points."},{"Start":"16:10.960 ","End":"16:15.865","Text":"I\u0027ll also be needing dx and dy might as well do it now,"},{"Start":"16:15.865 ","End":"16:21.670","Text":"so dx is equal to dt from here,"},{"Start":"16:21.670 ","End":"16:24.370","Text":"and dy is 0,"},{"Start":"16:24.370 ","End":"16:27.250","Text":"dt, or just plain 0."},{"Start":"16:27.250 ","End":"16:35.785","Text":"That already tells us that this part is going to cancel because dy is 0."},{"Start":"16:35.785 ","End":"16:39.385","Text":"I just have f dx, now,"},{"Start":"16:39.385 ","End":"16:43.250","Text":"f is written here,"},{"Start":"16:44.130 ","End":"16:47.650","Text":"what we get is the integral."},{"Start":"16:47.650 ","End":"16:53.214","Text":"Now, I need to do this in terms of t,"},{"Start":"16:53.214 ","End":"16:58.960","Text":"t goes from minus 2-2,"},{"Start":"16:58.960 ","End":"17:01.540","Text":"I just think I\u0027ll highlight this to make it easier to see,"},{"Start":"17:01.540 ","End":"17:04.975","Text":"this is the bit that I am going to be substituting in,"},{"Start":"17:04.975 ","End":"17:09.910","Text":"and I\u0027m going to be substituting x equals t and y equals 0."},{"Start":"17:09.910 ","End":"17:17.590","Text":"Minus 2e^2x is 2t,"},{"Start":"17:17.590 ","End":"17:20.405","Text":"y is just 0,"},{"Start":"17:20.405 ","End":"17:26.135","Text":"and then cosine of y, which is 0,"},{"Start":"17:26.135 ","End":"17:29.805","Text":"dx is d t,"},{"Start":"17:29.805 ","End":"17:39.730","Text":"and this is equal to the integral from minus 2-2."},{"Start":"17:39.730 ","End":"17:42.760","Text":"Cosine of 0 is 1,"},{"Start":"17:42.760 ","End":"17:49.760","Text":"and so all I get is minus 2e^2t,"},{"Start":"17:54.240 ","End":"17:57.585","Text":"and that\u0027s the dt."},{"Start":"17:57.585 ","End":"18:00.215","Text":"This is fairly straightforward."},{"Start":"18:00.215 ","End":"18:05.195","Text":"The integral of 2e^2t is almost e^2t,"},{"Start":"18:05.195 ","End":"18:08.510","Text":"I just have to divide by 2,"},{"Start":"18:08.510 ","End":"18:10.880","Text":"so that will cancel with the 2,"},{"Start":"18:10.880 ","End":"18:21.250","Text":"and I\u0027ll be left with minus e^2t between minus 2 and 2."},{"Start":"18:21.250 ","End":"18:25.525","Text":"Let\u0027s see, if I plug in 2,"},{"Start":"18:25.525 ","End":"18:31.300","Text":"I get minus e^4,"},{"Start":"18:31.300 ","End":"18:37.420","Text":"I plug in minus 2 and subtract to get minus minus,"},{"Start":"18:37.420 ","End":"18:43.980","Text":"which is plus e^ minus 4."},{"Start":"18:43.980 ","End":"18:49.010","Text":"So this is the answer for the integral over L,"},{"Start":"18:49.010 ","End":"18:57.860","Text":"I can plug that into here now and say that I\u0027ve got to take this minus this."},{"Start":"18:57.860 ","End":"19:03.740","Text":"Now, because it\u0027s a minus, I\u0027m subtracting this,"},{"Start":"19:03.740 ","End":"19:14.500","Text":"that will make this a plus e^4 and a minus e^ minus 4 because there\u0027s a minus here."},{"Start":"19:14.500 ","End":"19:17.784","Text":"This will be the final answer,"},{"Start":"19:17.784 ","End":"19:22.780","Text":"so that\u0027s it, we\u0027re done."}],"ID":8776},{"Watched":false,"Name":"Exercise 7 part 1","Duration":"3m 54s","ChapterTopicVideoID":8651,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.825","Text":"In this exercise, we have to prove that the area bounded by a simple closed curve C"},{"Start":"00:06.825 ","End":"00:09.078","Text":"is given as follows:"},{"Start":"00:09.078 ","End":"00:14.129","Text":"we have the area as a type 2 line integral."},{"Start":"00:14.129 ","End":"00:16.980","Text":"I\u0027ll just do a quick sketch."