Volume - Solids of Revolution
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Surface Area - Solids of Revolution
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Volume by Integrating Cross Sections
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[{"Name":"Volume - Solids of Revolution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Solids of Revolution","Duration":"17m 31s","ChapterTopicVideoID":4477,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/4477.jpeg","UploadDate":"2019-12-11T21:15:15.1070000","DurationForVideoObject":"PT17M31S","Description":null,"MetaTitle":"Solids of Revolution: Video + Workbook | Proprep","MetaDescription":"Applications of the Definite Integral - Volume and Surface Area - Volume - Solids of Revolution. 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/applications-of-the-definite-integral-_-volume-and-surface-area/volume-_-solids-of-revolution/vid4486","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this clip I\u0027m going to talk about solids of revolution,"},{"Start":"00:03.570 ","End":"00:07.080","Text":"and specifically the formulas for computing them."},{"Start":"00:07.080 ","End":"00:09.269","Text":"I\u0027m not going to get into any theory."},{"Start":"00:09.269 ","End":"00:16.380","Text":"Now, a solid of revolution could be the result of revolution about the x-axis,"},{"Start":"00:16.380 ","End":"00:19.890","Text":"or about the y-axis and we\u0027re going to start with the x-axis."},{"Start":"00:19.890 ","End":"00:21.060","Text":"There are 2 methods,"},{"Start":"00:21.060 ","End":"00:25.990","Text":"and 1 of them is called the disk method."},{"Start":"00:29.030 ","End":"00:34.410","Text":"The other one we\u0027ll get to it is called the shell method or cylindrical shell."},{"Start":"00:34.410 ","End":"00:36.930","Text":"Let\u0027s begin with this."},{"Start":"00:36.930 ","End":"00:44.290","Text":"We assume we have 2 curves and we want the area contained between the 2 curves."},{"Start":"00:44.290 ","End":"00:49.250","Text":"Let\u0027s say that 1 of them is y equals f of x,"},{"Start":"00:49.250 ","End":"00:53.805","Text":"and the other 1 is y equals g of x."},{"Start":"00:53.805 ","End":"00:59.215","Text":"Essentially, the area in between,"},{"Start":"00:59.215 ","End":"01:04.250","Text":"I can think of as a set of vertical lines."},{"Start":"01:04.250 ","End":"01:07.490","Text":"Mine don\u0027t look so vertical but they\u0027re meant to be."},{"Start":"01:07.490 ","End":"01:11.000","Text":"Using this idea, there is a formula for"},{"Start":"01:11.000 ","End":"01:14.794","Text":"computing the volume obtained by taking this area,"},{"Start":"01:14.794 ","End":"01:19.620","Text":"and revolving it about the x-axis."},{"Start":"01:19.620 ","End":"01:23.290","Text":"Then you get a hollow,"},{"Start":"01:23.290 ","End":"01:25.390","Text":"I don\u0027t know how to describe it."},{"Start":"01:25.390 ","End":"01:28.040","Text":"The formula is as follows, V,"},{"Start":"01:28.040 ","End":"01:32.610","Text":"that\u0027s the volume of the solid of revolution,"},{"Start":"01:32.610 ","End":"01:37.970","Text":"and this is equal to the integral from a to b,"},{"Start":"01:37.970 ","End":"01:41.250","Text":"where a and b is just would add it below."},{"Start":"01:41.870 ","End":"01:46.035","Text":"F of x squared,"},{"Start":"01:46.035 ","End":"01:51.210","Text":"minus g of x squared,"},{"Start":"01:51.210 ","End":"01:54.020","Text":"I\u0027m going to put another brackets here,"},{"Start":"01:54.020 ","End":"01:59.820","Text":"all this dx, that\u0027s not it, times Pi."},{"Start":"02:00.120 ","End":"02:06.150","Text":"I want you to pay special attention that it\u0027s f of x squared,"},{"Start":"02:06.150 ","End":"02:07.890","Text":"minus g of x squared,"},{"Start":"02:07.890 ","End":"02:13.709","Text":"because there is this tendency to get confused and to subtract first, and then square."},{"Start":"02:13.709 ","End":"02:15.825","Text":"Each 1 is squared separately,"},{"Start":"02:15.825 ","End":"02:18.190","Text":"and then there\u0027s a subtraction."},{"Start":"02:18.190 ","End":"02:26.470","Text":"This is the formula for the disk method of computing the solid of revolution."},{"Start":"02:26.470 ","End":"02:32.400","Text":"The other method is called the shell method,"},{"Start":"02:32.400 ","End":"02:37.060","Text":"although it\u0027s also sometimes called the cylindrical shell method."},{"Start":"02:37.060 ","End":"02:41.410","Text":"I suppose you could add the word cylindrical."},{"Start":"02:41.410 ","End":"02:51.685","Text":"It\u0027s based on a different way of viewing the area as more as horizontal slices."},{"Start":"02:51.685 ","End":"02:56.415","Text":"The theory of the formula is based on slicing it horizontally,"},{"Start":"02:56.415 ","End":"02:59.040","Text":"whereas here we did it vertically."},{"Start":"02:59.040 ","End":"03:01.840","Text":"I\u0027ll shade it this way."},{"Start":"03:01.910 ","End":"03:06.690","Text":"Also, what I need is instead of a and b,"},{"Start":"03:06.690 ","End":"03:11.500","Text":"I\u0027m going to have c and d here."},{"Start":"03:14.150 ","End":"03:18.200","Text":"Of course we have to have these functions,"},{"Start":"03:18.200 ","End":"03:21.470","Text":"but this time, instead of y in terms of x,"},{"Start":"03:21.470 ","End":"03:23.840","Text":"we have x in terms of y."},{"Start":"03:23.840 ","End":"03:31.340","Text":"Let\u0027s say that this curve is x equals a function u of y,"},{"Start":"03:31.340 ","End":"03:34.055","Text":"and this curve here, well,"},{"Start":"03:34.055 ","End":"03:38.640","Text":"this is not so good because this isn\u0027t really a function."},{"Start":"03:39.020 ","End":"03:44.330","Text":"X equals V of y. I\u0027m actually going to"},{"Start":"03:44.330 ","End":"03:49.505","Text":"alter it because this is not a function. That\u0027s better."},{"Start":"03:49.505 ","End":"03:51.920","Text":"Now, I can give you the formula."},{"Start":"03:51.920 ","End":"03:54.540","Text":"Again, V for volume equals,"},{"Start":"03:54.540 ","End":"03:57.315","Text":"instead of Pi, we have 2Pi."},{"Start":"03:57.315 ","End":"04:01.650","Text":"The integral now is from c to d,"},{"Start":"04:01.650 ","End":"04:04.675","Text":"and we have y."},{"Start":"04:04.675 ","End":"04:13.405","Text":"This is going to be an integral in y. Y times the right 1 minus the left 1,"},{"Start":"04:13.405 ","End":"04:20.385","Text":"V of y, minus u of y,"},{"Start":"04:20.385 ","End":"04:23.770","Text":"and all this dy."},{"Start":"04:24.440 ","End":"04:29.065","Text":"Of course it\u0027s supposed to come out the same answer as the other method,"},{"Start":"04:29.065 ","End":"04:30.520","Text":"and the question is,"},{"Start":"04:30.520 ","End":"04:34.105","Text":"why would we use 1 rather than another?"},{"Start":"04:34.105 ","End":"04:40.975","Text":"The answer to that is that sometimes it\u0027s easier to extract y in terms of x,"},{"Start":"04:40.975 ","End":"04:44.415","Text":"and sometimes it\u0027s easier to extract x in terms of y."},{"Start":"04:44.415 ","End":"04:47.510","Text":"For example these might be defined implicitly and it"},{"Start":"04:47.510 ","End":"04:50.870","Text":"might be hard to extract 1 in terms of the other."},{"Start":"04:50.870 ","End":"04:54.785","Text":"If we can easier write y as a function of x,"},{"Start":"04:54.785 ","End":"04:56.990","Text":"then we use the disk method."},{"Start":"04:56.990 ","End":"04:58.760","Text":"If we can get x in terms of y,"},{"Start":"04:58.760 ","End":"05:00.725","Text":"we use the shell method,"},{"Start":"05:00.725 ","End":"05:02.495","Text":"let me write that down."},{"Start":"05:02.495 ","End":"05:09.240","Text":"Use the disk method if it\u0027s easier to extract y in terms of x for both curves,"},{"Start":"05:09.240 ","End":"05:16.210","Text":"and use the shell method if it\u0027s easier to extract x in terms of y."},{"Start":"05:16.210 ","End":"05:21.425","Text":"This covers the revolution about the x-axis."},{"Start":"05:21.425 ","End":"05:26.495","Text":"Next up, we going to do the y-axis."},{"Start":"05:26.495 ","End":"05:30.845","Text":"Now, we\u0027re moving on to revolution about the y-axis."},{"Start":"05:30.845 ","End":"05:34.714","Text":"I\u0027ve pretty much reused the sketches from before,"},{"Start":"05:34.714 ","End":"05:37.550","Text":"but the big difference is the axes of rotation."},{"Start":"05:37.550 ","End":"05:39.140","Text":"Previously, it was the x-axis,"},{"Start":"05:39.140 ","End":"05:40.570","Text":"now it\u0027s the y-axis."},{"Start":"05:40.570 ","End":"05:42.150","Text":"Before it was the x-axis,"},{"Start":"05:42.150 ","End":"05:43.655","Text":"now it\u0027s the y-axis,"},{"Start":"05:43.655 ","End":"05:46.955","Text":"which also gives us a solid of revolution."},{"Start":"05:46.955 ","End":"05:54.800","Text":"The big difference is that this time when we\u0027re going around the y-axis,"},{"Start":"05:54.800 ","End":"05:58.259","Text":"this becomes the disk method,"},{"Start":"06:00.770 ","End":"06:06.725","Text":"and the other 1 becomes the shell method,"},{"Start":"06:06.725 ","End":"06:10.125","Text":"or cylindrical shell method."},{"Start":"06:10.125 ","End":"06:13.950","Text":"The formulas are very similar."},{"Start":"06:13.950 ","End":"06:15.485","Text":"In the disk method,"},{"Start":"06:15.485 ","End":"06:16.850","Text":"we get that V,"},{"Start":"06:16.850 ","End":"06:22.120","Text":"the volume of the solid of revolution, is equal Pi,"},{"Start":"06:22.120 ","End":"06:32.030","Text":"times the integral from c to d of V of y squared."},{"Start":"06:32.030 ","End":"06:33.770","Text":"Yes, that\u0027s the common mistake."},{"Start":"06:33.770 ","End":"06:35.180","Text":"and want to remind you again."},{"Start":"06:35.180 ","End":"06:37.640","Text":"It\u0027s not V minus u all squared."},{"Start":"06:37.640 ","End":"06:40.310","Text":"It\u0027s V separately squared,"},{"Start":"06:40.310 ","End":"06:47.110","Text":"and U separately squared and then dy."},{"Start":"06:48.410 ","End":"06:50.775","Text":"For the shell method,"},{"Start":"06:50.775 ","End":"06:57.774","Text":"we get that V is equal to 2Pi this time,"},{"Start":"06:57.774 ","End":"07:04.060","Text":"times the integral from a to b of x,"},{"Start":"07:04.060 ","End":"07:08.760","Text":"times the upper minus the lower;"},{"Start":"07:08.760 ","End":"07:16.035","Text":"f of x minus g of x, dx."},{"Start":"07:16.035 ","End":"07:19.955","Text":"Of course if you solve the problem with both techniques,"},{"Start":"07:19.955 ","End":"07:24.440","Text":"then they should give you the same answer of course."},{"Start":"07:24.440 ","End":"07:29.225","Text":"Just like before, when would you use 1 method and not another?"},{"Start":"07:29.225 ","End":"07:31.910","Text":"I\u0027ll just copy what I wrote before."},{"Start":"07:31.910 ","End":"07:33.995","Text":"Here\u0027s what I wrote before,"},{"Start":"07:33.995 ","End":"07:38.960","Text":"but I have to correct that because disk and shell have to be reversed,"},{"Start":"07:38.960 ","End":"07:47.130","Text":"and this is the now the shell method."},{"Start":"07:47.820 ","End":"07:52.555","Text":"The shell method, when I\u0027ve got y in terms of x,"},{"Start":"07:52.555 ","End":"07:55.855","Text":"these become shells when I rotate."},{"Start":"07:55.855 ","End":"07:58.690","Text":"When I\u0027ve got x in terms of y,"},{"Start":"07:58.690 ","End":"08:04.210","Text":"these become discs when I rotate around the y-axis."},{"Start":"08:04.210 ","End":"08:06.880","Text":"This is the disc method,"},{"Start":"08:06.880 ","End":"08:11.870","Text":"so it\u0027s just the reverse of what it was before."},{"Start":"08:13.620 ","End":"08:19.570","Text":"That\u0027s it with the revolution around the y-axis."},{"Start":"08:19.570 ","End":"08:25.420","Text":"In this clip, we\u0027re going to give an example of how to compute a solid of revolution."},{"Start":"08:25.420 ","End":"08:27.670","Text":"Up to now what we\u0027ve had are formulas,"},{"Start":"08:27.670 ","End":"08:29.815","Text":"it\u0027s time we get an actual example,"},{"Start":"08:29.815 ","End":"08:31.720","Text":"and here is 1."},{"Start":"08:31.720 ","End":"08:34.540","Text":"The area enclosed by the curves,"},{"Start":"08:34.540 ","End":"08:38.140","Text":"y equals x squared and y equals 2x,"},{"Start":"08:38.140 ","End":"08:40.495","Text":"revolves around the x-axis,"},{"Start":"08:40.495 ","End":"08:44.335","Text":"what is the volume of resulting solid of revolution?"},{"Start":"08:44.335 ","End":"08:46.945","Text":"You\u0027re asked to do it with 2 different methods,"},{"Start":"08:46.945 ","End":"08:50.500","Text":"the disc method and the cylindrical shell method."},{"Start":"08:50.500 ","End":"08:52.270","Text":"We probably want to sketch,"},{"Start":"08:52.270 ","End":"08:54.595","Text":"so let\u0027s start 1."},{"Start":"08:54.595 ","End":"08:57.310","Text":"Here are some axes."},{"Start":"08:57.310 ","End":"09:02.245","Text":"What I\u0027d like to know is where these curves intersect. That will help me."},{"Start":"09:02.245 ","End":"09:08.965","Text":"If I want to know that I just have to assign x squared equals 2x."},{"Start":"09:08.965 ","End":"09:14.950","Text":"It\u0027s easy to solve this that we get x equals 0 or x equals 2."},{"Start":"09:14.950 ","End":"09:18.160","Text":"If I substitute 0 and 2 in these,"},{"Start":"09:18.160 ","End":"09:20.020","Text":"it doesn\u0027t matter in which side,"},{"Start":"09:20.020 ","End":"09:25.465","Text":"I\u0027ll get if x is 0, y is 0 and if x is 2, y is 4."},{"Start":"09:25.465 ","End":"09:27.730","Text":"Now this goes with this and this goes with this."},{"Start":"09:27.730 ","End":"09:33.040","Text":"So we have 0,0 and we also have 2,4,"},{"Start":"09:33.040 ","End":"09:34.435","Text":"this will be here."},{"Start":"09:34.435 ","End":"09:39.890","Text":"Now we know how to draw a parabola and a straight line."},{"Start":"09:40.050 ","End":"09:44.365","Text":"We get a line in the curve, something like this."},{"Start":"09:44.365 ","End":"09:49.135","Text":"These are going to be rotated about the x-axis."},{"Start":"09:49.135 ","End":"09:53.410","Text":"We have the formulas above and I\u0027m going to copy"},{"Start":"09:53.410 ","End":"09:58.000","Text":"them because we\u0027re going to do it by both methods, so let\u0027s see."},{"Start":"09:58.000 ","End":"10:01.975","Text":"I\u0027ve shaded the area so we can see what we\u0027re talking about."},{"Start":"10:01.975 ","End":"10:05.260","Text":"It\u0027s actually easier or more natural to use"},{"Start":"10:05.260 ","End":"10:08.710","Text":"the disc method because when we have y in terms of x,"},{"Start":"10:08.710 ","End":"10:12.430","Text":"which we do, then this would be the formula to use."},{"Start":"10:12.430 ","End":"10:15.730","Text":"Let\u0027s get started with the disc method."},{"Start":"10:15.730 ","End":"10:18.490","Text":"In our case, f is the upper 1,"},{"Start":"10:18.490 ","End":"10:22.670","Text":"is the 2x, and g is the x squared function."},{"Start":"10:23.010 ","End":"10:26.740","Text":"I just added the points 4 and 2,"},{"Start":"10:26.740 ","End":"10:31.900","Text":"which came from here because I forgot earlier to do that."},{"Start":"10:31.900 ","End":"10:34.585","Text":"Now we\u0027ve got a and b,"},{"Start":"10:34.585 ","End":"10:37.465","Text":"which in our case is 0 and 2."},{"Start":"10:37.465 ","End":"10:48.115","Text":"What we get is Pi times the integral from 0-2 of the upper,"},{"Start":"10:48.115 ","End":"10:51.280","Text":"which is 2x minus the lower,"},{"Start":"10:51.280 ","End":"10:53.840","Text":"which is x squared."},{"Start":"10:54.750 ","End":"10:58.370","Text":"Each of these is squared,"},{"Start":"10:59.790 ","End":"11:06.950","Text":"f squared minus g squared and dx."},{"Start":"11:07.500 ","End":"11:11.020","Text":"Let\u0027s get some more room here."},{"Start":"11:11.020 ","End":"11:16.990","Text":"This equals Pi times the integral from"},{"Start":"11:16.990 ","End":"11:27.380","Text":"0-2 of 4x squared minus x to the fourth dx."},{"Start":"11:29.670 ","End":"11:34.045","Text":"Now the integral of 4x squared,"},{"Start":"11:34.045 ","End":"11:36.430","Text":"I raise the power by 1,"},{"Start":"11:36.430 ","End":"11:38.710","Text":"so it\u0027s x to the 3x cubed,"},{"Start":"11:38.710 ","End":"11:40.585","Text":"and then we divide that by 3."},{"Start":"11:40.585 ","End":"11:47.860","Text":"So we\u0027ve got 4/3x cubed minus x to the fifth over 5,"},{"Start":"11:47.860 ","End":"11:51.085","Text":"I\u0027ll write it as 1/5x to the fifth."},{"Start":"11:51.085 ","End":"11:57.730","Text":"All this taken between the bounds of 0 and 2."},{"Start":"11:57.730 ","End":"12:01.780","Text":"If I put in x equals 2,"},{"Start":"12:01.780 ","End":"12:10.330","Text":"then I\u0027ll get 4/3 times 2 cubed is 4/3 times 8 is 32/3."},{"Start":"12:10.330 ","End":"12:15.890","Text":"If I put here x equals 2, I\u0027ve got 32/5."},{"Start":"12:17.540 ","End":"12:21.180","Text":"If I put in 0, I get 0,"},{"Start":"12:21.180 ","End":"12:23.910","Text":"so there\u0027s nothing else that I need to do."},{"Start":"12:23.910 ","End":"12:32.275","Text":"This is equal to 64 Pi over 15."},{"Start":"12:32.275 ","End":"12:34.750","Text":"That\u0027s the answer using the disc method."},{"Start":"12:34.750 ","End":"12:38.680","Text":"Let\u0027s see if we can corroborate it using the shell method."},{"Start":"12:38.680 ","End":"12:39.880","Text":"For the shell method,"},{"Start":"12:39.880 ","End":"12:43.030","Text":"I\u0027m going to need x in terms of y,"},{"Start":"12:43.030 ","End":"12:48.760","Text":"so let\u0027s go back up here and see what we can do."},{"Start":"12:48.760 ","End":"12:51.100","Text":"When y equals 2x,"},{"Start":"12:51.100 ","End":"12:57.175","Text":"well it\u0027s very easy to see that this gives us that x equals y over 2."},{"Start":"12:57.175 ","End":"12:59.545","Text":"If y equals x squared,"},{"Start":"12:59.545 ","End":"13:04.135","Text":"then x has to be the square root of y."},{"Start":"13:04.135 ","End":"13:07.435","Text":"It\u0027s plus not minus because we\u0027re in the first quadrant."},{"Start":"13:07.435 ","End":"13:11.300","Text":"Now I can take these 2 functions down there,"},{"Start":"13:15.930 ","End":"13:20.695","Text":"and we can proceed with this method."