},{"Start":"00:16.980 ","End":"00:25.500","Text":"Here we have some curve C. So this is C. It doesn\u0027t say,"},{"Start":"00:25.500 ","End":"00:28.290","Text":"I should have written that the default is"},{"Start":"00:28.290 ","End":"00:34.530","Text":"the positive mathematical direction, which is anticlockwise,"},{"Start":"00:34.530 ","End":"00:39.945","Text":"it\u0027s important, maybe I should have said specifically, that C is anticlockwise,"},{"Start":"00:39.945 ","End":"00:43.135","Text":"it bound some region,"},{"Start":"00:43.135 ","End":"00:49.360","Text":"area, call it R, that\u0027s the inside."},{"Start":"00:49.360 ","End":"00:53.850","Text":"When the section on Green\u0027s theorem,"},{"Start":"00:53.850 ","End":"00:58.100","Text":"so obviously that\u0027s what we\u0027re supposed to use, in general,"},{"Start":"00:58.100 ","End":"01:07.910","Text":"the Green\u0027s theorem says that the line integral over a simple closed,"},{"Start":"01:07.910 ","End":"01:16.190","Text":"it\u0027s also got to be piecewise smooth."},{"Start":"01:16.190 ","End":"01:21.470","Text":"Anyway, let\u0027s assume all those good things of the curve C in"},{"Start":"01:21.470 ","End":"01:28.325","Text":"the positive direction of fdx plus gdy,"},{"Start":"01:28.325 ","End":"01:38.140","Text":"that this will equal the double integral over the region R of gy,"},{"Start":"01:38.140 ","End":"01:40.315","Text":"partial derivative that is,"},{"Start":"01:40.315 ","End":"01:44.730","Text":"minus f partial derivative with respect to xdA."},{"Start":"01:44.830 ","End":"01:48.905","Text":"Let\u0027s see if we can get this to look like this."},{"Start":"01:48.905 ","End":"01:53.840","Text":"Well, f is the part that goes with dx."},{"Start":"01:53.840 ","End":"01:56.090","Text":"The dx has a minus y,"},{"Start":"01:56.090 ","End":"01:57.935","Text":"but there\u0027s a half in front."},{"Start":"01:57.935 ","End":"02:08.540","Text":"So if I take f of xy is equal to minus a half y,"},{"Start":"02:08.540 ","End":"02:11.030","Text":"and if I take g to equal,"},{"Start":"02:11.030 ","End":"02:13.040","Text":"g is the bit in front of dy,"},{"Start":"02:13.040 ","End":"02:15.730","Text":"it\u0027s x but times a half,"},{"Start":"02:15.730 ","End":"02:18.720","Text":"so g is equal to a half x."},{"Start":"02:18.720 ","End":"02:21.485","Text":"Then I apply this formula,"},{"Start":"02:21.485 ","End":"02:27.890","Text":"then I\u0027ll get that the integral of, well,"},{"Start":"02:27.890 ","End":"02:31.459","Text":"if I did just did it literally would be minus a half"},{"Start":"02:31.459 ","End":"02:41.040","Text":"of ydx plus a half xdy."},{"Start":"02:41.810 ","End":"02:44.760","Text":"But this is the same as this,"},{"Start":"02:44.760 ","End":"02:47.825","Text":"I mean, it\u0027s just rewritten."},{"Start":"02:47.825 ","End":"02:53.000","Text":"This is equal to the double integral along"},{"Start":"02:53.000 ","End":"02:59.220","Text":"R. Let\u0027s see what is gy minus f with respect to x,"},{"Start":"02:59.220 ","End":"03:03.580","Text":"g with respect to,"},{"Start":"03:03.580 ","End":"03:06.635","Text":"wait a minute, whoops, I got these backwards."},{"Start":"03:06.635 ","End":"03:09.140","Text":"Sorry, I was doing it from memory."},{"Start":"03:09.140 ","End":"03:14.360","Text":"G with respect to x is just 1 half,"},{"Start":"03:14.360 ","End":"03:19.610","Text":"and f with respect to y is minus a half,"},{"Start":"03:19.610 ","End":"03:25.705","Text":"so it\u0027s minus, minus a half dA,"},{"Start":"03:25.705 ","End":"03:28.274","Text":"and this is just equal,"},{"Start":"03:28.274 ","End":"03:30.550","Text":"a half minus, minus a half is 1,"},{"Start":"03:30.550 ","End":"03:38.410","Text":"so it\u0027s just the double integral of over R of dA or we could keep the 1 in 1 dA."