},{"Start":"13:20.695 ","End":"13:24.880","Text":"V of y is the larger 1,"},{"Start":"13:24.880 ","End":"13:28.690","Text":"should have mentioned that\u0027s the square root,"},{"Start":"13:28.690 ","End":"13:31.675","Text":"and this is the y over 2."},{"Start":"13:31.675 ","End":"13:37.435","Text":"V of y, I\u0027ll just write this was the square root of y."},{"Start":"13:37.435 ","End":"13:41.020","Text":"The other 1, u of y was y over 2."},{"Start":"13:41.020 ","End":"13:48.475","Text":"Again, we can take a look up there that this 1 is the higher 1 and this 1 is the lower 1,"},{"Start":"13:48.475 ","End":"13:51.140","Text":"the left most 1."},{"Start":"13:51.720 ","End":"13:55.465","Text":"We\u0027re going to take the integral from 0-4."},{"Start":"13:55.465 ","End":"13:57.385","Text":"Go back down here."},{"Start":"13:57.385 ","End":"14:07.960","Text":"This is equal to 2 Pi times the integral from 0-4 of y."},{"Start":"14:07.960 ","End":"14:17.380","Text":"Then square root of y minus y over 2, all this dy."},{"Start":"14:17.380 ","End":"14:20.155","Text":"Let\u0027s see what we\u0027re going to do here."},{"Start":"14:20.155 ","End":"14:25.975","Text":"I suggest using exponents and converting the square roots to exponents."},{"Start":"14:25.975 ","End":"14:29.679","Text":"This is 2 Pi times the integral."},{"Start":"14:29.679 ","End":"14:36.100","Text":"Y times the square root of y is y to the power of 3/2."},{"Start":"14:36.100 ","End":"14:43.525","Text":"This is minus 1/2 and y times y is y squared,"},{"Start":"14:43.525 ","End":"14:48.625","Text":"all this dy and from 0-4."},{"Start":"14:48.625 ","End":"14:51.640","Text":"Now we just have exponents."},{"Start":"14:51.640 ","End":"14:55.315","Text":"If we want y to the 3/2,"},{"Start":"14:55.315 ","End":"15:03.280","Text":"its integral is y to the 5/2 if I raise the power by 1."},{"Start":"15:03.280 ","End":"15:05.890","Text":"I also have to divide by 5/2,"},{"Start":"15:05.890 ","End":"15:09.805","Text":"which I prefer to do as multiplying by 2/5,"},{"Start":"15:09.805 ","End":"15:11.965","Text":"multiplying by the reciprocal."},{"Start":"15:11.965 ","End":"15:16.240","Text":"Here I get y to the power of 3 and then over 3,"},{"Start":"15:16.240 ","End":"15:20.755","Text":"so altogether it\u0027s 1/6y to the power of 3."},{"Start":"15:20.755 ","End":"15:22.900","Text":"We\u0027re no longer in an integral,"},{"Start":"15:22.900 ","End":"15:30.470","Text":"so it\u0027s just this expression evaluated between 0 and 4."},{"Start":"15:31.140 ","End":"15:34.750","Text":"If y is 0,"},{"Start":"15:34.750 ","End":"15:36.865","Text":"clearly we get 0 here."},{"Start":"15:36.865 ","End":"15:41.365","Text":"So all we need is to substitute y equals 4,"},{"Start":"15:41.365 ","End":"15:47.680","Text":"so we get 2 Pi times 2/5."},{"Start":"15:47.680 ","End":"15:56.365","Text":"4 to the power of 5/2 is the square root of 4 to the power of"},{"Start":"15:56.365 ","End":"16:05.725","Text":"5 is 32 times 32 and y cubed is"},{"Start":"16:05.725 ","End":"16:09.700","Text":"64 because it\u0027s 4 to"},{"Start":"16:09.700 ","End":"16:16.720","Text":"the power of 3/6, 64/6."},{"Start":"16:16.720 ","End":"16:19.315","Text":"Let\u0027s see what this equals;"},{"Start":"16:19.315 ","End":"16:30.610","Text":"2 Pi times 64/5 minus 64/6."},{"Start":"16:30.610 ","End":"16:37.325","Text":"If you figure out at the side that 1/5 minus 1/6 equals 1/30,"},{"Start":"16:37.325 ","End":"16:47.600","Text":"then this is equal to 2 Pi times 64/30."},{"Start":"16:47.600 ","End":"16:52.405","Text":"If I cancel out the 2 from the 30,"},{"Start":"16:52.405 ","End":"16:54.220","Text":"I get 15 here."},{"Start":"16:54.220 ","End":"16:56.650","Text":"That\u0027s exactly what I got here,"},{"Start":"16:56.650 ","End":"17:05.430","Text":"64 Pi over 15."},{"Start":"17:05.430 ","End":"17:08.860","Text":"I\u0027m I glad that these 2 came out the same."},{"Start":"17:10.470 ","End":"17:15.145","Text":"It\u0027s hard to say which was really easier."},{"Start":"17:15.145 ","End":"17:17.160","Text":"More natural to abuse the disc method,"},{"Start":"17:17.160 ","End":"17:20.180","Text":"we didn\u0027t have to convert 2 functions of y,"},{"Start":"17:20.180 ","End":"17:21.810","Text":"but once we did that,"},{"Start":"17:21.810 ","End":"17:23.945","Text":"then it was just as easy."},{"Start":"17:23.945 ","End":"17:27.290","Text":"Anyway, you have both methods and you can choose which is"},{"Start":"17:27.290 ","End":"17:32.580","Text":"easiest in each particular case. We are done."}],"ID":4486},{"Watched":false,"Name":"Exercise 1","Duration":"15m 52s","ChapterTopicVideoID":4710,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.225","Text":"In this exercise, we have to compute the volume of a solid of revolution,"},{"Start":"00:06.225 ","End":"00:10.950","Text":"and we start off with 2 functions in the x-y plane."},{"Start":"00:10.950 ","End":"00:13.770","Text":"y equals x squared and y equals 2x."},{"Start":"00:13.770 ","End":"00:17.170","Text":"It would be nice to have a picture."},{"Start":"00:17.210 ","End":"00:21.795","Text":"Here\u0027s a picture I borrowed from somewhere."},{"Start":"00:21.795 ","End":"00:25.655","Text":"There\u0027s 2 curves here with a curve and a line."},{"Start":"00:25.655 ","End":"00:31.785","Text":"This one must be the y equal x squared,"},{"Start":"00:31.785 ","End":"00:34.560","Text":"which is an upward-facing parabola,"},{"Start":"00:34.560 ","End":"00:38.510","Text":"and this one would be y equals 2x,"},{"Start":"00:38.510 ","End":"00:41.210","Text":"which is a straight line through the origin."},{"Start":"00:41.210 ","End":"00:45.280","Text":"We can identify so we know which is which,"},{"Start":"00:45.280 ","End":"00:53.995","Text":"and I\u0027ve already done some of the work for you because I needed to know these points."},{"Start":"00:53.995 ","End":"01:00.470","Text":"These points were obtained by solving 2 equations and 2 unknowns,"},{"Start":"01:00.470 ","End":"01:03.450","Text":"and here it was,"},{"Start":"01:03.450 ","End":"01:10.955","Text":"let\u0027s say y equals x squared and y equals 2x."},{"Start":"01:10.955 ","End":"01:14.540","Text":"This is too simple for me to go into any detail."},{"Start":"01:14.540 ","End":"01:19.175","Text":"But what we do is we equate x squared equals 2x,"},{"Start":"01:19.175 ","End":"01:20.929","Text":"and then we get 2 solutions,"},{"Start":"01:20.929 ","End":"01:23.665","Text":"x equals 0 and x equals 2."},{"Start":"01:23.665 ","End":"01:27.935","Text":"Each of them we can substitute into one of the 2 equations."},{"Start":"01:27.935 ","End":"01:31.070","Text":"Let\u0027s say the 0, it doesn\u0027t matter which one I substituted,"},{"Start":"01:31.070 ","End":"01:33.485","Text":"I get y equals 0."},{"Start":"01:33.485 ","End":"01:35.750","Text":"If I substitute x equals 2,"},{"Start":"01:35.750 ","End":"01:38.630","Text":"I\u0027ll get 4 either here or here."},{"Start":"01:38.630 ","End":"01:41.550","Text":"This is 2,4."},{"Start":"01:41.680 ","End":"01:48.289","Text":"The area bounded by them is the one that\u0027s been shaded in yellow,"},{"Start":"01:48.289 ","End":"01:51.020","Text":"and now we have our basic shape."},{"Start":"01:51.020 ","End":"01:57.010","Text":"What we do with this is we revolve it around the x-axis."},{"Start":"01:57.010 ","End":"02:00.510","Text":"This yellow is revolved around."},{"Start":"02:00.510 ","End":"02:05.565","Text":"That gives us a volume of,"},{"Start":"02:05.565 ","End":"02:08.685","Text":"a solid of revolution."},{"Start":"02:08.685 ","End":"02:12.185","Text":"There are 2 main methods of solving it."},{"Start":"02:12.185 ","End":"02:15.095","Text":"One is essentially based on"},{"Start":"02:15.095 ","End":"02:20.719","Text":"vertical slices where y is a function of x and the other is based on horizontal slices,"},{"Start":"02:20.719 ","End":"02:22.825","Text":"where x is a function of y."},{"Start":"02:22.825 ","End":"02:25.445","Text":"The vertical slice method is"},{"Start":"02:25.445 ","End":"02:31.795","Text":"the disk method and the other one is the cylindrical shell method."},{"Start":"02:31.795 ","End":"02:37.355","Text":"Let me give the formula for both because we\u0027re just working straight off a formula."},{"Start":"02:37.355 ","End":"02:39.740","Text":"Let me give myself a bit more room."},{"Start":"02:39.740 ","End":"02:43.610","Text":"I\u0027m going to just move this picture a bit,"},{"Start":"02:43.610 ","End":"02:50.260","Text":"maybe shrink it a little bit, and should do."},{"Start":"02:50.360 ","End":"02:56.695","Text":"Now I can bring in the formulas I need,"},{"Start":"02:56.695 ","End":"02:58.535","Text":"and here they are,"},{"Start":"02:58.535 ","End":"03:00.570","Text":"but it\u0027s going off the page,"},{"Start":"03:00.570 ","End":"03:02.480","Text":"I\u0027ll make some adjustments."},{"Start":"03:02.480 ","End":"03:05.740","Text":"I\u0027ll move the labels here."},{"Start":"03:05.740 ","End":"03:10.550","Text":"Now I realize that the top one is the 2x and the bottom one is the x squared."},{"Start":"03:10.550 ","End":"03:14.620","Text":"I think it\u0027s better for right than the other way round here."},{"Start":"03:14.620 ","End":"03:17.330","Text":"Actually, I don\u0027t really need this little picture,"},{"Start":"03:17.330 ","End":"03:22.010","Text":"but I do want to remember that it\u0027s 0,0-2,4."},{"Start":"03:22.010 ","End":"03:25.160","Text":"I\u0027ll just write those points here,"},{"Start":"03:25.160 ","End":"03:26.630","Text":"just to remind myself,"},{"Start":"03:26.630 ","End":"03:34.260","Text":"from 0,0-2,4, and there we are."},{"Start":"03:34.260 ","End":"03:37.050","Text":"Okay. Everything fits in now."},{"Start":"03:37.050 ","End":"03:40.070","Text":"We\u0027re going to do the 2 methods."},{"Start":"03:40.070 ","End":"03:44.955","Text":"We\u0027ll start with method A, the disk method."},{"Start":"03:44.955 ","End":"03:54.170","Text":"We have the formula here that v equals Pi times the integral from a to b,"},{"Start":"03:54.170 ","End":"03:59.825","Text":"the upper function squared minus the lower function squared dx."},{"Start":"03:59.825 ","End":"04:05.000","Text":"I just want to point out that it\u0027s important to remember where the squares are."},{"Start":"04:05.000 ","End":"04:12.090","Text":"Because sometimes people get confused and they write f of x minus g of x all squared."},{"Start":"04:12.090 ","End":"04:13.190","Text":"Don\u0027t make that mistake."},{"Start":"04:13.190 ","End":"04:16.070","Text":"Each one is squared separately and then subtracted."},{"Start":"04:16.070 ","End":"04:18.275","Text":"In our particular case,"},{"Start":"04:18.275 ","End":"04:23.035","Text":"we have that a is 0,"},{"Start":"04:23.035 ","End":"04:27.750","Text":"b is 2, we\u0027re going to go from 0-2."},{"Start":"04:27.750 ","End":"04:32.510","Text":"The f of x is the upper one,"},{"Start":"04:32.510 ","End":"04:34.990","Text":"it\u0027s going to be the 2x,"},{"Start":"04:34.990 ","End":"04:38.060","Text":"and the g of x, which is the lower one,"},{"Start":"04:38.060 ","End":"04:41.045","Text":"is the x squared."},{"Start":"04:41.045 ","End":"04:43.760","Text":"Now we have a, b, f,"},{"Start":"04:43.760 ","End":"04:46.220","Text":"and g in our particular case."},{"Start":"04:46.220 ","End":"04:49.430","Text":"We can do the computation."},{"Start":"04:49.430 ","End":"04:53.825","Text":"I will have that the volume V is,"},{"Start":"04:53.825 ","End":"04:58.880","Text":"and I\u0027m copying Pi times the integral from a to b,"},{"Start":"04:58.880 ","End":"05:01.030","Text":"which is from 0-2."},{"Start":"05:01.030 ","End":"05:03.930","Text":"F of x is 2x,"},{"Start":"05:03.930 ","End":"05:08.205","Text":"I have 2x squared,"},{"Start":"05:08.205 ","End":"05:11.400","Text":"let me put a square bracket here, minus,"},{"Start":"05:11.400 ","End":"05:13.500","Text":"g of x is x squared,"},{"Start":"05:13.500 ","End":"05:19.095","Text":"it\u0027s x squared, all this dx."},{"Start":"05:19.095 ","End":"05:23.630","Text":"Now, this is a pretty straightforward integral."},{"Start":"05:23.630 ","End":"05:25.785","Text":"Let\u0027s just do it. I\u0027ll scroll down,"},{"Start":"05:25.785 ","End":"05:28.150","Text":"we\u0027ll have plenty of space."},{"Start":"05:28.150 ","End":"05:37.200","Text":"We get that this is equal to Pi that leave that as a constant."},{"Start":"05:37.200 ","End":"05:42.090","Text":"The integral from 0-2 of 4 x"},{"Start":"05:42.090 ","End":"05:49.260","Text":"squared minus x to the 4th dx,"},{"Start":"05:49.260 ","End":"05:52.220","Text":"which is equal to Pi times."},{"Start":"05:52.220 ","End":"06:02.250","Text":"Now the integral of this is 4x cubed over 3."},{"Start":"06:05.000 ","End":"06:13.430","Text":"The integral of this one is x to the 5th over 5 with a minus there."},{"Start":"06:13.430 ","End":"06:20.080","Text":"All this has to be taken between 0 and 2."},{"Start":"06:20.900 ","End":"06:25.890","Text":"What we get is,"},{"Start":"06:25.890 ","End":"06:27.795","Text":"let\u0027s substitute the 2."},{"Start":"06:27.795 ","End":"06:36.585","Text":"We get Pi times 4 times 2 cubed over 3,"},{"Start":"06:36.585 ","End":"06:41.860","Text":"is 4 times 8 over 3 is 32 over 3."},{"Start":"06:42.350 ","End":"06:47.670","Text":"Here 2 to the 5th is 32."},{"Start":"06:47.670 ","End":"06:50.975","Text":"It\u0027s minus 32 over 5."},{"Start":"06:50.975 ","End":"06:53.420","Text":"That\u0027s the bit for 2."},{"Start":"06:53.420 ","End":"06:59.110","Text":"But for 0, if I substituted all these things come out 0."},{"Start":"06:59.110 ","End":"07:02.300","Text":"Just, I\u0027ll indicate that I haven\u0027t forgotten the 0,"},{"Start":"07:02.300 ","End":"07:05.025","Text":"it\u0027s just not contributing anything,"},{"Start":"07:05.025 ","End":"07:08.155","Text":"and this is equal to,"},{"Start":"07:08.155 ","End":"07:10.920","Text":"I can take 32 outside the brackets,"},{"Start":"07:10.920 ","End":"07:12.315","Text":"I\u0027ll work with fractions."},{"Start":"07:12.315 ","End":"07:18.030","Text":"I have 32 Pi times a 1/3 minus 1/5,"},{"Start":"07:18.030 ","End":"07:21.090","Text":"1/3 minus a 1/5."},{"Start":"07:21.090 ","End":"07:27.905","Text":"This bit is 2/15 if you cross multiply and do a common denominator."},{"Start":"07:27.905 ","End":"07:33.675","Text":"Altogether, what I get is"},{"Start":"07:33.675 ","End":"07:44.950","Text":"32 Pi times 2/15 gives me 64 Pi over 15."},{"Start":"07:47.940 ","End":"07:55.120","Text":"We have our answer for the first part for the disk method,"},{"Start":"07:55.120 ","End":"07:56.965","Text":"and this is it."},{"Start":"07:56.965 ","End":"07:59.290","Text":"Now, I will go over to the other method,"},{"Start":"07:59.290 ","End":"08:01.840","Text":"and we\u0027ll use this as a check because we should get"},{"Start":"08:01.840 ","End":"08:05.320","Text":"the same answer if we do it by a different method."},{"Start":"08:05.320 ","End":"08:08.005","Text":"I\u0027m going back up here."},{"Start":"08:08.005 ","End":"08:14.710","Text":"What we have to do in the cylindrical shell method"},{"Start":"08:14.710 ","End":"08:23.420","Text":"is to put x in terms of y instead of y in terms of x everywhere."},{"Start":"08:23.550 ","End":"08:29.980","Text":"For example, well, I could go all the way back to the original diagram."},{"Start":"08:29.980 ","End":"08:31.645","Text":"Why don\u0027t I do that?"},{"Start":"08:31.645 ","End":"08:34.315","Text":"Let me go all the way back up here."},{"Start":"08:34.315 ","End":"08:36.160","Text":"What I have to do."},{"Start":"08:36.160 ","End":"08:39.670","Text":"Is reverse the roles of x and y. I want x in terms of y."},{"Start":"08:39.670 ","End":"08:42.220","Text":"Instead of y equals 2x,"},{"Start":"08:42.220 ","End":"08:44.005","Text":"we can easily see that,"},{"Start":"08:44.005 ","End":"08:50.049","Text":"that is the same as x equals 1/2 of y,"},{"Start":"08:50.049 ","End":"08:53.725","Text":"and instead of y equals x squared,"},{"Start":"08:53.725 ","End":"08:59.605","Text":"I\u0027ll get x equals the square root of y."},{"Start":"08:59.605 ","End":"09:01.930","Text":"No need for plus or minus because as you can see,"},{"Start":"09:01.930 ","End":"09:04.565","Text":"everything is in the first quadrant."},{"Start":"09:04.565 ","End":"09:06.660","Text":"That\u0027s for this."},{"Start":"09:06.660 ","End":"09:08.440","Text":"Then"},{"Start":"09:17.190 ","End":"09:18.820","Text":"I\u0027ll just write"},{"Start":"09:18.820 ","End":"09:22.840","Text":"the equivalent here that we have the 2 equations,"},{"Start":"09:22.840 ","End":"09:27.700","Text":"x equals 1/2, y and"},{"Start":"09:27.700 ","End":"09:34.870","Text":"x equals the square root of y."},{"Start":"09:34.870 ","End":"09:38.530","Text":"But actually I want them the other way around."},{"Start":"09:38.530 ","End":"09:40.090","Text":"The we are."},{"Start":"09:40.090 ","End":"09:43.975","Text":"Let me just shift them with all right."},{"Start":"09:43.975 ","End":"09:46.240","Text":"Now again, I don\u0027t need the small picture,"},{"Start":"09:46.240 ","End":"09:53.905","Text":"I just need the information that we have to translate."},{"Start":"09:53.905 ","End":"09:56.530","Text":"Here we have c and d,"},{"Start":"09:56.530 ","End":"09:57.640","Text":"and in this case,"},{"Start":"09:57.640 ","End":"09:59.650","Text":"c and d is going to be 0,"},{"Start":"09:59.650 ","End":"10:02.395","Text":"and, was it 0 and 4?"},{"Start":"10:02.395 ","End":"10:11.680","Text":"Yeah. We have that c equals 0, d equals 4."},{"Start":"10:11.680 ","End":"10:15.625","Text":"As for the functions, the higher 1,"},{"Start":"10:15.625 ","End":"10:17.950","Text":"the rightmost 1 which is bigger,"},{"Start":"10:17.950 ","End":"10:22.990","Text":"this 1 is going to be the square root of y."},{"Start":"10:22.990 ","End":"10:25.225","Text":"We have v of y,"},{"Start":"10:25.225 ","End":"10:29.335","Text":"which is the square root of y,"},{"Start":"10:29.335 ","End":"10:32.050","Text":"and the other 1, I know it\u0027s a straight line,"},{"Start":"10:32.050 ","End":"10:34.195","Text":"but this is a general picture."},{"Start":"10:34.195 ","End":"10:38.020","Text":"This 1 is equal to the linear 1,"},{"Start":"10:38.020 ","End":"10:40.300","Text":"x equals 1/2 y."