},{"Start":"03:38.410 ","End":"03:41.105","Text":"But this is well-known."},{"Start":"03:41.105 ","End":"03:46.385","Text":"We\u0027ve seen this before that this is exactly equal to the area of"},{"Start":"03:46.385 ","End":"03:54.550","Text":"R. That\u0027s what we had to prove, and we\u0027re done."}],"ID":8777},{"Watched":false,"Name":"Exercise 7 part 2","Duration":"6m 34s","ChapterTopicVideoID":8652,"CourseChapterTopicPlaylistID":4959,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.445","Text":"This exercise is the second part of a 2 part exercise, where in the first part,"},{"Start":"00:05.445 ","End":"00:07.650","Text":"we showed that, in general,"},{"Start":"00:07.650 ","End":"00:15.180","Text":"the area of a region bounded by a simple closed curve is given as follows."},{"Start":"00:15.180 ","End":"00:20.910","Text":"The little sketch will help to explain because otherwise, it\u0027s a bit hard to understand."},{"Start":"00:20.910 ","End":"00:30.740","Text":"We took a simple closed curve in the positive orientation."},{"Start":"00:30.740 ","End":"00:35.275","Text":"Positive meaning counterclockwise."},{"Start":"00:35.275 ","End":"00:38.465","Text":"This bounds a certain region,"},{"Start":"00:38.465 ","End":"00:40.280","Text":"and we\u0027ll call this R,"},{"Start":"00:40.280 ","End":"00:47.400","Text":"we usually use R or D. What we showed is that,"},{"Start":"00:47.690 ","End":"00:57.080","Text":"if we take A to be the area of the region R that C goes around,"},{"Start":"00:57.080 ","End":"00:59.885","Text":"then we showed that it\u0027s given by"},{"Start":"00:59.885 ","End":"01:03.470","Text":"this Type 2 line"},{"Start":"01:03.470 ","End":"01:08.585","Text":"integral over the closed curve C. That\u0027s why we have a little circle here."},{"Start":"01:08.585 ","End":"01:13.100","Text":"I put in an extra picture, because in our case, it doesn\u0027t quite look like this."},{"Start":"01:13.100 ","End":"01:15.410","Text":"The ellipse looks more like this is,"},{"Start":"01:15.410 ","End":"01:17.300","Text":"here\u0027s the C, here\u0027s R,"},{"Start":"01:17.300 ","End":"01:19.255","Text":"this is a standard ellipse,"},{"Start":"01:19.255 ","End":"01:22.065","Text":"a here, b here."},{"Start":"01:22.065 ","End":"01:26.030","Text":"There\u0027s a standard parametrization of the ellipse."},{"Start":"01:26.030 ","End":"01:33.390","Text":"The ellipse, meaning just the contour, the line."},{"Start":"01:33.460 ","End":"01:40.100","Text":"The parametrization for this is as follows."},{"Start":"01:40.100 ","End":"01:47.555","Text":"It is x equals a cosine t,"},{"Start":"01:47.555 ","End":"01:53.830","Text":"y equals b sine t,"},{"Start":"01:53.830 ","End":"02:00.030","Text":"and t goes from 0-2 Pi."},{"Start":"02:00.030 ","End":"02:03.410","Text":"You can check certain points, for example,"},{"Start":"02:03.410 ","End":"02:09.830","Text":"when t is 0, we get the point a comma 0."},{"Start":"02:09.830 ","End":"02:13.790","Text":"When t is Pi over 2 or 90 degrees,"},{"Start":"02:13.790 ","End":"02:18.500","Text":"then we get that this is 0 and y is b,"},{"Start":"02:18.500 ","End":"02:26.080","Text":"so that\u0027s here.180 degrees or Pi gives you this and so on."},{"Start":"02:26.080 ","End":"02:30.600","Text":"This is the parametrization we\u0027re going to use."},{"Start":"02:33.940 ","End":"02:39.855","Text":"Well, maybe I should also compute dx and dy I see I\u0027m going to need them,"},{"Start":"02:39.855 ","End":"02:47.410","Text":"dx is going to be minus a sine tdt,"},{"Start":"02:47.600 ","End":"02:56.305","Text":"and dy is going to equal b cosine of t, dt."