},{"Start":"10:40.300 ","End":"10:45.340","Text":"You have y is 1/2 of y."},{"Start":"10:45.340 ","End":"10:49.975","Text":"Now we\u0027ve got everything that we need to apply this formula,"},{"Start":"10:49.975 ","End":"10:52.840","Text":"which has a v and a u instead of an f and a g,"},{"Start":"10:52.840 ","End":"10:54.820","Text":"and a c and a d instead of an a and a b,"},{"Start":"10:54.820 ","End":"10:57.430","Text":"so let\u0027s just keep going."},{"Start":"10:57.430 ","End":"11:04.450","Text":"After I substitute, I get V equals."},{"Start":"11:04.450 ","End":"11:07.705","Text":"Now, I\u0027m just looking at this formula and substituting everything."},{"Start":"11:07.705 ","End":"11:12.670","Text":"2pi times the integral from c to d,"},{"Start":"11:12.670 ","End":"11:16.555","Text":"which is 0-4 y,"},{"Start":"11:16.555 ","End":"11:22.270","Text":"times v of y is"},{"Start":"11:22.270 ","End":"11:28.870","Text":"square root of y and minus u of y,"},{"Start":"11:28.870 ","End":"11:31.525","Text":"which is a half y,"},{"Start":"11:31.525 ","End":"11:35.390","Text":"and all this dy."},{"Start":"11:35.970 ","End":"11:40.120","Text":"It\u0027s just a straightforward definite integral."},{"Start":"11:40.120 ","End":"11:42.620","Text":"Let\u0027s see what we get."},{"Start":"11:44.280 ","End":"11:47.890","Text":"We get that, let\u0027s see,"},{"Start":"11:47.890 ","End":"11:48.910","Text":"I\u0027ll go even more."},{"Start":"11:48.910 ","End":"11:52.640","Text":"I\u0027m going to see everything and I\u0027m hoping to get to this."},{"Start":"11:53.340 ","End":"11:55.900","Text":"This is equal to"},{"Start":"11:55.900 ","End":"12:02.080","Text":"2pi times"},{"Start":"12:02.080 ","End":"12:07.090","Text":"the integral from 0-4."},{"Start":"12:07.090 ","End":"12:14.515","Text":"Now, y times square root of y is y to the power of 1/2 or 3/2."},{"Start":"12:14.515 ","End":"12:16.970","Text":"I\u0027ll work with fractions."},{"Start":"12:17.520 ","End":"12:21.190","Text":"y times y is y squared,"},{"Start":"12:21.190 ","End":"12:27.980","Text":"so I have minus a 1/2 y squared dy,"},{"Start":"12:27.980 ","End":"12:32.130","Text":"which equals, now I\u0027m actually going to do the integration."},{"Start":"12:32.130 ","End":"12:36.745","Text":"It\u0027s 2pi times."},{"Start":"12:36.745 ","End":"12:38.980","Text":"If I integrate this,"},{"Start":"12:38.980 ","End":"12:41.170","Text":"I raise the power by 1,"},{"Start":"12:41.170 ","End":"12:45.700","Text":"so it becomes 5/2, or 2.5."},{"Start":"12:45.700 ","End":"12:53.350","Text":"It\u0027s y^5/2, but also divide by the new exponent."},{"Start":"12:53.350 ","End":"12:58.150","Text":"It means I multiply by 2/5 that are dividing by 5/2."},{"Start":"12:58.150 ","End":"13:04.480","Text":"Here I have the integral of y squared is y cubed over 3,"},{"Start":"13:04.480 ","End":"13:11.520","Text":"the 1/3 with the 1/2 gives me 1/6 y cubed."},{"Start":"13:11.520 ","End":"13:14.445","Text":"All this, this is now the integral."},{"Start":"13:14.445 ","End":"13:17.205","Text":"I just need to take it between its limits,"},{"Start":"13:17.205 ","End":"13:21.550","Text":"which is 0-4, and this equals,"},{"Start":"13:21.550 ","End":"13:25.435","Text":"let\u0027s first of all substitute 4, the 2pi stays."},{"Start":"13:25.435 ","End":"13:28.580","Text":"If I substitute 4,"},{"Start":"13:29.820 ","End":"13:37.630","Text":"4^5/2 means the square root of 4 and then raised to the power of 5, 2^5, 32."},{"Start":"13:37.630 ","End":"13:47.870","Text":"I have 2/5 times 32 minus 1/6 and 4 cubed is 64."},{"Start":"13:48.450 ","End":"13:53.140","Text":"This is what I get when I substitute 4,"},{"Start":"13:53.140 ","End":"13:57.355","Text":"when I substitute 0, this is 0 and this is 0."},{"Start":"13:57.355 ","End":"14:00.550","Text":"Even if I multiply by 2pi, it\u0027s still 0,"},{"Start":"14:00.550 ","End":"14:05.470","Text":"but I\u0027ll put minus 0 just to show I haven\u0027t forgotten that I need to subtract something."},{"Start":"14:05.470 ","End":"14:11.650","Text":"Now this is equal to, let\u0027s see,"},{"Start":"14:11.650 ","End":"14:16.930","Text":"if I put it over a common denominator,"},{"Start":"14:16.930 ","End":"14:24.610","Text":"well, I\u0027ll just do a little small cancellation first 6 and 64 both divide by 2."},{"Start":"14:24.610 ","End":"14:26.905","Text":"Let me put a 3 here,"},{"Start":"14:26.905 ","End":"14:30.670","Text":"and a 32 here."},{"Start":"14:30.670 ","End":"14:32.965","Text":"That\u0027s going to help me,"},{"Start":"14:32.965 ","End":"14:36.685","Text":"because now I have something over 5 and something over 3,"},{"Start":"14:36.685 ","End":"14:44.330","Text":"so I can put this bit as something over 5 times 3, which is 15."},{"Start":"14:44.400 ","End":"14:49.240","Text":"Here I\u0027m missing, because of the 5,"},{"Start":"14:49.240 ","End":"14:54.670","Text":"I\u0027m missing a 3, so I get 3 times 2 times 32."},{"Start":"14:54.670 ","End":"14:58.765","Text":"Here, I\u0027m missing a factor of 5,"},{"Start":"14:58.765 ","End":"15:03.460","Text":"so I have to put a 5 here,"},{"Start":"15:03.460 ","End":"15:09.865","Text":"and it\u0027s still 2pi, and this equals."},{"Start":"15:09.865 ","End":"15:14.770","Text":"Now let\u0027s see. That\u0027s pretty straightforward,"},{"Start":"15:14.770 ","End":"15:17.050","Text":"because here I have another multiply it all out here,"},{"Start":"15:17.050 ","End":"15:20.499","Text":"I have 6 times 32 minus 5 times 32,"},{"Start":"15:20.499 ","End":"15:23.740","Text":"so 6 of them minus 5 of them is 1 of them."},{"Start":"15:23.740 ","End":"15:30.145","Text":"It\u0027s just 2pi times 32 over 15."},{"Start":"15:30.145 ","End":"15:33.235","Text":"If I multiply 2 by 32,"},{"Start":"15:33.235 ","End":"15:35.455","Text":"I just get 64."},{"Start":"15:35.455 ","End":"15:41.695","Text":"It\u0027s 64pi over 15."},{"Start":"15:41.695 ","End":"15:46.955","Text":"Let me highlight this. Aren\u0027t we lucky?"},{"Start":"15:46.955 ","End":"15:52.680","Text":"We did get the same answer both ways and we are done."}],"ID":4723},{"Watched":false,"Name":"Exercise 2","Duration":"8m 26s","ChapterTopicVideoID":4711,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:04.470","Text":"The previous exercise was exactly the same as this."},{"Start":"00:04.470 ","End":"00:09.660","Text":"The only difference is that here we have the y-axis and there it was the x-axis"},{"Start":"00:09.660 ","End":"00:16.215","Text":"so I\u0027m going to reuse the diagram from the previous exercise."},{"Start":"00:16.215 ","End":"00:19.470","Text":"Now there\u0027s one small thing I want to change and that is"},{"Start":"00:19.470 ","End":"00:26.170","Text":"this little arrow which now we\u0027re revolving around the y-axis,"},{"Start":"00:26.170 ","End":"00:28.870","Text":"so well, draw it like this."},{"Start":"00:29.570 ","End":"00:37.620","Text":"Just like before, we brought the general formula in the solution,"},{"Start":"00:37.620 ","End":"00:41.995","Text":"I borrowed it from the tutorial and I\u0027m going to do the same here."},{"Start":"00:41.995 ","End":"00:44.210","Text":"Now we can\u0027t see the whole thing,"},{"Start":"00:44.210 ","End":"00:47.030","Text":"but in a moment I\u0027ll just make some space."},{"Start":"00:47.030 ","End":"00:50.000","Text":"I just need the vital information from here."},{"Start":"00:50.000 ","End":"00:57.150","Text":"The main thing is that this goes from 0-2 in the x-direction and 0-4 in the y-direction."},{"Start":"00:57.150 ","End":"01:03.790","Text":"Now I\u0027ll just to make a note to myself that a here is 0,"},{"Start":"01:03.790 ","End":"01:05.990","Text":"when I\u0027m going in the x-direction,"},{"Start":"01:05.990 ","End":"01:12.665","Text":"I\u0027m going from 0-2, and when I\u0027m going in the y-direction dy,"},{"Start":"01:12.665 ","End":"01:16.280","Text":"I\u0027m going from 0-4."},{"Start":"01:16.280 ","End":"01:24.450","Text":"The other thing I need to know is that the f of x here,"},{"Start":"01:25.250 ","End":"01:29.350","Text":"I\u0027ll just scroll a little bit now,"},{"Start":"01:31.160 ","End":"01:37.320","Text":"and I could still see that f"},{"Start":"01:37.320 ","End":"01:43.395","Text":"of x was the one that\u0027s above."},{"Start":"01:43.395 ","End":"01:47.160","Text":"I\u0027m looking at the blue writing now that\u0027s the 2x,"},{"Start":"01:47.160 ","End":"01:52.085","Text":"and the g of x would be the lower one is the x squared."},{"Start":"01:52.085 ","End":"01:56.644","Text":"For here what I need is looking at the orange,"},{"Start":"01:56.644 ","End":"01:58.820","Text":"where x is in terms of y,"},{"Start":"01:58.820 ","End":"02:01.115","Text":"that the rightmost one,"},{"Start":"02:01.115 ","End":"02:02.720","Text":"which is the higher one,"},{"Start":"02:02.720 ","End":"02:05.360","Text":"is x equals square root of"},{"Start":"02:05.360 ","End":"02:15.855","Text":"y and that the other one is x equals 1/2 of y."},{"Start":"02:15.855 ","End":"02:21.955","Text":"Now I don\u0027t need any of that diagram."},{"Start":"02:21.955 ","End":"02:28.615","Text":"We\u0027ve got a bit more space now and a bit more."},{"Start":"02:28.615 ","End":"02:32.245","Text":"We just have to evaluate couple of integrals,"},{"Start":"02:32.245 ","End":"02:35.560","Text":"that\u0027s all and hope that the answer comes out the same."},{"Start":"02:35.560 ","End":"02:38.810","Text":"It will be a confirmation."},{"Start":"02:39.760 ","End":"02:44.240","Text":"I\u0027ll start with this one, v,"},{"Start":"02:44.240 ","End":"02:51.840","Text":"which is 2 Pi times the integral from 0-2,"},{"Start":"02:51.840 ","End":"02:57.060","Text":"x times, f of x is 2x minus,"},{"Start":"02:57.060 ","End":"03:04.390","Text":"g of x is x squared dx."},{"Start":"03:04.910 ","End":"03:08.660","Text":"You know what? I\u0027ll write the other one here while I\u0027m at"},{"Start":"03:08.660 ","End":"03:12.150","Text":"it and then we\u0027ll just develop one than the other."},{"Start":"03:12.150 ","End":"03:17.210","Text":"Here what I\u0027m going to get is that the volume of revolution"},{"Start":"03:17.210 ","End":"03:23.800","Text":"of the same thing is going to be the integral."},{"Start":"03:23.800 ","End":"03:28.515","Text":"Sorry, there is a Pi there from"},{"Start":"03:28.515 ","End":"03:35.235","Text":"0-4 of this thing squared,"},{"Start":"03:35.235 ","End":"03:38.155","Text":"square root of y squared,"},{"Start":"03:38.155 ","End":"03:43.325","Text":"minus the lower one or the leftmost one squared,"},{"Start":"03:43.325 ","End":"03:49.450","Text":"1/2 y squared and all this is dy."},{"Start":"03:49.910 ","End":"03:54.380","Text":"Now it all boils down to just mechanically computing a couple of"},{"Start":"03:54.380 ","End":"03:59.245","Text":"integrals that don\u0027t need any diagrams, just computation."},{"Start":"03:59.245 ","End":"04:02.330","Text":"We begin with the one on the left,"},{"Start":"04:02.330 ","End":"04:06.105","Text":"which is 2 Pi,"},{"Start":"04:06.105 ","End":"04:09.345","Text":"that\u0027s just to expand open brackets first,"},{"Start":"04:09.345 ","End":"04:19.020","Text":"2x squared minus x cubed dx from 0-2,"},{"Start":"04:19.020 ","End":"04:21.000","Text":"which is 2 Pi."},{"Start":"04:21.000 ","End":"04:26.430","Text":"Integral of 2x squared is 2x cubed over"},{"Start":"04:26.430 ","End":"04:36.885","Text":"3 and integral of x cubed is x^4 over 4,"},{"Start":"04:36.885 ","End":"04:40.960","Text":"all this taken from 0-2."},{"Start":"04:41.620 ","End":"04:45.920","Text":"We substitute 2 first, then 0,"},{"Start":"04:45.920 ","End":"04:48.780","Text":"then subtract the 2 I get,"},{"Start":"04:48.780 ","End":"04:52.410","Text":"2 Pi, 2 cubed is 8,"},{"Start":"04:52.410 ","End":"04:55.920","Text":"so it\u0027s 16 over 3,"},{"Start":"04:55.920 ","End":"05:01.170","Text":"2^4 is 16 over 4 minus 4,"},{"Start":"05:01.170 ","End":"05:04.200","Text":"that\u0027s for the 2 and now for the 0."},{"Start":"05:04.200 ","End":"05:05.900","Text":"These all come out to 0,"},{"Start":"05:05.900 ","End":"05:10.740","Text":"multiply by 2 Pi still 0. What do we get?"},{"Start":"05:10.740 ","End":"05:16.035","Text":"16 over 3 is 5 1/3 I will just write that down."},{"Start":"05:16.035 ","End":"05:22.455","Text":"We get 5 1/3 minus 4 is 1 1/3."},{"Start":"05:22.455 ","End":"05:26.640","Text":"1 1/3 is 4/3,"},{"Start":"05:26.640 ","End":"05:28.665","Text":"4/3 times 2 Pi,"},{"Start":"05:28.665 ","End":"05:30.140","Text":"let\u0027s write that here."},{"Start":"05:30.140 ","End":"05:34.595","Text":"This altogether as 4 over 3 times the 2 Pi,"},{"Start":"05:34.595 ","End":"05:38.280","Text":"I get 8 Pi over 3."},{"Start":"05:38.410 ","End":"05:42.950","Text":"I\u0027m not going to look it up on a calculator for an approximation,"},{"Start":"05:42.950 ","End":"05:45.230","Text":"this is good enough,"},{"Start":"05:45.230 ","End":"05:48.425","Text":"and I will highlight it,"},{"Start":"05:48.425 ","End":"05:50.745","Text":"8 Pi over 3."},{"Start":"05:50.745 ","End":"05:53.400","Text":"Now we can go onto the other one."},{"Start":"05:53.400 ","End":"05:58.770","Text":"This is now equal to Pi times the integral."},{"Start":"05:58.770 ","End":"06:02.770","Text":"Simplify. What\u0027s in the integral?"},{"Start":"06:02.770 ","End":"06:08.090","Text":"Square root of y squared is just y minus,"},{"Start":"06:08.090 ","End":"06:14.115","Text":"this comes out to be 1/4 of y squared,"},{"Start":"06:14.115 ","End":"06:18.105","Text":"all this dy from 0-4."},{"Start":"06:18.105 ","End":"06:24.905","Text":"For y I get y squared over 2 or 1/2 y squared."},{"Start":"06:24.905 ","End":"06:27.480","Text":"Here I get y cubed over 3,"},{"Start":"06:27.480 ","End":"06:31.830","Text":"the 3 with the 4 is 12 so 1/12 cubed."},{"Start":"06:31.830 ","End":"06:34.845","Text":"This from 0-4."},{"Start":"06:34.845 ","End":"06:39.020","Text":"As usual, the 0 is not going to give me anything so I\u0027ll"},{"Start":"06:39.020 ","End":"06:42.860","Text":"just substitute the 4 and I\u0027ve got Pi times,"},{"Start":"06:42.860 ","End":"06:46.719","Text":"4 squared is 16 over 2 is 8,"},{"Start":"06:46.719 ","End":"06:56.189","Text":"4 cubed is 64 so leave it as 64 over 12 for the moment."},{"Start":"06:57.080 ","End":"07:01.370","Text":"I\u0027ll put it in the minus 0 to show I haven\u0027t forgotten to substitute 0,"},{"Start":"07:01.370 ","End":"07:03.320","Text":"but I don\u0027t get anything."},{"Start":"07:03.320 ","End":"07:07.964","Text":"64 over 12 is,"},{"Start":"07:07.964 ","End":"07:09.540","Text":"I can\u0027t really cancel, yeah,"},{"Start":"07:09.540 ","End":"07:13.770","Text":"I can cancel by 4, 4 into 12 goes 3,"},{"Start":"07:13.770 ","End":"07:17.050","Text":"4 into this goes 16."},{"Start":"07:17.210 ","End":"07:24.390","Text":"I\u0027ve got Pi times 8 minus 16 over 3."},{"Start":"07:24.390 ","End":"07:31.030","Text":"No, we had 16 over 3 here, was 5 1/3,"},{"Start":"07:32.630 ","End":"07:42.210","Text":"I wrote it even here so 8 minus 5 1/3,"},{"Start":"07:42.210 ","End":"07:51.945","Text":"that comes out to be 2 2/3."},{"Start":"07:51.945 ","End":"07:56.270","Text":"But I\u0027d better do it to be consistent with what they call an"},{"Start":"07:56.270 ","End":"08:00.935","Text":"improper fraction with a numerator greater than the denominator."},{"Start":"08:00.935 ","End":"08:08.960","Text":"So I do 2 times 3 plus 2 is 8 over 3."},{"Start":"08:08.960 ","End":"08:12.480","Text":"The Pi stays Pi."},{"Start":"08:12.520 ","End":"08:15.490","Text":"Aren\u0027t we lucky?"},{"Start":"08:15.490 ","End":"08:20.030","Text":"I\u0027m always actually a bit surprised to tell you the truth that mathematics works,"},{"Start":"08:20.030 ","End":"08:22.010","Text":"that if you do something by two different methods,"},{"Start":"08:22.010 ","End":"08:23.615","Text":"you get the same answer."},{"Start":"08:23.615 ","End":"08:25.980","Text":"Then we\u0027re done."}],"ID":4724},{"Watched":false,"Name":"Exercise 3 part a","Duration":"4m 44s","ChapterTopicVideoID":4712,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.275","Text":"In this exercise, we have to compute the solid of revolution,"},{"Start":"00:05.275 ","End":"00:10.329","Text":"and the axis it revolves around varies."},{"Start":"00:10.329 ","End":"00:12.990","Text":"It\u0027s actually 6 in 1 exercises here,"},{"Start":"00:12.990 ","End":"00:14.790","Text":"and we\u0027ll do each 1 as a separate clip."},{"Start":"00:14.790 ","End":"00:16.825","Text":"Let\u0027s start with part a,"},{"Start":"00:16.825 ","End":"00:21.450","Text":"where we have this area that\u0027s shaded in the figure,"},{"Start":"00:21.450 ","End":"00:25.090","Text":"and we have to revolve it around the x-axis,"},{"Start":"00:25.090 ","End":"00:27.249","Text":"which are marked in yellow."},{"Start":"00:27.249 ","End":"00:31.465","Text":"The function, in all cases,"},{"Start":"00:31.465 ","End":"00:37.015","Text":"this function here, maybe I\u0027ll even highlight that."},{"Start":"00:37.015 ","End":"00:41.590","Text":"This is the function which is given"},{"Start":"00:41.590 ","End":"00:48.560","Text":"by y equals or f of x equals 1 minus x cubed."},{"Start":"00:48.560 ","End":"00:55.320","Text":"What I\u0027d like to know is where it cuts the axis."},{"Start":"00:56.480 ","End":"01:02.450","Text":"Let\u0027s just call it also y equals to a market here again,"},{"Start":"01:02.450 ","End":"01:06.125","Text":"y equals 1 minus x cubed, that\u0027s the yellow."},{"Start":"01:06.125 ","End":"01:08.525","Text":"If x is 0,"},{"Start":"01:08.525 ","End":"01:12.385","Text":"then y is 1, clearly."},{"Start":"01:12.385 ","End":"01:18.150","Text":"If y is 0, then 1 minus x cubed is 0,"},{"Start":"01:18.150 ","End":"01:20.400","Text":"so x cubed is 1, so x is 1,"},{"Start":"01:20.400 ","End":"01:24.765","Text":"so this is also 1 and this is the origin."},{"Start":"01:24.765 ","End":"01:29.210","Text":"Now, we basically have everything we need, because in the first 1,"},{"Start":"01:29.210 ","End":"01:33.185","Text":"we have a formula for revolving around the x-axis."},{"Start":"01:33.185 ","End":"01:40.060","Text":"Normally, it\u0027s the difference of 2 functions and the formula is given as follows."},{"Start":"01:40.060 ","End":"01:45.380","Text":"This is the general formula for rotation about the x-axis using the disk method."},{"Start":"01:45.380 ","End":"01:47.405","Text":"However, in our case,"},{"Start":"01:47.405 ","End":"01:52.910","Text":"the x-axis is y equals 0."},{"Start":"01:52.910 ","End":"01:55.415","Text":"In this case, we use a simplified formula."},{"Start":"01:55.415 ","End":"02:00.605","Text":"Basically, we just let g of x equals 0 and what we get is,"},{"Start":"02:00.605 ","End":"02:02.930","Text":"now in our case,"},{"Start":"02:02.930 ","End":"02:13.210","Text":"what we have is that a is the 0 and b is the 1."},{"Start":"02:13.210 ","End":"02:15.210","Text":"That\u0027s this 1 here,"},{"Start":"02:15.210 ","End":"02:19.860","Text":"so we get Pi times the integral from 0 to 1."},{"Start":"02:19.860 ","End":"02:23.805","Text":"F of x is 1 minus x cubed."},{"Start":"02:23.805 ","End":"02:30.585","Text":"Here, I need to have it squared and dx."},{"Start":"02:30.585 ","End":"02:33.194","Text":"From now on, it\u0027s just technical."},{"Start":"02:33.194 ","End":"02:34.985","Text":"Let\u0027s do this integration."},{"Start":"02:34.985 ","End":"02:37.860","Text":"I\u0027ll get myself some space here."},{"Start":"02:39.370 ","End":"02:43.400","Text":"What we\u0027ll do is we\u0027ll just square"},{"Start":"02:43.400 ","End":"02:48.635","Text":"this expression using the formula for a minus b squared,"},{"Start":"02:48.635 ","End":"02:54.325","Text":"which is a squared minus 2 ab plus b squared."},{"Start":"02:54.325 ","End":"03:00.380","Text":"What we get is 1 squared is 1 minus twice this times this is"},{"Start":"03:00.380 ","End":"03:08.225","Text":"minus 2x cubed plus x cubed squared is x to the 6th dx."},{"Start":"03:08.225 ","End":"03:14.930","Text":"Straightforward polynomial integral, integral of 1 is x,"},{"Start":"03:14.930 ","End":"03:17.585","Text":"integral of minus 2x cubed."},{"Start":"03:17.585 ","End":"03:19.580","Text":"I raise the power by 1, it\u0027s 4,"},{"Start":"03:19.580 ","End":"03:22.610","Text":"divide by 4 minus 2 over 4,"},{"Start":"03:22.610 ","End":"03:31.215","Text":"which is 1.5 x to the 4th plus raise this at 7 divide by 7,"},{"Start":"03:31.215 ","End":"03:34.515","Text":"1 7th x to the 7th."},{"Start":"03:34.515 ","End":"03:38.490","Text":"This is taken between 0 and 1,"},{"Start":"03:38.490 ","End":"03:41.940","Text":"and so we get Pi for substitute"},{"Start":"03:41.940 ","End":"03:50.530","Text":"1 minus a half plus 1 7th."},{"Start":"03:52.430 ","End":"03:55.250","Text":"If we put in 0,"},{"Start":"03:55.250 ","End":"03:57.080","Text":"all these terms come out 0,"},{"Start":"03:57.080 ","End":"03:59.180","Text":"so we end up with nothing."},{"Start":"03:59.180 ","End":"04:04.010","Text":"Now, it\u0027s just an exercise with fractions."},{"Start":"04:04.010 ","End":"04:06.410","Text":"What we get is,"},{"Start":"04:06.410 ","End":"04:12.060","Text":"if I put it all in terms of 14, let\u0027s see."},{"Start":"04:12.060 ","End":"04:13.220","Text":"There\u0027s a common denominator here,"},{"Start":"04:13.220 ","End":"04:17.930","Text":"I have 14 over 14 minus 7 over 14 plus 2 over"},{"Start":"04:17.930 ","End":"04:27.390","Text":"14 and 14 plus 2 minus 7, I make that 9."},{"Start":"04:27.390 ","End":"04:36.270","Text":"I have 9 Pi over 14 for my answer,"},{"Start":"04:36.270 ","End":"04:44.020","Text":"and we can just highlight that and we\u0027re done with part a."}],"ID":4725},{"Watched":false,"Name":"Exercise 3 part b","Duration":"6m 16s","ChapterTopicVideoID":4713,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"Now we come to part b of the same exercise."},{"Start":"00:04.320 ","End":"00:07.245","Text":"The curve is the same,"},{"Start":"00:07.245 ","End":"00:10.349","Text":"y equals 1 minus x cubed."},{"Start":"00:10.349 ","End":"00:13.380","Text":"This is the Point 1 on the x-axis,"},{"Start":"00:13.380 ","End":"00:15.705","Text":"this is the Point 1 on the y-axis."},{"Start":"00:15.705 ","End":"00:18.570","Text":"The difference is that in part a,"},{"Start":"00:18.570 ","End":"00:20.760","Text":"we revolved around the x-axis,"},{"Start":"00:20.760 ","End":"00:24.085","Text":"which is in fact y equals 0."},{"Start":"00:24.085 ","End":"00:29.135","Text":"This time we\u0027re going to revolve around the axis y equals minus 1."},{"Start":"00:29.135 ","End":"00:33.140","Text":"Now we haven\u0027t learned how to revolve about anything other than the x-axis"},{"Start":"00:33.140 ","End":"00:37.130","Text":"or the y-axis but there\u0027s a standard trick that we use."},{"Start":"00:37.130 ","End":"00:42.020","Text":"What we\u0027re going to do is everything that you see in yellow."},{"Start":"00:42.020 ","End":"00:45.465","Text":"I mean, well, the shape,"},{"Start":"00:45.465 ","End":"00:51.270","Text":"I\u0027ll just complete the shape so you\u0027ll see what I mean."},{"Start":"00:51.270 ","End":"00:57.065","Text":"This bit, the area and the axis,"},{"Start":"00:57.065 ","End":"01:04.350","Text":"and then erase everything upwards by 1 unit upwards."},{"Start":"01:04.350 ","End":"01:10.220","Text":"What we\u0027ll end up with is that this axis will fall on the x-axis and this will be a bit"},{"Start":"01:10.220 ","End":"01:12.290","Text":"higher and then we\u0027ll be able to use"},{"Start":"01:12.290 ","End":"01:16.725","Text":"a standard case of rotational revolution about the x-axis."},{"Start":"01:16.725 ","End":"01:19.270","Text":"Let me draw what I mean."},{"Start":"01:19.480 ","End":"01:21.530","Text":"Here\u0027s what I mean."},{"Start":"01:21.530 ","End":"01:26.220","Text":"You raise everything by 1 unit upwards."},{"Start":"01:26.220 ","End":"01:28.980","Text":"What\u0027s here 0 and 1,"},{"Start":"01:28.980 ","End":"01:31.560","Text":"it becomes 1 and 2."},{"Start":"01:31.560 ","End":"01:40.040","Text":"This y equals negative 1 becomes y equals 0,"},{"Start":"01:40.040 ","End":"01:43.715","Text":"which is the x-axis and the x-axis,"},{"Start":"01:43.715 ","End":"01:48.090","Text":"well, I could continue the line a bit."},{"Start":"01:51.210 ","End":"01:55.690","Text":"What was the x-axis or y equals 0,"},{"Start":"01:55.690 ","End":"02:00.560","Text":"now becomes y equals 1."},{"Start":"02:00.920 ","End":"02:04.440","Text":"This curve also changes."},{"Start":"02:04.440 ","End":"02:12.010","Text":"Here it was y equals 1 minus x cubed and when we raise something upwards by 1,"},{"Start":"02:12.010 ","End":"02:13.690","Text":"then we just add 1."},{"Start":"02:13.690 ","End":"02:22.535","Text":"This would be the function y equals 1 minus x cubed plus 1 so it\u0027s 2 minus x cubed."},{"Start":"02:22.535 ","End":"02:29.170","Text":"Now we are reduced to the case where we actually do have a difference between 2 curves."},{"Start":"02:29.170 ","End":"02:34.980","Text":"This is 1 of them and this is the other one, a constant."},{"Start":"02:34.980 ","End":"02:41.195","Text":"I\u0027ll bring that formula back for the disk method"},{"Start":"02:41.195 ","End":"02:47.660","Text":"for revolving around the x-axis and here\u0027s that formula."},{"Start":"02:47.660 ","End":"02:55.850","Text":"In our case, we know everything the a and the b are still going to be 0 and 1."},{"Start":"02:55.850 ","End":"03:00.105","Text":"This is going to be 0, this is going to be 1."},{"Start":"03:00.105 ","End":"03:06.305","Text":"F of x is the upper one is going to be 2 minus x cubed."},{"Start":"03:06.305 ","End":"03:09.140","Text":"G of x is going to be the lower one,"},{"Start":"03:09.140 ","End":"03:14.100","Text":"which is just the constant 1 as a function of x."},{"Start":"03:14.100 ","End":"03:19.245","Text":"Now we just have to get technical and compute this integral."},{"Start":"03:19.245 ","End":"03:22.385","Text":"First of all, I\u0027ll just write it in our case,"},{"Start":"03:22.385 ","End":"03:29.000","Text":"v equals Pi times the integral from 0 to 1 of"},{"Start":"03:29.000 ","End":"03:34.650","Text":"2 minus x cubed squared"},{"Start":"03:34.650 ","End":"03:43.965","Text":"minus the constant function 1 squared dx."},{"Start":"03:43.965 ","End":"03:46.995","Text":"This is equal to, let\u0027s see,"},{"Start":"03:46.995 ","End":"03:49.780","Text":"Pi stays integral from 0 to 1."},{"Start":"03:49.780 ","End":"03:57.140","Text":"I suggest we open the brackets again using the a minus b squared formula,"},{"Start":"03:57.140 ","End":"04:00.170","Text":"which is a squared minus 2ab plus b squared."},{"Start":"04:00.170 ","End":"04:10.250","Text":"We get 2 squared minus 2 times, I\u0027ll just write it as 4."},{"Start":"04:10.250 ","End":"04:18.230","Text":"So 2 squared is 4 minus twice this times this is"},{"Start":"04:18.230 ","End":"04:23.479","Text":"minus 4x cubed plus this thing squared"},{"Start":"04:23.479 ","End":"04:29.830","Text":"is x^6 and minus 1."},{"Start":"04:29.830 ","End":"04:33.190","Text":"You know what? Just for laziness,"},{"Start":"04:33.190 ","End":"04:34.930","Text":"instead of writing a new line,"},{"Start":"04:34.930 ","End":"04:37.120","Text":"instead of the minus 1,"},{"Start":"04:37.120 ","End":"04:43.450","Text":"I\u0027ll just take the 4 and the minus 1 and write it as 3."},{"Start":"04:44.570 ","End":"04:47.910","Text":"The 4 with the minus 3 gave me 3,"},{"Start":"04:47.910 ","End":"04:51.600","Text":"saves me a line, dx."},{"Start":"04:51.600 ","End":"04:54.870","Text":"Now let\u0027s do the actual integral."},{"Start":"04:54.870 ","End":"04:57.900","Text":"We get Pi times,"},{"Start":"04:57.900 ","End":"05:05.455","Text":"now the integral of 3 is 3x minus raise the power by 1,"},{"Start":"05:05.455 ","End":"05:14.020","Text":"x^4 and divide by 4, 4 over 4 so it\u0027s just x^4 and then plus x^7 over"},{"Start":"05:14.020 ","End":"05:24.285","Text":"7 so 1/7x^7 and this between 0 and 1."},{"Start":"05:24.285 ","End":"05:26.900","Text":"If we put in 1,"},{"Start":"05:26.900 ","End":"05:35.780","Text":"we get Pi times 3 minus 1 plus 1/7."},{"Start":"05:35.780 ","End":"05:41.755","Text":"If we put in 0, all these terms are 0 so everything comes out to be 0."},{"Start":"05:41.755 ","End":"05:49.870","Text":"What I end up with is 3 minus 1 is 2 plus 1/7 is 2 and 1/7 Pi."},{"Start":"05:51.260 ","End":"05:54.560","Text":"I prefer a mixed, sorry,"},{"Start":"05:54.560 ","End":"05:55.655","Text":"this is a mixed fraction,"},{"Start":"05:55.655 ","End":"06:00.720","Text":"an improper fraction, which is 2 times 7 plus 1 is 15."},{"Start":"06:02.380 ","End":"06:08.640","Text":"15/7 Pi, or 15 Pi/7 and this"},{"Start":"06:08.640 ","End":"06:16.450","Text":"would be the answer and we\u0027re done."}],"ID":4726},{"Watched":false,"Name":"Exercise 3 part c","Duration":"5m 22s","ChapterTopicVideoID":4714,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.395","Text":"Now we\u0027re up to Part c of that same exercise."},{"Start":"00:04.395 ","End":"00:07.485","Text":"It\u0027s very similar to Part b."},{"Start":"00:07.485 ","End":"00:10.050","Text":"We already learned the trick."},{"Start":"00:10.050 ","End":"00:12.960","Text":"It\u0027s just applied slightly differently."},{"Start":"00:12.960 ","End":"00:17.550","Text":"In Part b, we revolved around the line y equals 1 and"},{"Start":"00:17.550 ","End":"00:22.905","Text":"our trick was just to raise everything by 1."},{"Start":"00:22.905 ","End":"00:31.180","Text":"This time we\u0027re revolving around y equals 2 and our trick will be to lower everything"},{"Start":"00:33.290 ","End":"00:40.640","Text":"down by 2 units downwards and then this axis of"},{"Start":"00:40.640 ","End":"00:47.520","Text":"revolution will become the x-axis and this will become 2 units lower and so on,"},{"Start":"00:47.520 ","End":"00:51.605","Text":"and I\u0027ll just draw what it looks like after we move everything."},{"Start":"00:51.605 ","End":"00:53.900","Text":"Basically what\u0027s in yellow,"},{"Start":"00:53.900 ","End":"00:56.670","Text":"will move 2 units down."},{"Start":"00:57.110 ","End":"01:01.940","Text":"Here I am after I moved everything 2 units down,"},{"Start":"01:01.940 ","End":"01:06.934","Text":"the y equals 2 becomes y equals 0, the x axis."},{"Start":"01:06.934 ","End":"01:11.180","Text":"If we move these 2 down, this becomes, well,"},{"Start":"01:11.180 ","End":"01:14.165","Text":"this is still 0, but this is minus 1,"},{"Start":"01:14.165 ","End":"01:16.115","Text":"this is minus 2,"},{"Start":"01:16.115 ","End":"01:19.080","Text":"this is still the point"},{"Start":"01:20.600 ","End":"01:29.220","Text":"1 and instead of y equals 0,"},{"Start":"01:29.220 ","End":"01:32.434","Text":"this is the line y equals negative 2,"},{"Start":"01:32.434 ","End":"01:38.135","Text":"and this curve which was y equals 1 minus x cubed from here,"},{"Start":"01:38.135 ","End":"01:40.985","Text":"after we\u0027ve taken it 2 down,"},{"Start":"01:40.985 ","End":"01:44.960","Text":"it becomes y equals this minus 2,"},{"Start":"01:44.960 ","End":"01:50.600","Text":"which is minus 1 minus x cubed."},{"Start":"01:50.600 ","End":"01:53.300","Text":"We have an upper function,"},{"Start":"01:53.300 ","End":"01:55.670","Text":"which in the formula is f,"},{"Start":"01:55.670 ","End":"01:58.250","Text":"and we have a lower function, a constant function,"},{"Start":"01:58.250 ","End":"01:59.885","Text":"which in the formula is g,"},{"Start":"01:59.885 ","End":"02:03.020","Text":"I\u0027ll write down that formula again."},{"Start":"02:03.020 ","End":"02:06.530","Text":"Here it is, this formula,"},{"Start":"02:06.530 ","End":"02:10.500","Text":"and let\u0027s interpret it in our case."},{"Start":"02:10.500 ","End":"02:14.950","Text":"In our case, we have V equals Pi integral."},{"Start":"02:14.950 ","End":"02:19.700","Text":"Now A and B are still 0 and 1."},{"Start":"02:20.600 ","End":"02:24.060","Text":"But f of x, like we said, at the upper 1,"},{"Start":"02:24.060 ","End":"02:31.005","Text":"is minus 1 minus x cubed and that\u0027s squared."},{"Start":"02:31.005 ","End":"02:34.440","Text":"Less g of x is the constant function,"},{"Start":"02:34.440 ","End":"02:43.090","Text":"negative 2 squared and all this, dx."},{"Start":"02:43.460 ","End":"02:48.110","Text":"Everything\u0027s now just mechanical standard."},{"Start":"02:48.110 ","End":"02:55.715","Text":"We get that this is equal to this here and I spared you the algebraic details."},{"Start":"02:55.715 ","End":"03:02.595","Text":"Anyway, we can combine the 1 and the minus 4 and get minus 3,"},{"Start":"03:02.595 ","End":"03:09.190","Text":"here it is, dx,"},{"Start":"03:10.100 ","End":"03:15.010","Text":"and now this is equal to Pi."},{"Start":"03:15.010 ","End":"03:17.210","Text":"Now we can actually do the integral."},{"Start":"03:17.210 ","End":"03:24.300","Text":"Integral is minus 3x plus,"},{"Start":"03:24.300 ","End":"03:27.340","Text":"I make this to the power of 4 and divide by 4,"},{"Start":"03:27.340 ","End":"03:35.830","Text":"so I get 1/2x^4 and here 1/7x^7,"},{"Start":"03:35.830 ","End":"03:38.875","Text":"between 0 and 1,"},{"Start":"03:38.875 ","End":"03:44.939","Text":"and now we substitute x equals 1,"},{"Start":"03:44.939 ","End":"03:55.330","Text":"so we get Pi times minus 3 plus 1/2 plus 1/7 and when we put in 0,"},{"Start":"03:55.330 ","End":"04:00.200","Text":"everything becomes 0, so it\u0027s minus 0."},{"Start":"04:00.390 ","End":"04:03.400","Text":"Let\u0027s see this fraction,"},{"Start":"04:03.400 ","End":"04:08.940","Text":"1/2 and 1/7, if I put it over,"},{"Start":"04:08.940 ","End":"04:13.900","Text":"14 is 7 plus 2 is 9/14,"},{"Start":"04:13.900 ","End":"04:19.160","Text":"I\u0027ll just write it as minus 42 plus"},{"Start":"04:19.160 ","End":"04:27.460","Text":"7 plus 2 over 14 times Pi."},{"Start":"04:27.500 ","End":"04:36.110","Text":"We get minus 33 Pi over 14 but what I"},{"Start":"04:36.110 ","End":"04:45.455","Text":"forgot to say was that in the case that we have the area below the x-axis,"},{"Start":"04:45.455 ","End":"04:53.065","Text":"then we have to expect a negative and we take the absolute value,"},{"Start":"04:53.065 ","End":"04:59.075","Text":"so we just make this plus 33 Pi over 14."},{"Start":"04:59.075 ","End":"05:06.545","Text":"We are done and the answer for this is 33 Pi over 14."},{"Start":"05:06.545 ","End":"05:10.370","Text":"Not a simple exercise and I would like to add,"},{"Start":"05:10.370 ","End":"05:12.380","Text":"although we\u0027re done, that not to worry,"},{"Start":"05:12.380 ","End":"05:17.315","Text":"this is very rarely given in exams or tests,"},{"Start":"05:17.315 ","End":"05:20.270","Text":"but just in case you encounter it,"},{"Start":"05:20.270 ","End":"05:23.130","Text":"you\u0027ll have some idea what to do."}],"ID":4727},{"Watched":false,"Name":"Exercise 3 part d","Duration":"3m 47s","ChapterTopicVideoID":4715,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.415","Text":"Now we come to Part D of this long question."},{"Start":"00:05.415 ","End":"00:09.285","Text":"This time it\u0027s the y-axis,"},{"Start":"00:09.285 ","End":"00:12.390","Text":"which is the axis of revolution,"},{"Start":"00:12.390 ","End":"00:15.480","Text":"the same shape as before."},{"Start":"00:15.480 ","End":"00:21.560","Text":"We have the top function which is the 1 minus x cubed,"},{"Start":"00:21.560 ","End":"00:27.180","Text":"and here, the x axis,"},{"Start":"00:27.180 ","End":"00:29.970","Text":"which is y equals 0."},{"Start":"00:29.970 ","End":"00:35.160","Text":"We\u0027ll just write it here, y equals 1 minus x cubed."},{"Start":"00:35.160 ","End":"00:42.040","Text":"This time, we\u0027ll use the shell method for"},{"Start":"00:42.040 ","End":"00:49.580","Text":"revolving around the y-axis because that\u0027s what we use when y is a function of x."},{"Start":"00:49.580 ","End":"00:51.830","Text":"When we\u0027re given y in terms of x,"},{"Start":"00:51.830 ","End":"00:56.530","Text":"it\u0027s the most convenient one and let me write it down."},{"Start":"00:56.530 ","End":"01:01.745","Text":"Here it is, but I just forgot to write some of the values."},{"Start":"01:01.745 ","End":"01:06.995","Text":"This is 0, this is 1, and this is 1,"},{"Start":"01:06.995 ","End":"01:14.200","Text":"and the top 1 is the function f of x."