},{"Start":"02:56.305 ","End":"02:59.100","Text":"I think that\u0027s everything we need."},{"Start":"02:59.100 ","End":"03:04.010","Text":"So a, which is in our case, the area of the ellipse,"},{"Start":"03:04.010 ","End":"03:11.940","Text":"that area A is equal to 1/2 the integral."},{"Start":"03:11.940 ","End":"03:21.240","Text":"Now, see, I\u0027m using the parametrization of the parameter goes from 0-2Pi,"},{"Start":"03:21.240 ","End":"03:24.620","Text":"that\u0027s the parameter t. Then I"},{"Start":"03:24.620 ","End":"03:28.085","Text":"need to plug everything in along the curve, along the ellipse,"},{"Start":"03:28.085 ","End":"03:37.430","Text":"x is a cosine t. Then dy is"},{"Start":"03:37.430 ","End":"03:47.610","Text":"b cosine t dt minus y,"},{"Start":"03:47.610 ","End":"03:54.765","Text":"dx minus y is b sine t,"},{"Start":"03:54.765 ","End":"04:05.410","Text":"and dx is minus a sine tdt."},{"Start":"04:08.720 ","End":"04:12.500","Text":"Let\u0027s see if we can simplify this."},{"Start":"04:12.500 ","End":"04:15.930","Text":"What we get is"},{"Start":"04:17.560 ","End":"04:24.530","Text":"1/2 of the integral from 0-2Pi."},{"Start":"04:24.530 ","End":"04:27.515","Text":"Now, let\u0027s see, from the first bit,"},{"Start":"04:27.515 ","End":"04:36.750","Text":"we get ab cosine squared tdt,"},{"Start":"04:37.220 ","End":"04:40.410","Text":"but I\u0027ll put the dt at the end."},{"Start":"04:40.410 ","End":"04:43.910","Text":"From the next bit, we have a minus and a minus,"},{"Start":"04:43.910 ","End":"04:45.785","Text":"so it\u0027s a plus."},{"Start":"04:45.785 ","End":"04:51.905","Text":"I\u0027ve got ba, which I\u0027ll write as ab,"},{"Start":"04:51.905 ","End":"04:54.095","Text":"because I want it to look like this,"},{"Start":"04:54.095 ","End":"05:02.540","Text":"so ba or ab sine squared t and then the dt."},{"Start":"05:03.130 ","End":"05:12.380","Text":"What I have is, I can take ab outside the brackets here, and all the way in front,"},{"Start":"05:12.380 ","End":"05:19.480","Text":"so I get 1/2 ab times the integral from"},{"Start":"05:19.520 ","End":"05:25.340","Text":"0-2Pi of cosine squared t"},{"Start":"05:25.340 ","End":"05:31.340","Text":"plus sine squared t dt."},{"Start":"05:31.340 ","End":"05:38.575","Text":"But cosine squared plus sine squared is equal to 1,"},{"Start":"05:38.575 ","End":"05:46.230","Text":"so the integral of 1 is just the upper limit minus the lower limit, which is 2Pi."},{"Start":"05:46.900 ","End":"05:57.265","Text":"This is going to equal 1/2 ab times 2Pi,"},{"Start":"05:57.265 ","End":"06:05.145","Text":"and simplified, it\u0027s just equal to Pi ab."},{"Start":"06:05.145 ","End":"06:08.060","Text":"I\u0027ll highlight this, and this is actually"},{"Start":"06:08.060 ","End":"06:12.350","Text":"the correct formula for the area of the ellipse."},{"Start":"06:12.350 ","End":"06:15.740","Text":"Just to give you a, for example, we are done,"},{"Start":"06:15.740 ","End":"06:18.620","Text":"but I\u0027m just saying if a and b were equal,"},{"Start":"06:18.620 ","End":"06:20.285","Text":"say they\u0027re both equal to R,"},{"Start":"06:20.285 ","End":"06:22.100","Text":"we\u0027d get Pi RR,"},{"Start":"06:22.100 ","End":"06:23.965","Text":"which is Pi R-squared."},{"Start":"06:23.965 ","End":"06:25.920","Text":"For a circle it makes sense,"},{"Start":"06:25.920 ","End":"06:29.555","Text":"and in general, this is the correct formula."},{"Start":"06:29.555 ","End":"06:34.470","Text":"We use Green\u0027s theorem. Now, I\u0027m really done."}],"ID":8778}],"Thumbnail":null,"ID":4959}]

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