},{"Start":"01:14.200 ","End":"01:15.100","Text":"Well, we have it here."},{"Start":"01:15.100 ","End":"01:17.180","Text":"What I did want to say though,"},{"Start":"01:17.180 ","End":"01:19.249","Text":"is that because the lower 1,"},{"Start":"01:19.249 ","End":"01:23.420","Text":"the g of x is just the x axis, which is y equals 0."},{"Start":"01:23.420 ","End":"01:26.930","Text":"We can use a simplified version of this."},{"Start":"01:26.930 ","End":"01:29.270","Text":"In fact, if I just let g equals 0,"},{"Start":"01:29.270 ","End":"01:31.770","Text":"I\u0027ll get the simplified version."},{"Start":"01:32.080 ","End":"01:35.720","Text":"Well, here it is. It looks a lot simpler."},{"Start":"01:35.720 ","End":"01:39.200","Text":"Now that we don\u0027t have g of x,"},{"Start":"01:39.200 ","End":"01:41.225","Text":"we only have an f of x."},{"Start":"01:41.225 ","End":"01:46.060","Text":"Well, what remains is to substitute the known quantities."},{"Start":"01:46.060 ","End":"01:51.870","Text":"Let\u0027s see. V is equal to 2 pi times the integral."},{"Start":"01:51.870 ","End":"01:56.655","Text":"Now, A and B are 0 and 1 respectively."},{"Start":"01:56.655 ","End":"02:03.765","Text":"X is x, f of x is 1 minus x cubed."},{"Start":"02:03.765 ","End":"02:10.615","Text":"1 minus x cubed dx."},{"Start":"02:10.615 ","End":"02:14.125","Text":"This is just a straightforward integration and computation."},{"Start":"02:14.125 ","End":"02:18.130","Text":"Don\u0027t need the diagrams anymore. Let\u0027s see."},{"Start":"02:18.130 ","End":"02:23.995","Text":"This is equal to 2 pi integral from 0 to 1."},{"Start":"02:23.995 ","End":"02:25.390","Text":"I\u0027ll open brackets."},{"Start":"02:25.390 ","End":"02:30.730","Text":"We\u0027ve got x minus x to the fourth dx,"},{"Start":"02:30.730 ","End":"02:32.370","Text":"which is equal to 2 pi."},{"Start":"02:32.370 ","End":"02:38.050","Text":"Now, the integral of x is x squared over 2."},{"Start":"02:38.050 ","End":"02:47.200","Text":"Here we have minus x to the fifth over 5 evaluated between 0 and 1."},{"Start":"02:47.390 ","End":"02:51.105","Text":"Let\u0027s substitute 1 first."},{"Start":"02:51.105 ","End":"02:58.515","Text":"We\u0027ve got 2 pi times 1/2 minus 1/5."},{"Start":"02:58.515 ","End":"03:05.664","Text":"If we substitute 0, it\u0027s 2 pi times 0 minus 0 it\u0027s just 0."},{"Start":"03:05.664 ","End":"03:12.670","Text":"I write it in to show that I haven\u0027t forgotten that there\u0027s a 0 also."},{"Start":"03:12.670 ","End":"03:14.770","Text":"Now, let\u0027s see."},{"Start":"03:14.770 ","End":"03:19.105","Text":"If we put it in decimals or in tenths,"},{"Start":"03:19.105 ","End":"03:21.805","Text":"this is 5/10 minus 2/10."},{"Start":"03:21.805 ","End":"03:25.420","Text":"We have 3/10."},{"Start":"03:25.420 ","End":"03:30.750","Text":"3/10 times 2 is 6/10, so 3/5."},{"Start":"03:30.750 ","End":"03:33.375","Text":"I make it 3/5 pi,"},{"Start":"03:33.375 ","End":"03:37.140","Text":"or 3 pi over 5."},{"Start":"03:37.140 ","End":"03:42.830","Text":"That\u0027s the answer. We are done."},{"Start":"03:42.830 ","End":"03:44.315","Text":"I\u0027ll just highlight it."},{"Start":"03:44.315 ","End":"03:47.160","Text":"That\u0027ll be nice. Okay."}],"ID":4728},{"Watched":false,"Name":"Exercise 3 part e","Duration":"9m 46s","ChapterTopicVideoID":4716,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"Now, we come to part e. This time,"},{"Start":"00:05.730 ","End":"00:10.380","Text":"it\u0027s the line x equals minus 1 that we revolve about."},{"Start":"00:10.380 ","End":"00:14.505","Text":"I\u0027ll highlight the shape,"},{"Start":"00:14.505 ","End":"00:21.700","Text":"little area, and I\u0027ll highlight this line."},{"Start":"00:21.700 ","End":"00:26.149","Text":"Let\u0027s just recall the important values."},{"Start":"00:26.149 ","End":"00:30.429","Text":"The points, this was 0, this was 1, this was 1,"},{"Start":"00:30.429 ","End":"00:39.340","Text":"and this shape was y equals 1 minus x cubed, it\u0027s function."},{"Start":"00:39.350 ","End":"00:46.565","Text":"Now, we did a similar trick in part b,"},{"Start":"00:46.565 ","End":"00:51.800","Text":"and we need a trick because we only know how to revolve about the x-axis or the y-axis."},{"Start":"00:51.800 ","End":"00:54.535","Text":"Just as in the case of the x-axis,"},{"Start":"00:54.535 ","End":"00:57.535","Text":"we shifted the whole thing by 1 unit."},{"Start":"00:57.535 ","End":"00:59.585","Text":"We\u0027re going to do a shifting here."},{"Start":"00:59.585 ","End":"01:09.260","Text":"What I\u0027m going to do is take both the axis and the area,"},{"Start":"01:09.260 ","End":"01:13.110","Text":"both of them, I will shift, say,"},{"Start":"01:13.110 ","End":"01:19.395","Text":"plus 1 unit to the right and plus 1 unit to the right."},{"Start":"01:19.395 ","End":"01:21.305","Text":"If I do that,"},{"Start":"01:21.305 ","End":"01:23.570","Text":"the problem will be essentially the same."},{"Start":"01:23.570 ","End":"01:27.815","Text":"I\u0027ll draw you a quick sketch of what it will look like."},{"Start":"01:27.815 ","End":"01:29.850","Text":"I can do this freehand."},{"Start":"01:29.850 ","End":"01:35.550","Text":"Let\u0027s see. We have, y-axis and x-axis."},{"Start":"01:35.550 ","End":"01:38.330","Text":"If I shift the important things,"},{"Start":"01:38.330 ","End":"01:41.845","Text":"the line x equals minus 1 will"},{"Start":"01:41.845 ","End":"01:50.070","Text":"move to the y-axis."},{"Start":"01:50.070 ","End":"01:53.795","Text":"The x is minus 1 will become x equals 0."},{"Start":"01:53.795 ","End":"01:56.975","Text":"This shape will shift."},{"Start":"01:56.975 ","End":"02:04.190","Text":"Well, the y-axis will also shift 1 unit to the right."},{"Start":"02:04.190 ","End":"02:11.530","Text":"Our shape will be somewhere over here."},{"Start":"02:14.770 ","End":"02:21.770","Text":"Instead of 0 and 1,"},{"Start":"02:21.770 ","End":"02:24.760","Text":"it will be 1 and 2."},{"Start":"02:24.760 ","End":"02:29.260","Text":"This is still going to be equal to 1."},{"Start":"02:29.480 ","End":"02:32.860","Text":"I\u0027ll just shade it a bit."},{"Start":"02:34.870 ","End":"02:39.995","Text":"Shading just so you recognize it\u0027s the same thing."},{"Start":"02:39.995 ","End":"02:43.630","Text":"Just taking this and this."},{"Start":"02:43.630 ","End":"02:48.170","Text":"Go over it in yellow also just like over here."},{"Start":"02:48.170 ","End":"02:50.855","Text":"Yellow, it\u0027s got a bit squashed,"},{"Start":"02:50.855 ","End":"02:53.105","Text":"but it\u0027s the same thing."},{"Start":"02:53.105 ","End":"03:03.235","Text":"What we have to do is revolve this shape around the regular y-axis."},{"Start":"03:03.235 ","End":"03:10.895","Text":"Just like here, we had the x equals minus 1 axis."},{"Start":"03:10.895 ","End":"03:18.680","Text":"The thing we still don\u0027t know is the equation of this curve here."},{"Start":"03:18.680 ","End":"03:21.650","Text":"Over here, we had"},{"Start":"03:21.650 ","End":"03:30.540","Text":"y equals f of x which was 1 minus x cubed."},{"Start":"03:30.540 ","End":"03:37.990","Text":"Now, I want to shift it 1 unit to the right."},{"Start":"03:38.930 ","End":"03:44.025","Text":"There was a whole chapter on how to do this."},{"Start":"03:44.025 ","End":"03:46.790","Text":"When you shift 1 unit to the right,"},{"Start":"03:46.790 ","End":"03:55.135","Text":"what you do is you replace x by x minus 1."},{"Start":"03:55.135 ","End":"03:57.750","Text":"You might think to the right is plus 1,"},{"Start":"03:57.750 ","End":"04:00.420","Text":"but no, it is actually minus 1."},{"Start":"04:00.420 ","End":"04:03.680","Text":"If I do this, I\u0027ll get a new function, let\u0027s say,"},{"Start":"04:03.680 ","End":"04:05.535","Text":"I call it g of x,"},{"Start":"04:05.535 ","End":"04:07.565","Text":"y is g of x."},{"Start":"04:07.565 ","End":"04:10.690","Text":"Let\u0027s compute what g of x is."},{"Start":"04:10.690 ","End":"04:20.555","Text":"Essentially, what I\u0027m saying is that g of x is given by f of x minus 1."},{"Start":"04:20.555 ","End":"04:26.715","Text":"This is equal to 1 minus,"},{"Start":"04:26.715 ","End":"04:30.765","Text":"instead of x, I put x minus 1 cubed."},{"Start":"04:30.765 ","End":"04:32.745","Text":"Let\u0027s see what we get."},{"Start":"04:32.745 ","End":"04:35.820","Text":"We get 1 minus."},{"Start":"04:35.820 ","End":"04:43.955","Text":"Now, x cubed is a formula for a minus b cubed."},{"Start":"04:43.955 ","End":"04:46.760","Text":"You can look it up, I\u0027ll just write the answer."},{"Start":"04:46.760 ","End":"04:54.110","Text":"X cubed minus 3x squared plus 3x minus 1,"},{"Start":"04:54.110 ","End":"05:00.460","Text":"and this is equal to minus x cubed plus 3x"},{"Start":"05:00.460 ","End":"05:07.965","Text":"squared minus 3x plus 1 plus 1 plus 2."},{"Start":"05:07.965 ","End":"05:10.565","Text":"That is the equation of this."},{"Start":"05:10.565 ","End":"05:14.750","Text":"Once again, the lower function is y equals 0,"},{"Start":"05:14.750 ","End":"05:17.659","Text":"so we\u0027re going to use our simplified formula."},{"Start":"05:17.659 ","End":"05:23.360","Text":"Let me go and copy it from the previous exercise. Here it is."},{"Start":"05:23.360 ","End":"05:26.270","Text":"But just so we don\u0027t get confused, here,"},{"Start":"05:26.270 ","End":"05:29.240","Text":"we have g naught f. Now,"},{"Start":"05:29.240 ","End":"05:31.190","Text":"all we have to do is substitute."},{"Start":"05:31.190 ","End":"05:34.470","Text":"Notice that a and b are 1 and 2,"},{"Start":"05:34.470 ","End":"05:37.545","Text":"and g is what I have written here."},{"Start":"05:37.545 ","End":"05:46.070","Text":"What we get is that V is equal to 2Pi times the integral from 1 to"},{"Start":"05:46.070 ","End":"05:52.940","Text":"2 of x times g of x is what\u0027s right here minus"},{"Start":"05:52.940 ","End":"06:01.830","Text":"x cubed plus 3x squared minus 3x plus 2 dx."},{"Start":"06:01.830 ","End":"06:05.415","Text":"Now, it\u0027s just a computation. Let\u0027s see."},{"Start":"06:05.415 ","End":"06:08.400","Text":"2Pi integral from 1 to 2."},{"Start":"06:08.400 ","End":"06:10.500","Text":"Let\u0027s just multiply out."},{"Start":"06:10.500 ","End":"06:16.980","Text":"We get minus x^4 plus 3x cubed"},{"Start":"06:16.980 ","End":"06:27.510","Text":"minus 3x squared plus 2x dx."},{"Start":"06:27.510 ","End":"06:30.510","Text":"Let\u0027s see. We get 2Pi,"},{"Start":"06:30.510 ","End":"06:32.940","Text":"and now, we can actually do the integral,"},{"Start":"06:32.940 ","End":"06:37.095","Text":"and that is minus x^5 over"},{"Start":"06:37.095 ","End":"06:44.445","Text":"5 plus 3x^4 over 4,"},{"Start":"06:44.445 ","End":"06:49.575","Text":"minus 3x cubed over 3,"},{"Start":"06:49.575 ","End":"06:53.535","Text":"I\u0027ll write it straight away as x cubed because the 3 is cancel."},{"Start":"06:53.535 ","End":"06:56.565","Text":"Derivative of x cubed is 3x squared, of course."},{"Start":"06:56.565 ","End":"06:59.340","Text":"The 2x, same thing, x squared."},{"Start":"06:59.340 ","End":"07:06.010","Text":"2x squared over 2 is just x squared between 1 and 2."},{"Start":"07:07.130 ","End":"07:09.150","Text":"This is equal to,"},{"Start":"07:09.150 ","End":"07:12.730","Text":"now some arithmetic here, 2Pi."},{"Start":"07:12.980 ","End":"07:20.355","Text":"For 2, I get minus 32 over 5,"},{"Start":"07:20.355 ","End":"07:26.790","Text":"plus, 2^4 is 16 over 4,"},{"Start":"07:26.790 ","End":"07:30.405","Text":"is 4 times 3 is 12."},{"Start":"07:30.405 ","End":"07:33.075","Text":"2 cubed is 8."},{"Start":"07:33.075 ","End":"07:35.685","Text":"2 squared is 4."},{"Start":"07:35.685 ","End":"07:37.470","Text":"That\u0027s for the 2."},{"Start":"07:37.470 ","End":"07:42.060","Text":"Now, the part for 1 which is going to be"},{"Start":"07:42.060 ","End":"07:51.310","Text":"minus 1/5 plus 3/4 minus 1 plus 1."},{"Start":"07:51.500 ","End":"07:55.460","Text":"I won\u0027t bore you with all the details."},{"Start":"07:55.460 ","End":"08:02.890","Text":"This first part here comes out to 8/5,"},{"Start":"08:02.890 ","End":"08:09.540","Text":"and this part here comes out to 11/20."},{"Start":"08:11.750 ","End":"08:19.230","Text":"Let\u0027s see now. 8 over 5 is 32 over 20 so we get 21 over 20."},{"Start":"08:19.230 ","End":"08:24.000","Text":"This is 2Pi times 21 over 20,"},{"Start":"08:24.000 ","End":"08:26.820","Text":"2 into 20 goes 10 times."},{"Start":"08:26.820 ","End":"08:31.930","Text":"This is equal to 21 over 10 Pi."},{"Start":"08:31.930 ","End":"08:33.410","Text":"Or if you like,"},{"Start":"08:33.410 ","End":"08:38.355","Text":"it\u0027s exactly equal to 2.1 Pi."},{"Start":"08:38.355 ","End":"08:41.715","Text":"That is the answer."},{"Start":"08:41.715 ","End":"08:43.650","Text":"Before I say we\u0027re done,"},{"Start":"08:43.650 ","End":"08:47.580","Text":"I\u0027ll just do something I realized I should have mentioned,"},{"Start":"08:47.580 ","End":"08:55.175","Text":"that is that when I talked about shifting 1 unit to the right,"},{"Start":"08:55.175 ","End":"08:57.380","Text":"then I should have made it more general."},{"Start":"08:57.380 ","End":"09:02.685","Text":"In general, if I shift a units to the right,"},{"Start":"09:02.685 ","End":"09:05.840","Text":"then I replace x by x minus a."},{"Start":"09:05.840 ","End":"09:07.505","Text":"In our particular case,"},{"Start":"09:07.505 ","End":"09:09.860","Text":"a will happen to equal 1."},{"Start":"09:09.860 ","End":"09:14.305","Text":"Similarly, if it\u0027s to the left,"},{"Start":"09:14.305 ","End":"09:19.605","Text":"and I want to shift it a units to the left,"},{"Start":"09:19.605 ","End":"09:27.970","Text":"then I replace x by x plus a."},{"Start":"09:27.970 ","End":"09:31.530","Text":"This 1 is for the right,"},{"Start":"09:31.530 ","End":"09:33.155","Text":"and this 1 is for the left."},{"Start":"09:33.155 ","End":"09:35.540","Text":"But we covered all that in the chapter on"},{"Start":"09:35.540 ","End":"09:38.945","Text":"shifting functions up and down and left and right."},{"Start":"09:38.945 ","End":"09:42.330","Text":"Anyway, this, as I say, is our answer,"},{"Start":"09:42.330 ","End":"09:46.180","Text":"21 over 10 Pi, and we are done."}],"ID":4729},{"Watched":false,"Name":"Exercise 3 part f","Duration":"14m 38s","ChapterTopicVideoID":4717,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.434","Text":"Finally, we come to part F. In this part we\u0027re going to revolve"},{"Start":"00:06.434 ","End":"00:13.830","Text":"our shape around the vertical axis, x equals 2."},{"Start":"00:14.360 ","End":"00:18.194","Text":"Here\u0027s x equals 2 in yellow,"},{"Start":"00:18.194 ","End":"00:22.500","Text":"and here is our shape,"},{"Start":"00:22.500 ","End":"00:26.055","Text":"I\u0027ll just mark its contour."},{"Start":"00:26.055 ","End":"00:31.020","Text":"As before, I\u0027ll mark some of the details that we found."},{"Start":"00:31.020 ","End":"00:35.830","Text":"This was 0, this was 1, this was 1."},{"Start":"00:35.870 ","End":"00:40.200","Text":"This shape here was f of x,"},{"Start":"00:40.200 ","End":"00:45.165","Text":"which is 1 minus x cubed."},{"Start":"00:45.165 ","End":"00:49.790","Text":"We\u0027re going to do something here that\u0027s very similar to the previous exercise"},{"Start":"00:49.790 ","End":"00:54.170","Text":"where we had to find the revolution about this axis."},{"Start":"00:54.170 ","End":"01:04.470","Text":"What we did was simply shifted the area and the axis 1 unit to the left."},{"Start":"01:04.840 ","End":"01:09.380","Text":"Sorry, we moved the axis and the shape 1 unit to"},{"Start":"01:09.380 ","End":"01:13.610","Text":"the right over there so that this would land on top of the y-axis."},{"Start":"01:13.610 ","End":"01:17.045","Text":"Then we know how to revolve about the y-axis."},{"Start":"01:17.045 ","End":"01:19.625","Text":"This time we\u0027re going to do the opposite."},{"Start":"01:19.625 ","End":"01:21.005","Text":"We are going to move"},{"Start":"01:21.005 ","End":"01:31.765","Text":"both the axis and the shape 2 units to the left."},{"Start":"01:31.765 ","End":"01:39.090","Text":"Now, the way we move a function 2 units to the left is,"},{"Start":"01:39.090 ","End":"01:43.360","Text":"like I said before when we move to the right by a unit,"},{"Start":"01:43.360 ","End":"01:45.400","Text":"we replace x by x minus a,"},{"Start":"01:45.400 ","End":"01:47.800","Text":"and to the left, it\u0027s the opposite."},{"Start":"01:47.800 ","End":"01:50.260","Text":"In general, if I move,"},{"Start":"01:50.260 ","End":"01:51.730","Text":"let me just write that down,"},{"Start":"01:51.730 ","End":"02:01.000","Text":"if I want to move a function a units to the left,"},{"Start":"02:01.000 ","End":"02:10.665","Text":"then I replace, so I want to shift by a to the left."},{"Start":"02:10.665 ","End":"02:18.020","Text":"Then what I do is I replace x in the function by x plus a."},{"Start":"02:18.020 ","End":"02:21.080","Text":"That\u0027s the rule, just like with moving to the right,"},{"Start":"02:21.080 ","End":"02:26.180","Text":"it\u0027s x minus a. What do we have here?"},{"Start":"02:26.180 ","End":"02:31.040","Text":"First of all, give you a quick sketch of what it\u0027s going to look like after the shift."},{"Start":"02:31.040 ","End":"02:33.170","Text":"The important things are going to be seen,"},{"Start":"02:33.170 ","End":"02:34.865","Text":"this thing is going to land on top of this,"},{"Start":"02:34.865 ","End":"02:36.935","Text":"and this is going to go over here."},{"Start":"02:36.935 ","End":"02:43.115","Text":"What we\u0027re going to get after a shift of 2 units is the following."},{"Start":"02:43.115 ","End":"02:46.175","Text":"We\u0027ll have this thing will come on the y-axis."},{"Start":"02:46.175 ","End":"02:52.620","Text":"The y-axis will be 2 units to the left,"},{"Start":"02:52.620 ","End":"02:54.855","Text":"we still have an x-axis."},{"Start":"02:54.855 ","End":"03:03.410","Text":"The shape that was here is now going to be over here."},{"Start":"03:06.980 ","End":"03:11.565","Text":"Shade it, get the yellow out,"},{"Start":"03:11.565 ","End":"03:15.645","Text":"the yellow for the area,"},{"Start":"03:15.645 ","End":"03:21.430","Text":"and for this axis."},{"Start":"03:22.220 ","End":"03:26.745","Text":"Once again, I move I think mainly the axis"},{"Start":"03:26.745 ","End":"03:30.640","Text":"and the areas what I want to move 2 units to the left."},{"Start":"03:30.640 ","End":"03:32.800","Text":"This lands on top of the y-axis."},{"Start":"03:32.800 ","End":"03:40.035","Text":"This goes to, wherever was 1 is now minus 1,"},{"Start":"03:40.035 ","End":"03:42.810","Text":"whatever is 0 is now minus 2."},{"Start":"03:42.810 ","End":"03:46.860","Text":"This thing is still equal to 1."},{"Start":"03:46.860 ","End":"03:53.390","Text":"This is 0, and the important thing now is to find out,"},{"Start":"03:53.390 ","End":"03:55.520","Text":"what is this function?"},{"Start":"03:55.520 ","End":"04:02.250","Text":"This function which was f of x is now going to be some g of x,"},{"Start":"04:02.250 ","End":"04:08.655","Text":"this line here is y equals g of x."},{"Start":"04:08.655 ","End":"04:14.330","Text":"We get g by replacing x by x plus 2."},{"Start":"04:14.330 ","End":"04:16.145","Text":"In other words, in our case,"},{"Start":"04:16.145 ","End":"04:20.085","Text":"x goes to x plus 2."},{"Start":"04:20.085 ","End":"04:23.945","Text":"What we\u0027ll get is that our function,"},{"Start":"04:23.945 ","End":"04:25.850","Text":"and let\u0027s call this g,"},{"Start":"04:25.850 ","End":"04:33.255","Text":"so g of x is in general f of x plus 2."},{"Start":"04:33.255 ","End":"04:37.900","Text":"This is equal to 1 minus instead of x,"},{"Start":"04:37.900 ","End":"04:41.925","Text":"I put x plus 2 and it\u0027s cubed."},{"Start":"04:41.925 ","End":"04:48.600","Text":"This is equal to 1 minus x plus 2 cubed,"},{"Start":"04:48.600 ","End":"04:52.595","Text":"I would like to have the answer."},{"Start":"04:52.595 ","End":"04:59.240","Text":"There is a formula for raising to the power of 3 x cubed plus 3"},{"Start":"04:59.240 ","End":"05:07.150","Text":"times this squared times this 3 times x squared times 2."},{"Start":"05:08.300 ","End":"05:11.740","Text":"I\u0027ll just write it like that."},{"Start":"05:11.740 ","End":"05:21.960","Text":"Plus 3 times this times this squared plus this thing cubed."},{"Start":"05:21.960 ","End":"05:25.645","Text":"Never mind I\u0027ve gone into the diagram area."},{"Start":"05:25.645 ","End":"05:31.050","Text":"No big deal. Which is equal to, let\u0027s see."},{"Start":"05:31.050 ","End":"05:34.900","Text":"Now we have minus x cubed,"},{"Start":"05:35.660 ","End":"05:40.080","Text":"minus 3 times 2 is 6, sorry,"},{"Start":"05:40.080 ","End":"05:48.690","Text":"minus 6x squared, 3 times 2 squared is 12 minus 12x,"},{"Start":"05:48.690 ","End":"05:53.715","Text":"plus 8 plus 1 is plus 9."},{"Start":"05:53.715 ","End":"05:56.919","Text":"That is my g of x."},{"Start":"05:57.080 ","End":"06:03.565","Text":"As usual, because the lower function is y equals,"},{"Start":"06:03.565 ","End":"06:10.440","Text":"I mean, this part here is just y equals 0."},{"Start":"06:10.440 ","End":"06:18.030","Text":"We use the abbreviated formula where there\u0027s only 1 function between it and the x-axis."},{"Start":"06:18.030 ","End":"06:22.710","Text":"Now, I forgot to label y and x."},{"Start":"06:22.710 ","End":"06:29.230","Text":"Yes, I want to repeat the formula I used in the previous exercise."},{"Start":"06:29.230 ","End":"06:31.905","Text":"This is the formula."},{"Start":"06:31.905 ","End":"06:36.860","Text":"Now, just like what happened in part C,"},{"Start":"06:36.860 ","End":"06:41.870","Text":"is there we had a shape that was below"},{"Start":"06:41.870 ","End":"06:47.240","Text":"the axis on here we have a shape that\u0027s to the left of the y-axis."},{"Start":"06:47.240 ","End":"06:50.680","Text":"Actually, it\u0027s come out negative or reversed,"},{"Start":"06:50.680 ","End":"06:53.599","Text":"and if it does, we\u0027ll just take the absolute value."},{"Start":"06:53.599 ","End":"06:55.370","Text":"There was a proper way to handle this."},{"Start":"06:55.370 ","End":"06:59.030","Text":"I just do it whatever and if it comes out negative,"},{"Start":"06:59.030 ","End":"07:03.855","Text":"you know you\u0027re supposed to make it positive. Essentially,"},{"Start":"07:03.855 ","End":"07:08.680","Text":"what you\u0027re supposed to do is take the mirror image of this shape and"},{"Start":"07:08.680 ","End":"07:13.869","Text":"then to the right of the axis and it doesn\u0027t change the answer,"},{"Start":"07:13.869 ","End":"07:19.990","Text":"but the mirror image involves replacing x by minus x."},{"Start":"07:19.990 ","End":"07:22.840","Text":"But I\u0027m just going to do it naively,"},{"Start":"07:22.840 ","End":"07:25.150","Text":"the same formula and if it comes out negative,"},{"Start":"07:25.150 ","End":"07:27.025","Text":"I\u0027ll take absolute value."},{"Start":"07:27.025 ","End":"07:29.380","Text":"This is the formula,"},{"Start":"07:29.380 ","End":"07:33.175","Text":"and in our case,"},{"Start":"07:33.175 ","End":"07:40.675","Text":"what we\u0027ll get is that V is equal to 2Pi, the integral."},{"Start":"07:40.675 ","End":"07:46.720","Text":"Now from a to b would be from minus 2 to minus 1."},{"Start":"07:46.720 ","End":"07:49.795","Text":"Again, it doesn\u0027t matter if I take them the wrong way round."},{"Start":"07:49.795 ","End":"07:53.530","Text":"All that can happen is it will make"},{"Start":"07:53.530 ","End":"07:57.955","Text":"the answer negative and we\u0027ll just throw out the negative if there is one at the end,"},{"Start":"07:57.955 ","End":"08:02.830","Text":"so x times now g of x is what I wrote here,"},{"Start":"08:02.830 ","End":"08:08.320","Text":"which is the minus x cubed minus 6x squared"},{"Start":"08:08.320 ","End":"08:11.485","Text":"minus 12x"},{"Start":"08:11.485 ","End":"08:19.900","Text":"and there\u0027s something wrong here."},{"Start":"08:19.900 ","End":"08:24.130","Text":"Yes, I can see I made a mistake when I took 1 minus this,"},{"Start":"08:24.130 ","End":"08:26.680","Text":"there should be minus 8 plus 1."},{"Start":"08:26.680 ","End":"08:29.290","Text":"This has got to be minus 7."},{"Start":"08:29.290 ","End":"08:31.765","Text":"I\u0027ll write it here first,"},{"Start":"08:31.765 ","End":"08:35.590","Text":"minus 7 and just let me quickly,"},{"Start":"08:35.590 ","End":"08:43.570","Text":"I\u0027ll take an eraser and then back to the pan and minus 7,"},{"Start":"08:43.570 ","End":"08:46.640","Text":"no harm done just yet."},{"Start":"08:47.490 ","End":"08:52.780","Text":"Continuing though really I could have taken the minus outside the brackets."},{"Start":"08:52.780 ","End":"08:55.190","Text":"I can still do that."},{"Start":"08:56.520 ","End":"08:59.125","Text":"Well, we can still do that."},{"Start":"08:59.125 ","End":"09:08.425","Text":"This is equal to minus 2Pi integral from minus 2 to minus 1,"},{"Start":"09:08.425 ","End":"09:14.035","Text":"and all of these become plus and also multiply them now by x."},{"Start":"09:14.035 ","End":"09:21.370","Text":"We have x^4 plus 6x cubed plus 12x"},{"Start":"09:21.370 ","End":"09:29.470","Text":"squared plus 7x dx,"},{"Start":"09:29.470 ","End":"09:32.185","Text":"which equals minus 2."},{"Start":"09:32.185 ","End":"09:34.750","Text":"Now we actually get to do the integral,"},{"Start":"09:34.750 ","End":"09:45.084","Text":"so x^5 over 5 plus 6x^4 over 4,"},{"Start":"09:45.084 ","End":"09:50.365","Text":"plus 12x cubed over 3,"},{"Start":"09:50.365 ","End":"09:57.385","Text":"plus 7x squared over 2,"},{"Start":"09:57.385 ","End":"10:02.870","Text":"between minus 2 and minus 1."},{"Start":"10:04.770 ","End":"10:08.860","Text":"I\u0027m going to use a well-known trick here."},{"Start":"10:08.860 ","End":"10:10.780","Text":"When I have a negative number,"},{"Start":"10:10.780 ","End":"10:13.510","Text":"I can throw out the negative and reverse the order"},{"Start":"10:13.510 ","End":"10:16.330","Text":"of the limits of integration, so that\u0027s what I\u0027m going to do."},{"Start":"10:16.330 ","End":"10:20.860","Text":"Throw out this negative and then switch the 1 and the 2"},{"Start":"10:20.860 ","End":"10:28.760","Text":"around so that I have the second,"},{"Start":"10:30.000 ","End":"10:35.680","Text":"the 2 on the top and the 1 on the bottom."},{"Start":"10:35.680 ","End":"10:38.950","Text":"Now I\u0027m going to expand,"},{"Start":"10:38.950 ","End":"10:42.790","Text":"so let\u0027s do the 2 first."},{"Start":"10:42.790 ","End":"10:45.190","Text":"Let me leave a big 2 outside,"},{"Start":"10:45.190 ","End":"10:48.880","Text":"and then now I\u0027m going to substitute negative 2 here."},{"Start":"10:48.880 ","End":"10:54.505","Text":"I\u0027m going to get alternating signs, minus-plus, minus-plus."},{"Start":"10:54.505 ","End":"11:00.760","Text":"So 2^5 over 5 is 32 over 5."},{"Start":"11:00.760 ","End":"11:02.019","Text":"That\u0027s with a minus."},{"Start":"11:02.019 ","End":"11:04.460","Text":"Then I\u0027m going to get a plus."},{"Start":"11:05.460 ","End":"11:16.780","Text":"2^4 is 16 over 4 is 4 times 6 is 24."},{"Start":"11:16.780 ","End":"11:19.195","Text":"Then I\u0027m going to get a minus."},{"Start":"11:19.195 ","End":"11:22.585","Text":"Well, 12 over 3 already is 4,"},{"Start":"11:22.585 ","End":"11:27.970","Text":"4 times 2 cubed is 8,"},{"Start":"11:27.970 ","End":"11:31.540","Text":"that\u0027s 32 and we already said it\u0027s minus."},{"Start":"11:31.540 ","End":"11:34.795","Text":"Here we\u0027re going to get a plus."},{"Start":"11:34.795 ","End":"11:41.260","Text":"2 squared over 2 is just 2 times 7 is 14."},{"Start":"11:41.260 ","End":"11:43.975","Text":"That\u0027s the minus 2 part,"},{"Start":"11:43.975 ","End":"11:46.135","Text":"and now the minus 1 part."},{"Start":"11:46.135 ","End":"11:48.430","Text":"Again, I\u0027m going to get alternating sign,"},{"Start":"11:48.430 ","End":"11:50.020","Text":"so this time it\u0027s easier."},{"Start":"11:50.020 ","End":"12:00.520","Text":"It\u0027s minus 1/5 plus 6/4 minus 12 over 3 plus 7 over 2."},{"Start":"12:03.540 ","End":"12:07.585","Text":"Now let\u0027s see what cancels here."},{"Start":"12:07.585 ","End":"12:13.725","Text":"I just noticed it\u0027s supposed to be 14 here, 7 times 2."},{"Start":"12:13.725 ","End":"12:22.860","Text":"This bit, the negatives that we have are 30, the whole numbers,"},{"Start":"12:22.860 ","End":"12:28.095","Text":"24 and 14 is 38 minus 32 is 66,"},{"Start":"12:28.095 ","End":"12:32.530","Text":"and this thing is 5."},{"Start":"12:32.910 ","End":"12:37.135","Text":"Sorry, this is 6 and 2/5,"},{"Start":"12:37.135 ","End":"12:43.285","Text":"so 6 minus 6 of 2/5 is just minus 2/5,"},{"Start":"12:43.285 ","End":"12:47.200","Text":"so this first part comes out to be minus 2/5."},{"Start":"12:47.200 ","End":"12:51.310","Text":"The second part comes out to be,"},{"Start":"12:51.310 ","End":"12:54.640","Text":"well, actually it would be easier if I cancel a bit first."},{"Start":"12:54.640 ","End":"12:57.670","Text":"12 over 3 is 4,"},{"Start":"12:57.670 ","End":"13:04.300","Text":"6 over 4 is like 3 over 2. Let\u0027s see."},{"Start":"13:04.300 ","End":"13:10.795","Text":"I get 3 over 2 and 7 over 2 is altogether 10 over 2, which is 5."},{"Start":"13:10.795 ","End":"13:13.840","Text":"5 minus 4 is 1,"},{"Start":"13:13.840 ","End":"13:19.225","Text":"1 minus 1/5 is 4/5."},{"Start":"13:19.225 ","End":"13:22.810","Text":"What I get here is 4/5,"},{"Start":"13:22.810 ","End":"13:27.880","Text":"so I get this, minus this,"},{"Start":"13:27.880 ","End":"13:35.530","Text":"and altogether 2 and so what I get is"},{"Start":"13:35.530 ","End":"13:44.710","Text":"minus 2/5 minus 4/5 is minus 6/5 times 2 is minus 12/5."},{"Start":"13:44.710 ","End":"13:49.930","Text":"I said that we were going to get a minus probably and we\u0027d have to reverse it,"},{"Start":"13:49.930 ","End":"13:55.735","Text":"so let\u0027s just say this is equal 2 but because we\u0027re on the left of the axis,"},{"Start":"13:55.735 ","End":"13:59.065","Text":"it should be 12 over 5,"},{"Start":"13:59.065 ","End":"14:06.625","Text":"which is 2 and 2/5 or 2.4."},{"Start":"14:06.625 ","End":"14:10.160","Text":"Somewhere or other, I lost the Pi."},{"Start":"14:10.410 ","End":"14:12.800","Text":"Yes, there it is."},{"Start":"14:12.800 ","End":"14:16.515","Text":"The Pi, the Pi, the Pi,"},{"Start":"14:16.515 ","End":"14:24.490","Text":"Pi, Pi, Pi, and Pi, and I\u0027ll highlight."},{"Start":"14:24.490 ","End":"14:28.580","Text":"I\u0027ll take the decimal answer 2.4Pi."},{"Start":"14:29.880 ","End":"14:32.650","Text":"I think we\u0027re done."},{"Start":"14:32.650 ","End":"14:34.840","Text":"Let me also highlight this one."},{"Start":"14:34.840 ","End":"14:38.720","Text":"You know what, we are done."}],"ID":4730},{"Watched":false,"Name":"Exercise 4","Duration":"5m 55s","ChapterTopicVideoID":4718,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"Here we have another problem involving a solid of revolution."},{"Start":"00:04.680 ","End":"00:09.450","Text":"This time, we have the graph of y equals sine of x squared,"},{"Start":"00:09.450 ","End":"00:12.640","Text":"and this will be my f of x."},{"Start":"00:13.280 ","End":"00:17.435","Text":"The lines, our vertical lines,"},{"Start":"00:17.435 ","End":"00:22.710","Text":"this is x equals square root of Pi over 6,"},{"Start":"00:22.710 ","End":"00:31.475","Text":"and this is where x equals the square root of Pi over 3 and y equals 0 is,"},{"Start":"00:31.475 ","End":"00:35.515","Text":"in effect this is my g of x, the lower function."},{"Start":"00:35.515 ","End":"00:38.700","Text":"The areas already shaded."},{"Start":"00:38.700 ","End":"00:43.669","Text":"All we have to do is to apply the formula."},{"Start":"00:43.669 ","End":"00:47.945","Text":"I copied it from the tutorial on this topic."},{"Start":"00:47.945 ","End":"00:54.020","Text":"When we have y in terms of x and we revolve around the y-axis,"},{"Start":"00:54.020 ","End":"00:58.010","Text":"this is a cylindrical shell formula that we\u0027re going to use."},{"Start":"00:58.010 ","End":"01:01.075","Text":"Because g of x is 0,"},{"Start":"01:01.075 ","End":"01:06.025","Text":"so to simplify here by removing the g of x,"},{"Start":"01:06.025 ","End":"01:12.770","Text":"and I\u0027ll just put the dx a bit closer here."},{"Start":"01:13.640 ","End":"01:15.300","Text":"Let\u0027s see."},{"Start":"01:15.300 ","End":"01:20.445","Text":"What we get is that V equals 2 Pi times the integral,"},{"Start":"01:20.445 ","End":"01:24.465","Text":"a is root Pi over 6,"},{"Start":"01:24.465 ","End":"01:28.930","Text":"b is root Pi over 3."},{"Start":"01:30.210 ","End":"01:40.420","Text":"X is just x and f of x is sine of x squared and dx."},{"Start":"01:40.760 ","End":"01:43.090","Text":"Now we have an integral to do."},{"Start":"01:43.090 ","End":"01:46.945","Text":"First of all, let\u0027s see how we find the indefinite integral of this."},{"Start":"01:46.945 ","End":"01:51.865","Text":"I can\u0027t help noticing that the derivative of x squared,"},{"Start":"01:51.865 ","End":"01:55.225","Text":"which is 2x, is almost what I have here."},{"Start":"01:55.225 ","End":"02:01.210","Text":"In fact, I could even make it 2x if I borrow the 2 from here and here and put it here."},{"Start":"02:01.210 ","End":"02:06.234","Text":"I\u0027m saying this because there is 1 of those series of formulas which says,"},{"Start":"02:06.234 ","End":"02:11.365","Text":"the indefinite integral of sine of"},{"Start":"02:11.365 ","End":"02:17.800","Text":"some function of x times the derivative of that function,"},{"Start":"02:17.800 ","End":"02:22.800","Text":"dx is equal to"},{"Start":"02:22.800 ","End":"02:30.895","Text":"minus the cosine of that function plus the constant of integration."},{"Start":"02:30.895 ","End":"02:35.210","Text":"Getting back to what we have here,"},{"Start":"02:37.970 ","End":"02:41.355","Text":"we get that v is equal 2."},{"Start":"02:41.355 ","End":"02:45.735","Text":"Now I\u0027m putting this 2 here,"},{"Start":"02:45.735 ","End":"02:49.955","Text":"so I have 2x here and Pi outside."},{"Start":"02:49.955 ","End":"02:54.265","Text":"So I get Pi times,"},{"Start":"02:54.265 ","End":"02:56.900","Text":"and now using this formula,"},{"Start":"02:56.900 ","End":"03:03.930","Text":"I have minus cosine sine of x squared."},{"Start":"03:05.080 ","End":"03:09.990","Text":"I don\u0027t need the plus C with a definite integral."},{"Start":"03:10.030 ","End":"03:16.590","Text":"Taking between the limits of the top it\u0027s root Pi over 3,"},{"Start":"03:16.590 ","End":"03:21.845","Text":"and here it\u0027s root Pi over 6."},{"Start":"03:21.845 ","End":"03:24.320","Text":"I\u0027ll just briefly go over the last bit again."},{"Start":"03:24.320 ","End":"03:29.060","Text":"I took the 2 from outside the integral sign and put it inside."},{"Start":"03:29.060 ","End":"03:30.800","Text":"Then in the purple formula,"},{"Start":"03:30.800 ","End":"03:38.285","Text":"I let f of x equals x squared and the f prime in the formula is just equal to 2x."},{"Start":"03:38.285 ","End":"03:41.820","Text":"Then I substituted, as I explained before."},{"Start":"03:43.750 ","End":"03:52.835","Text":"I\u0027m going to use 1 of the tricks that I like to use is that when I have a negative,"},{"Start":"03:52.835 ","End":"03:54.740","Text":"instead of dealing with all the negatives,"},{"Start":"03:54.740 ","End":"03:56.360","Text":"I can just reverse the order of"},{"Start":"03:56.360 ","End":"03:59.840","Text":"the limits that are subtracting this from this, I subtract this from this,"},{"Start":"03:59.840 ","End":"04:08.915","Text":"and then I get that this is Pi times cosine of x squared without the negative."},{"Start":"04:08.915 ","End":"04:17.280","Text":"But reversing here, root Pi over 6 and here root of Pi over 3."},{"Start":"04:18.590 ","End":"04:23.520","Text":"Now I\u0027m going to substitute and get Pi times,"},{"Start":"04:23.520 ","End":"04:25.700","Text":"I\u0027ll put a big square brackets or first of all,"},{"Start":"04:25.700 ","End":"04:28.925","Text":"substitute the root Pi over 6."},{"Start":"04:28.925 ","End":"04:33.230","Text":"Root Pi over 6 squared is just Pi over 6,"},{"Start":"04:33.230 ","End":"04:38.655","Text":"so I get cosine of Pi over 6 minus,"},{"Start":"04:38.655 ","End":"04:42.330","Text":"the second term is root Pi over 3 all squared,"},{"Start":"04:42.330 ","End":"04:43.905","Text":"that\u0027s just Pi over 3."},{"Start":"04:43.905 ","End":"04:48.580","Text":"So cosine of Pi over 3."},{"Start":"04:49.610 ","End":"04:55.265","Text":"Now, all we have to do is to compute the cosines."},{"Start":"04:55.265 ","End":"04:59.285","Text":"Now Pi over 6 is 180 over 6,"},{"Start":"04:59.285 ","End":"05:04.115","Text":"which is 30 degrees because I like doing cosine in degrees,"},{"Start":"05:04.115 ","End":"05:11.210","Text":"and Pi over 3 is 180 over 3 is 60 degrees."},{"Start":"05:11.210 ","End":"05:20.095","Text":"What I get is the cosine of 30 degrees is the square root of 3 over 2,"},{"Start":"05:20.095 ","End":"05:22.970","Text":"and the cosine of 60 degrees is a 1/2."},{"Start":"05:22.970 ","End":"05:28.865","Text":"There are some standard tables of cosines for well-known angles."},{"Start":"05:28.865 ","End":"05:31.280","Text":"This is actually our answer."},{"Start":"05:31.280 ","End":"05:32.975","Text":"I could simplify it."},{"Start":"05:32.975 ","End":"05:36.980","Text":"I suppose if I say that this is equal to"},{"Start":"05:36.980 ","End":"05:42.440","Text":"maybe take the 2 out and square root of 3 minus 1."},{"Start":"05:42.440 ","End":"05:43.970","Text":"I don\u0027t know if that\u0027s much simpler."},{"Start":"05:43.970 ","End":"05:49.085","Text":"In any event, this is going to be my answer that"},{"Start":"05:49.085 ","End":"05:56.040","Text":"the volume is this and we are done."}],"ID":4731},{"Watched":false,"Name":"Exercise 5","Duration":"4m 36s","ChapterTopicVideoID":4719,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"Here we have another 1 of those solid of revolution problems."},{"Start":"00:04.200 ","End":"00:12.510","Text":"This time, this is the area and the revolution is around the y-axis."},{"Start":"00:12.510 ","End":"00:19.380","Text":"What we\u0027re given is the function which is y equals,"},{"Start":"00:19.380 ","End":"00:22.019","Text":"let\u0027s call it f of x,"},{"Start":"00:22.019 ","End":"00:26.715","Text":"which is e to the power of x squared."},{"Start":"00:26.715 ","End":"00:33.210","Text":"We have f, and this is x equals 0 when x equals 1."},{"Start":"00:33.210 ","End":"00:35.565","Text":"This is 0, this is 1,"},{"Start":"00:35.565 ","End":"00:40.320","Text":"and y equals 0 is the x-axis."},{"Start":"00:40.320 ","End":"00:44.500","Text":"When we have an area between a function on the x-axis,"},{"Start":"00:44.500 ","End":"00:47.060","Text":"there is a simplified formula."},{"Start":"00:47.060 ","End":"00:51.035","Text":"First thing I want to say is which formula we\u0027re going to use."},{"Start":"00:51.035 ","End":"00:55.895","Text":"We have y in terms of x and we revolve around the y-axis."},{"Start":"00:55.895 ","End":"00:59.755","Text":"This is 1 of the cylindrical shell formulas."},{"Start":"00:59.755 ","End":"01:04.620","Text":"What it\u0027s going to equal is the volume, you can write it down,"},{"Start":"01:04.620 ","End":"01:11.260","Text":"the volume is equal to Pi times the integral."},{"Start":"01:12.170 ","End":"01:16.035","Text":"I meant 2Pi, that\u0027s what happen when you do it from memory."},{"Start":"01:16.035 ","End":"01:20.295","Text":"2Pi times the integral from a to b, this is the general."},{"Start":"01:20.295 ","End":"01:22.530","Text":"When we only have 1 function,"},{"Start":"01:22.530 ","End":"01:25.215","Text":"then it\u0027s f of x,"},{"Start":"01:25.215 ","End":"01:31.260","Text":"but we have to multiply that by x and dx."},{"Start":"01:32.170 ","End":"01:36.795","Text":"In our case, a and b are 0 and 1,"},{"Start":"01:36.795 ","End":"01:38.700","Text":"f of x is this."},{"Start":"01:38.700 ","End":"01:44.600","Text":"Let\u0027s just go ahead and substitute that in a different color."},{"Start":"01:44.600 ","End":"01:55.475","Text":"We\u0027ll take V as equal to 2Pi times the integral from 0 to 1 x,"},{"Start":"01:55.475 ","End":"01:59.554","Text":"f of x is this thing here."},{"Start":"01:59.554 ","End":"02:01.235","Text":"That\u0027s my f of x,"},{"Start":"02:01.235 ","End":"02:07.340","Text":"e to the x squared and dx."},{"Start":"02:07.340 ","End":"02:12.725","Text":"Now we want to know about the indefinite integral of this."},{"Start":"02:12.725 ","End":"02:17.285","Text":"What would help me is I\u0027m going to use 1 of these formulas."},{"Start":"02:17.285 ","End":"02:19.385","Text":"If I had a 2x here,"},{"Start":"02:19.385 ","End":"02:22.370","Text":"that would be the derivative of x squared."},{"Start":"02:22.370 ","End":"02:24.740","Text":"I\u0027ll explain where I\u0027m going with this."},{"Start":"02:24.740 ","End":"02:31.580","Text":"There\u0027s a general formula that the integral of e to the power of some function of"},{"Start":"02:31.580 ","End":"02:35.720","Text":"x times f prime dx is"},{"Start":"02:35.720 ","End":"02:40.825","Text":"just e to the power of f plus the constant of integration, of course."},{"Start":"02:40.825 ","End":"02:45.090","Text":"Slight problem, the letter f is already taken,"},{"Start":"02:45.090 ","End":"02:47.755","Text":"we\u0027ve used it for our function and we\u0027ve used it here."},{"Start":"02:47.755 ","End":"02:49.510","Text":"I got to change this to another letter,"},{"Start":"02:49.510 ","End":"02:53.000","Text":"maybe I\u0027ll change it to g. Just hang on."},{"Start":"02:53.000 ","End":"02:57.519","Text":"I\u0027ve replaced f by g. In our case,"},{"Start":"02:57.519 ","End":"03:03.519","Text":"what we have is that the g of x here is going to be x squared,"},{"Start":"03:03.519 ","End":"03:07.125","Text":"and g prime of x is equal to 2x."},{"Start":"03:07.125 ","End":"03:10.115","Text":"That\u0027s what I was saying it\u0027s a pity that it\u0027s x but not 2x,"},{"Start":"03:10.115 ","End":"03:12.760","Text":"but that\u0027s not really a problem at all because I can throw this 2"},{"Start":"03:12.760 ","End":"03:17.020","Text":"inside the integral and it will go with the 2x."},{"Start":"03:17.020 ","End":"03:20.560","Text":"What I\u0027ll get if I put the 2 inside and also I switch"},{"Start":"03:20.560 ","End":"03:23.585","Text":"the order of the 2x and e to the x squared."},{"Start":"03:23.585 ","End":"03:30.855","Text":"What I get is Pi times the integral from 0 to 1."},{"Start":"03:30.855 ","End":"03:34.069","Text":"Now, the 2 is going inside,"},{"Start":"03:34.069 ","End":"03:41.925","Text":"but I\u0027m also putting the 2x after the e to the x squared dx."},{"Start":"03:41.925 ","End":"03:46.640","Text":"Now I can really use this formula because it just really fits this with x"},{"Start":"03:46.640 ","End":"03:53.730","Text":"squared being g. I get that this is equal to Pi times,"},{"Start":"03:53.730 ","End":"03:55.085","Text":"now the indefinite integral."},{"Start":"03:55.085 ","End":"03:59.420","Text":"I don\u0027t need the c for definite integral but I need d to the g,"},{"Start":"03:59.420 ","End":"04:02.495","Text":"which is e to the x squared."},{"Start":"04:02.495 ","End":"04:06.544","Text":"I just take this between"},{"Start":"04:06.544 ","End":"04:13.994","Text":"0 and 1 and that will give me Pi times."},{"Start":"04:13.994 ","End":"04:17.085","Text":"Now I plug in 1, plug in 0 and subtract, I put in 1,"},{"Start":"04:17.085 ","End":"04:20.670","Text":"e to the 1 squared e to the 1 is e. Put in 0,"},{"Start":"04:20.670 ","End":"04:25.230","Text":"0 squared is 0, e^0 is 1, and I subtract."},{"Start":"04:25.230 ","End":"04:30.245","Text":"The volume is Pi times e minus 1."},{"Start":"04:30.245 ","End":"04:35.400","Text":"Actually, that\u0027s the answer. We are done."}],"ID":4732},{"Watched":false,"Name":"Exercise 6","Duration":"17m 30s","ChapterTopicVideoID":4720,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.229","Text":"Here we have yet another solid of revolution problem."},{"Start":"00:04.229 ","End":"00:13.020","Text":"This time it\u0027s this thin-shaded area that is going to be revolved around the x-axis."},{"Start":"00:13.020 ","End":"00:16.620","Text":"The general strategy will be,"},{"Start":"00:16.620 ","End":"00:18.300","Text":"just like with areas,"},{"Start":"00:18.300 ","End":"00:20.955","Text":"we use the concept of subtraction."},{"Start":"00:20.955 ","End":"00:23.430","Text":"We\u0027re going to find the equation of the tangent."},{"Start":"00:23.430 ","End":"00:26.850","Text":"We\u0027re going to find this point and this point and this point."},{"Start":"00:26.850 ","End":"00:33.980","Text":"But ultimately, we don\u0027t have the regular setup for the difference between 2 functions"},{"Start":"00:33.980 ","End":"00:35.530","Text":"that is revolved because"},{"Start":"00:35.530 ","End":"00:40.880","Text":"the curved part starts from here and the straight line part starts from here,"},{"Start":"00:40.880 ","End":"00:42.200","Text":"although they both end here."},{"Start":"00:42.200 ","End":"00:46.730","Text":"What we\u0027re actually going to do is drop a perpendicular"},{"Start":"00:46.730 ","End":"00:49.490","Text":"here and we\u0027re going to"},{"Start":"00:49.490 ","End":"00:52.880","Text":"compute the volume of revolution as the difference between 2 volumes."},{"Start":"00:52.880 ","End":"00:55.670","Text":"There\u0027s going to be an outer volume and then there\u0027s going to"},{"Start":"00:55.670 ","End":"00:58.850","Text":"be the volume of revolving this triangle."},{"Start":"00:58.850 ","End":"01:00.425","Text":"Let me shade the triangle."},{"Start":"01:00.425 ","End":"01:03.170","Text":"I think it\u0027s best if we label the points,"},{"Start":"01:03.170 ","End":"01:06.665","Text":"then we can a bit more easily refer to different shapes."},{"Start":"01:06.665 ","End":"01:08.080","Text":"Let\u0027s say this is A,"},{"Start":"01:08.080 ","End":"01:10.510","Text":"this is B, this is C,"},{"Start":"01:10.510 ","End":"01:18.984","Text":"and this is D. What I\u0027m saying is that the volume that we want,"},{"Start":"01:18.984 ","End":"01:23.290","Text":"volume of revolution, let\u0027s call it VR,"},{"Start":"01:23.290 ","End":"01:33.625","Text":"of area ACD is equal to the volume of revolution"},{"Start":"01:33.625 ","End":"01:43.775","Text":"of the area ABC minus"},{"Start":"01:43.775 ","End":"01:45.475","Text":"the volume of revolution"},{"Start":"01:45.475 ","End":"01:47.450","Text":"of the triangle"},{"Start":"01:47.450 ","End":"01:57.010","Text":"ABD."},{"Start":"01:57.010 ","End":"02:03.830","Text":"I\u0027ll just better put an asterisk because it\u0027s not a standard abbreviation"},{"Start":"02:03.830 ","End":"02:11.915","Text":"that VR is volume of revolution find."},{"Start":"02:11.915 ","End":"02:14.200","Text":"That\u0027s the general strategy."},{"Start":"02:14.200 ","End":"02:20.005","Text":"Now let\u0027s introduce some details into here."},{"Start":"02:20.005 ","End":"02:23.730","Text":"What we have is a function."},{"Start":"02:23.730 ","End":"02:26.000","Text":"Well, they\u0027re very close together,"},{"Start":"02:26.000 ","End":"02:31.310","Text":"but let\u0027s say this curved part, a function of x,"},{"Start":"02:31.310 ","End":"02:40.615","Text":"is equal to e^x squared and it\u0027s also y."},{"Start":"02:40.615 ","End":"02:44.195","Text":"The tangent line we haven\u0027t found yet,"},{"Start":"02:44.195 ","End":"02:49.720","Text":"but I\u0027ll just call it for the moment, tangent."},{"Start":"02:49.720 ","End":"02:52.265","Text":"We\u0027ll find out what it is."},{"Start":"02:52.265 ","End":"02:56.750","Text":"Then we have this point A. This we\u0027re given."},{"Start":"02:56.750 ","End":"02:59.150","Text":"This actually is our point A,"},{"Start":"02:59.150 ","End":"03:02.980","Text":"so this is e, e."},{"Start":"03:02.980 ","End":"03:09.560","Text":"I just noticed that I wrote down the wrong function in this exercise."},{"Start":"03:09.560 ","End":"03:12.395","Text":"It\u0027s not too late because we haven\u0027t used it yet,"},{"Start":"03:12.395 ","End":"03:13.910","Text":"but it\u0027s not e^x squared,"},{"Start":"03:13.910 ","End":"03:17.195","Text":"it\u0027s x natural log of x. I\u0027ve got to alter it here."},{"Start":"03:17.195 ","End":"03:21.230","Text":"The reason I stumbled onto it is because I realized that the point e,"},{"Start":"03:21.230 ","End":"03:22.955","Text":"e wasn\u0027t on the graph."},{"Start":"03:22.955 ","End":"03:26.345","Text":"Now it is okay because if x is e,"},{"Start":"03:26.345 ","End":"03:31.020","Text":"then e times natural log of e is e times 1,"},{"Start":"03:31.020 ","End":"03:34.840","Text":"which is e, so we\u0027re all okay."},{"Start":"03:35.210 ","End":"03:38.485","Text":"I\u0027ll just continue with some of the labeling."},{"Start":"03:38.485 ","End":"03:40.610","Text":"As I said, A is the point e,"},{"Start":"03:40.610 ","End":"03:43.445","Text":"e. B is directly below it,"},{"Start":"03:43.445 ","End":"03:48.910","Text":"so this would be the point e, 0."},{"Start":"03:48.910 ","End":"03:59.150","Text":"C is on the graph where y equals 0. x natural log of x is 0."},{"Start":"03:59.150 ","End":"04:01.760","Text":"I can do this at the side."},{"Start":"04:01.760 ","End":"04:05.750","Text":"x natural log of x equals 0."},{"Start":"04:05.750 ","End":"04:10.310","Text":"The function is not defined when x is 0 because natural log of 0 is not defined."},{"Start":"04:10.310 ","End":"04:12.260","Text":"If x is not 0,"},{"Start":"04:12.260 ","End":"04:18.900","Text":"then natural log of x is 0. x theoretically could have been 0 or this could have been 0,"},{"Start":"04:18.900 ","End":"04:20.120","Text":"1 or the other."},{"Start":"04:20.120 ","End":"04:23.210","Text":"This case has been ruled out,"},{"Start":"04:23.210 ","End":"04:30.960","Text":"so natural log of x is 0 and the number whose natural log is 0 is 1."},{"Start":"04:30.960 ","End":"04:34.995","Text":"So we get that x equals 1 or you can just take e^0."},{"Start":"04:34.995 ","End":"04:40.745","Text":"If x is 1, that means that C is the point 1, 0."},{"Start":"04:40.745 ","End":"04:43.440","Text":"I don\u0027t have room here."},{"Start":"04:43.440 ","End":"04:47.145","Text":"I\u0027ll write it on the left, 1, 0."},{"Start":"04:47.145 ","End":"04:49.890","Text":"We have A, we have B,"},{"Start":"04:49.890 ","End":"04:53.655","Text":"we have C. D we don\u0027t have."},{"Start":"04:53.655 ","End":"04:58.840","Text":"We know that D is the point something,"},{"Start":"04:58.840 ","End":"05:04.235","Text":"0 and it\u0027s this something that we need to find out."},{"Start":"05:04.235 ","End":"05:09.605","Text":"But first of all, we\u0027ve got to find the equation of the tangent."},{"Start":"05:09.605 ","End":"05:16.315","Text":"We\u0027ll use the standard equation of the tangent to a curve at a point,"},{"Start":"05:16.315 ","End":"05:25.025","Text":"is y minus y_1 equals f prime of x_1 times x minus x_1,"},{"Start":"05:25.025 ","End":"05:29.285","Text":"where x_1, y_1 is the point of contact."},{"Start":"05:29.285 ","End":"05:33.280","Text":"In our case, x_1 and y_1 are both e,"},{"Start":"05:33.280 ","End":"05:40.085","Text":"but we need f prime of e. Let\u0027s do some differentiation and substitution."},{"Start":"05:40.085 ","End":"05:43.460","Text":"Since f of x is x natural log of x,"},{"Start":"05:43.460 ","End":"05:44.885","Text":"I just copied it,"},{"Start":"05:44.885 ","End":"05:50.770","Text":"f prime of x by the product rule is derivative of this."},{"Start":"05:50.770 ","End":"05:57.175","Text":"1 times this underived plus vice versa, this as is,"},{"Start":"05:57.175 ","End":"06:04.820","Text":"natural log derivative and this is equal to natural log of x plus 1."},{"Start":"06:04.820 ","End":"06:08.015","Text":"So f prime of e,"},{"Start":"06:08.015 ","End":"06:09.784","Text":"which is what we want,"},{"Start":"06:09.784 ","End":"06:16.640","Text":"is equal to, just put in x equals e natural log of e plus 1."},{"Start":"06:16.640 ","End":"06:19.400","Text":"Natural log of e is 1, 1 plus 1,"},{"Start":"06:19.400 ","End":"06:23.335","Text":"let me do it on my calculator, comes out 2."},{"Start":"06:23.335 ","End":"06:25.410","Text":"Now I have all the quantities."},{"Start":"06:25.410 ","End":"06:28.890","Text":"I have y_1, I have f prime of x_1, and I have x_1."},{"Start":"06:28.890 ","End":"06:32.059","Text":"Let\u0027s put all the quantities we know into here."},{"Start":"06:32.059 ","End":"06:40.130","Text":"Now we have y minus e is equal to f prime of x_1,"},{"Start":"06:40.130 ","End":"06:49.890","Text":"is 2, times x minus x_1 is e. This is the equation of the tangent,"},{"Start":"06:49.890 ","End":"06:52.295","Text":"but I\u0027d like to just simplify it."},{"Start":"06:52.295 ","End":"06:55.070","Text":"Let\u0027s keep y on 1 side."},{"Start":"06:55.070 ","End":"07:01.740","Text":"y equals to 2x minus 2e,"},{"Start":"07:01.740 ","End":"07:07.790","Text":"but plus e, so y equals 2x minus e. That\u0027s an intermediate result."},{"Start":"07:07.790 ","End":"07:10.010","Text":"I\u0027ll just make a note of that."},{"Start":"07:10.010 ","End":"07:13.020","Text":"That is our tangent."},{"Start":"07:13.030 ","End":"07:16.055","Text":"We almost have everything."},{"Start":"07:16.055 ","End":"07:19.775","Text":"We still have this little question mark here."},{"Start":"07:19.775 ","End":"07:25.970","Text":"The question mark here is where the tangent cuts the x-axis,"},{"Start":"07:25.970 ","End":"07:28.870","Text":"so I just have to substitute in the tangent."},{"Start":"07:28.870 ","End":"07:30.704","Text":"y equals 0."},{"Start":"07:30.704 ","End":"07:35.955","Text":"This will give me the 2x minus e equals 0."},{"Start":"07:35.955 ","End":"07:41.280","Text":"So x will be equal to e over 2."},{"Start":"07:41.280 ","End":"07:47.140","Text":"Finally, I can replace the question mark by e over 2."},{"Start":"07:47.140 ","End":"07:49.650","Text":"Good, we\u0027re progressing well."},{"Start":"07:49.650 ","End":"07:54.680","Text":"The next thing we\u0027ll need is the formula for volume of"},{"Start":"07:54.680 ","End":"08:01.890","Text":"revolution about the x-axis when y is given in terms of x."},{"Start":"08:03.910 ","End":"08:10.460","Text":"This is the formula for when we only have 1 function and the x-axis is the other."},{"Start":"08:10.460 ","End":"08:16.159","Text":"Perhaps I should not use the letter f because it\u0027s already taken."},{"Start":"08:16.159 ","End":"08:23.080","Text":"Let me just replace f with h. Let\u0027s go with the ABC,"},{"Start":"08:23.080 ","End":"08:25.250","Text":"which is the big area."},{"Start":"08:25.250 ","End":"08:32.810","Text":"That\u0027s exactly what\u0027s under the f of x from C to B,"},{"Start":"08:32.810 ","End":"08:42.185","Text":"which is from 1-e. V of the area ABC,"},{"Start":"08:42.185 ","End":"08:45.460","Text":"I\u0027ll just call it V_ABC."},{"Start":"08:45.460 ","End":"08:48.200","Text":"ABC is this thing,"},{"Start":"08:48.200 ","End":"08:56.060","Text":"which is our function f between 1 and e. This is equal to"},{"Start":"08:56.060 ","End":"09:01.310","Text":"Pi times the integral from x equals 1 to x"},{"Start":"09:01.310 ","End":"09:08.190","Text":"equals e of our function x natural log of x."},{"Start":"09:10.380 ","End":"09:16.935","Text":"All this squared dx."},{"Start":"09:16.935 ","End":"09:20.910","Text":"Now it\u0027s just a mechanical computation."},{"Start":"09:20.910 ","End":"09:23.430","Text":"Now I don\u0027t want to interrupt the main flow with"},{"Start":"09:23.430 ","End":"09:25.980","Text":"the computation of the indefinite integral."},{"Start":"09:25.980 ","End":"09:28.980","Text":"Let me just write it at the side."},{"Start":"09:28.980 ","End":"09:32.790","Text":"I\u0027ll write the solution and then at the end I\u0027ll justify it."},{"Start":"09:32.790 ","End":"09:38.025","Text":"The indefinite integral of x,"},{"Start":"09:38.025 ","End":"09:47.785","Text":"natural log of x squared dx is equal to this nasty expression,"},{"Start":"09:47.785 ","End":"09:50.350","Text":"and of course, there\u0027s a constant of"},{"Start":"09:50.350 ","End":"09:54.925","Text":"integration which we won\u0027t need because we\u0027re going to use definite integrals."},{"Start":"09:54.925 ","End":"09:57.265","Text":"This looks a bit of a mess."},{"Start":"09:57.265 ","End":"10:01.870","Text":"I\u0027ll simplify it a bit by taking x cubed over 27 outside the brackets,"},{"Start":"10:01.870 ","End":"10:03.655","Text":"and this is what we\u0027ll get."},{"Start":"10:03.655 ","End":"10:05.770","Text":"Now, this is the indefinite integral."},{"Start":"10:05.770 ","End":"10:07.390","Text":"Remember it came from here,"},{"Start":"10:07.390 ","End":"10:08.845","Text":"from a definite integral."},{"Start":"10:08.845 ","End":"10:11.890","Text":"We\u0027re going to have to evaluate this with"},{"Start":"10:11.890 ","End":"10:17.180","Text":"the Pi and between 1 and e. Let\u0027s see what we get."},{"Start":"10:21.990 ","End":"10:26.830","Text":"I could take the 27 outside,"},{"Start":"10:26.830 ","End":"10:28.930","Text":"so Pi over 27."},{"Start":"10:28.930 ","End":"10:32.140","Text":"Now I\u0027ll need what\u0027s left here,"},{"Start":"10:32.140 ","End":"10:41.410","Text":"which is x cubed times natural log squared of x minus 12,"},{"Start":"10:41.410 ","End":"10:44.590","Text":"natural log of x plus 2,"},{"Start":"10:44.590 ","End":"10:54.655","Text":"no need for the constant between 1 and e. Let\u0027s see what we get if we can simplify this."},{"Start":"10:54.655 ","End":"11:04.690","Text":"Let\u0027s put in first of all x equals e. The Pi over 27 says,"},{"Start":"11:04.690 ","End":"11:10.480","Text":"when x equals e, we get e cubed times,"},{"Start":"11:10.480 ","End":"11:12.895","Text":"the natural log of e is 1."},{"Start":"11:12.895 ","End":"11:19.795","Text":"So we have 1 squared minus 12 times 1 plus 2."},{"Start":"11:19.795 ","End":"11:22.630","Text":"That\u0027s what happens when x equals e,"},{"Start":"11:22.630 ","End":"11:32.365","Text":"less what happens when x is 1 is we have 1 cubed and the natural log of 1 is 0."},{"Start":"11:32.365 ","End":"11:39.410","Text":"We get 0 minus 0 plus 2."},{"Start":"11:40.770 ","End":"11:44.665","Text":"Let\u0027s see what we get."},{"Start":"11:44.665 ","End":"11:50.060","Text":"This is equal to Pi over 27,"},{"Start":"11:50.520 ","End":"11:56.005","Text":"1 plus 2 minus 12,"},{"Start":"11:56.005 ","End":"12:01.390","Text":"13 minus 1, minus 9e cubed."},{"Start":"12:01.390 ","End":"12:11.545","Text":"Over here we have minus 1 times 2 is just minus 2."},{"Start":"12:11.545 ","End":"12:16.390","Text":"This has come out negative and I\u0027ll just check into that."},{"Start":"12:16.390 ","End":"12:20.020","Text":"I goofed. The 9 here is missing here."},{"Start":"12:20.020 ","End":"12:21.925","Text":"Let me try and fix that."},{"Start":"12:21.925 ","End":"12:26.335","Text":"I need to put this 9 into here, fine."},{"Start":"12:26.335 ","End":"12:28.975","Text":"Now this 9 has to go into here."},{"Start":"12:28.975 ","End":"12:32.035","Text":"Next mistake I discovered I miscopied,"},{"Start":"12:32.035 ","End":"12:35.740","Text":"and this is actually a 2."},{"Start":"12:35.740 ","End":"12:42.340","Text":"This 12 becomes a 6 and the same here,"},{"Start":"12:42.340 ","End":"12:48.325","Text":"this 12 and this 12 both become 6\u0027s,"},{"Start":"12:48.325 ","End":"12:53.560","Text":"so that\u0027s a 6 here and that\u0027s a 6 here."},{"Start":"12:53.560 ","End":"12:56.605","Text":"That makes this into a 5."},{"Start":"12:56.605 ","End":"12:58.585","Text":"That\u0027s the first part."},{"Start":"12:58.585 ","End":"13:03.490","Text":"That\u0027s the volume of the ABC part,"},{"Start":"13:03.490 ","End":"13:06.475","Text":"which is this bit here."},{"Start":"13:06.475 ","End":"13:12.310","Text":"Now we need the volume of revolution of the triangle."},{"Start":"13:12.310 ","End":"13:17.050","Text":"We could do it with integration with the"},{"Start":"13:17.050 ","End":"13:22.180","Text":"solid of revolution of the tangent line about the x-axis."},{"Start":"13:22.180 ","End":"13:23.890","Text":"But just for a change,"},{"Start":"13:23.890 ","End":"13:28.780","Text":"I think I\u0027ll go with the geometry where this volume of"},{"Start":"13:28.780 ","End":"13:35.570","Text":"revolution is actually a cone, a triangle ABD."},{"Start":"13:43.560 ","End":"13:48.145","Text":"Now when I put it on its side,"},{"Start":"13:48.145 ","End":"13:55.130","Text":"what it\u0027s going to look like is something like this."},{"Start":"13:55.950 ","End":"14:04.825","Text":"This is the cone and this bit is e. Look,"},{"Start":"14:04.825 ","End":"14:07.045","Text":"go back to the picture again."},{"Start":"14:07.045 ","End":"14:11.575","Text":"There\u0027s another half to it here."},{"Start":"14:11.575 ","End":"14:17.530","Text":"The base is 2e and the height from e over 2"},{"Start":"14:17.530 ","End":"14:24.235","Text":"to e is a half e. Base is 2e height is e. Let\u0027s get back there."},{"Start":"14:24.235 ","End":"14:28.390","Text":"Base is 2e, which means that this is e and this is"},{"Start":"14:28.390 ","End":"14:33.280","Text":"e and the height is 1 half e and it\u0027s a cone."},{"Start":"14:33.280 ","End":"14:35.770","Text":"Not great with the sketches."},{"Start":"14:35.770 ","End":"14:38.140","Text":"Anyway, it\u0027s a mess, but it\u0027s a cone."},{"Start":"14:38.140 ","End":"14:43.525","Text":"Radius of the base is e,"},{"Start":"14:43.525 ","End":"14:46.675","Text":"the height is e over 2,"},{"Start":"14:46.675 ","End":"14:54.520","Text":"the volume of a cone is 1/3 Pi r squared h,"},{"Start":"14:54.520 ","End":"15:00.460","Text":"which in our case is 1/3 Pi r squared is e squared,"},{"Start":"15:00.460 ","End":"15:07.450","Text":"h is a half e,"},{"Start":"15:07.450 ","End":"15:09.490","Text":"which is just the e over 2."},{"Start":"15:09.490 ","End":"15:11.260","Text":"Yeah, I\u0027ll put it as a half e,"},{"Start":"15:11.260 ","End":"15:16.540","Text":"I don\u0027t know why I wrote it as e times a half. That\u0027s fine."},{"Start":"15:16.540 ","End":"15:22.750","Text":"What we get is Pi e cubed over"},{"Start":"15:22.750 ","End":"15:28.795","Text":"4 or 1/4 Pi e cubed."},{"Start":"15:28.795 ","End":"15:36.520","Text":"What we have is that the volume ABC is this,"},{"Start":"15:36.520 ","End":"15:43.000","Text":"and the volume ABD is this."},{"Start":"15:43.000 ","End":"15:44.875","Text":"Now I have to,"},{"Start":"15:44.875 ","End":"15:46.645","Text":"like it says here,"},{"Start":"15:46.645 ","End":"15:51.220","Text":"take the ABC minus the ABD and that will give us what we want,"},{"Start":"15:51.220 ","End":"15:53.230","Text":"which is the volume of, what was it?"},{"Start":"15:53.230 ","End":"15:56.590","Text":"ACD, which is that shaded bit."},{"Start":"15:56.590 ","End":"16:06.894","Text":"The volume of ACD is equal to this bit here,"},{"Start":"16:06.894 ","End":"16:09.655","Text":"less this bit here."},{"Start":"16:09.655 ","End":"16:13.330","Text":"That just isn\u0027t my day for little mistakes."},{"Start":"16:13.330 ","End":"16:17.650","Text":"I miscopied the third to be a half."},{"Start":"16:17.650 ","End":"16:18.955","Text":"Well, let me fix that."},{"Start":"16:18.955 ","End":"16:27.050","Text":"This is 1/3 and this is going to be 1/6."},{"Start":"16:28.050 ","End":"16:30.655","Text":"I could leave it like this,"},{"Start":"16:30.655 ","End":"16:33.970","Text":"but I\u0027d like to put a common denominator."},{"Start":"16:33.970 ","End":"16:40.420","Text":"The common denominator for 6 and 27 is 54."},{"Start":"16:40.420 ","End":"16:49.270","Text":"Let\u0027s take Pi/54 outside"},{"Start":"16:49.270 ","End":"16:53.290","Text":"and then what we\u0027re left with is this."},{"Start":"16:53.290 ","End":"17:00.025","Text":"I have Pi/54 and"},{"Start":"17:00.025 ","End":"17:07.190","Text":"e cubed minus 4 highlighted."},{"Start":"17:07.620 ","End":"17:14.260","Text":"We took it on trust that the indefinite integral of this is this."},{"Start":"17:14.260 ","End":"17:19.270","Text":"I\u0027ll just leave it as an extra homework exercise because"},{"Start":"17:19.270 ","End":"17:24.445","Text":"we\u0027ve already gone well over time into this clip."},{"Start":"17:24.445 ","End":"17:27.174","Text":"We\u0027ll just take it on faith."},{"Start":"17:27.174 ","End":"17:31.280","Text":"We are done."}],"ID":4733},{"Watched":false,"Name":"Exercise 7","Duration":"6m 21s","ChapterTopicVideoID":4730,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this exercise, we have to state and prove"},{"Start":"00:03.090 ","End":"00:06.735","Text":"the formula for computing the volume of a cylinder."},{"Start":"00:06.735 ","End":"00:10.620","Text":"Well, I\u0027m not great at sketches,"},{"Start":"00:10.620 ","End":"00:14.880","Text":"so I decided to find a picture from the Internet,"},{"Start":"00:14.880 ","End":"00:17.280","Text":"and here it is."},{"Start":"00:17.280 ","End":"00:21.800","Text":"Here\u0027s our cylinder that the only 2 parameters to it,"},{"Start":"00:21.800 ","End":"00:27.020","Text":"we have to know the radius of the base and the height."},{"Start":"00:27.020 ","End":"00:33.890","Text":"In general, there are other cylinder which are not circular."},{"Start":"00:33.890 ","End":"00:37.985","Text":"In general, the volume is the area of the base times height."},{"Start":"00:37.985 ","End":"00:40.610","Text":"Now, in our particular case,"},{"Start":"00:40.610 ","End":"00:42.080","Text":"the base is a circle,"},{"Start":"00:42.080 ","End":"00:48.920","Text":"so the area of a circle we all know is Pi r squared and that\u0027s the area of the base and"},{"Start":"00:48.920 ","End":"00:53.820","Text":"the height is h so we get Pi r squared h."},{"Start":"00:53.820 ","End":"00:57.210","Text":"I think I\u0027ll highlight that,"},{"Start":"00:59.540 ","End":"01:02.400","Text":"where r is this,"},{"Start":"01:02.400 ","End":"01:05.910","Text":"h is that and Pi is the famous constant,"},{"Start":"01:05.910 ","End":"01:08.554","Text":"and so that\u0027s the state part."},{"Start":"01:08.554 ","End":"01:10.400","Text":"Now the proof part."},{"Start":"01:10.400 ","End":"01:16.410","Text":"We\u0027re going to prove it using solids of the revolution."},{"Start":"01:16.850 ","End":"01:21.730","Text":"I\u0027ll start with drawing some axes."},{"Start":"01:21.730 ","End":"01:24.750","Text":"Let\u0027s say that\u0027s the y-axis,"},{"Start":"01:24.750 ","End":"01:27.490","Text":"and here we have the x-axis."},{"Start":"01:27.490 ","End":"01:30.840","Text":"It doesn\u0027t have to be a work of art."},{"Start":"01:31.160 ","End":"01:35.519","Text":"What I\u0027ll do is I\u0027ll sketch"},{"Start":"01:35.519 ","End":"01:42.650","Text":"the cross-section of the cylinder or more simply,"},{"Start":"01:42.650 ","End":"01:47.850","Text":"I\u0027ll just take the line which is horizontal,"},{"Start":"01:47.850 ","End":"01:49.950","Text":"in other words, y equals r."},{"Start":"01:49.950 ","End":"01:50.870","Text":"It\u0027s a function of x,"},{"Start":"01:50.870 ","End":"01:53.460","Text":"but it\u0027s a constant function."},{"Start":"01:54.100 ","End":"02:02.850","Text":"I\u0027ll take it from x equals 0 to x equals h."},{"Start":"02:02.850 ","End":"02:08.840","Text":"Of course, this is the y-axis and this is the x-axis"},{"Start":"02:08.840 ","End":"02:13.610","Text":"and this is where y equals r."},{"Start":"02:13.610 ","End":"02:19.145","Text":"If I rotate this rectangle,"},{"Start":"02:19.145 ","End":"02:20.795","Text":"this is r area,"},{"Start":"02:20.795 ","End":"02:25.480","Text":"if I rotate this rectangle around the x-axis,"},{"Start":"02:25.480 ","End":"02:30.185","Text":"I will get a solid cylinder."},{"Start":"02:30.185 ","End":"02:33.530","Text":"I\u0027m not great with 3D drawing, so again,"},{"Start":"02:33.530 ","End":"02:37.160","Text":"I went to the Internet and found a nice little picture."},{"Start":"02:37.160 ","End":"02:43.525","Text":"Here\u0027s our picture, I\u0027ll just scroll up a bit and then we can see it better."},{"Start":"02:43.525 ","End":"02:48.490","Text":"Once again, we have r and we have h,"},{"Start":"02:48.490 ","End":"02:50.449","Text":"just like in my drawing,"},{"Start":"02:50.449 ","End":"02:56.630","Text":"we have f of x or y equals r and we have this as h."},{"Start":"02:56.630 ","End":"03:03.425","Text":"If you rotate this then this is what you get and you can see that we get a cylinder."},{"Start":"03:03.425 ","End":"03:05.990","Text":"Now I have to apply the standard formula."},{"Start":"03:05.990 ","End":"03:12.285","Text":"Well, I have to choose between the cylindrical shell method or the disk method,"},{"Start":"03:12.285 ","End":"03:14.940","Text":"and you\u0027d think cylinder is cylindrical shell."},{"Start":"03:14.940 ","End":"03:20.600","Text":"No, here the disk method is the best where we take vertical slices"},{"Start":"03:20.600 ","End":"03:27.019","Text":"and we\u0027re going to slice it just to illustrate that vertically,"},{"Start":"03:27.019 ","End":"03:35.490","Text":"so to speak and each of these is rotated around the x-axis and forms a disk."},{"Start":"03:38.390 ","End":"03:46.490","Text":"I\u0027m using the abbreviated formula when the lower function is just the x-axis,"},{"Start":"03:46.490 ","End":"03:48.410","Text":"this is the upper function,"},{"Start":"03:48.410 ","End":"03:53.085","Text":"which is y equals f of x,"},{"Start":"03:53.085 ","End":"04:00.715","Text":"f of x is my constant function so let me bring the formula."},{"Start":"04:00.715 ","End":"04:08.860","Text":"Here\u0027s the formula for the volume in the disk method due to Cavalieri."},{"Start":"04:08.860 ","End":"04:12.800","Text":"This works when there\u0027s no second function and"},{"Start":"04:12.800 ","End":"04:16.925","Text":"it\u0027s just between the function and the x-axis."},{"Start":"04:16.925 ","End":"04:22.280","Text":"We have everything f of x is r,"},{"Start":"04:22.280 ","End":"04:25.515","Text":"and I\u0027ve marked a is 0,"},{"Start":"04:25.515 ","End":"04:31.015","Text":"b is h, so we just have to substitute."},{"Start":"04:31.015 ","End":"04:37.350","Text":"We get all our volume V is equal to Pi times the"},{"Start":"04:37.350 ","End":"04:41.490","Text":"integral from 0 to h."},{"Start":"04:41.490 ","End":"04:44.850","Text":"F of x is just the constant r,"},{"Start":"04:44.850 ","End":"04:50.130","Text":"so it\u0027s r squared times dx."},{"Start":"04:50.130 ","End":"04:57.365","Text":"I can write this as Pi r squared because r is a constant,"},{"Start":"04:57.365 ","End":"04:59.135","Text":"it doesn\u0027t depend on x,"},{"Start":"04:59.135 ","End":"05:07.460","Text":"Pi r squared times the integral from 0 to h of dx or if you want to think of it as 1dx,"},{"Start":"05:07.460 ","End":"05:11.015","Text":"either way, the integral of this is x."},{"Start":"05:11.015 ","End":"05:13.325","Text":"We get Pi r squared,"},{"Start":"05:13.325 ","End":"05:17.840","Text":"but the x has to be taken between 0 and h,"},{"Start":"05:17.840 ","End":"05:19.100","Text":"it means substitute this,"},{"Start":"05:19.100 ","End":"05:21.205","Text":"substitute that, and subtract."},{"Start":"05:21.205 ","End":"05:24.410","Text":"What I get is if I put in h,"},{"Start":"05:24.410 ","End":"05:26.600","Text":"I get Pi r squared h."},{"Start":"05:26.600 ","End":"05:33.500","Text":"If I put in 0, I get minus Pi r squared 0,"},{"Start":"05:33.500 ","End":"05:36.120","Text":"which is just 0."},{"Start":"05:36.200 ","End":"05:46.620","Text":"Ultimately, what I get is the formula, I\u0027ll highlight it,"},{"Start":"05:46.620 ","End":"05:50.400","Text":"that V which is the volume,"},{"Start":"05:50.400 ","End":"05:59.760","Text":"what I called here is equal to Pi r squared h"},{"Start":"05:59.760 ","End":"06:03.380","Text":"and that is exactly what is in yellow"},{"Start":"06:03.380 ","End":"06:09.245","Text":"here and so we\u0027ve proved it using solids of revolution,"},{"Start":"06:09.245 ","End":"06:12.545","Text":"where our function happened to be particular simple,"},{"Start":"06:12.545 ","End":"06:22.320","Text":"a straight line passing through y equals r and we\u0027re done."}],"ID":4738},{"Watched":false,"Name":"Exercise 8","Duration":"6m 23s","ChapterTopicVideoID":4731,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this exercise, we have to state and prove"},{"Start":"00:02.730 ","End":"00:06.225","Text":"the formula for computing the volume of a ball."},{"Start":"00:06.225 ","End":"00:08.200","Text":"I wrote the word ball deliberately."},{"Start":"00:08.200 ","End":"00:12.465","Text":"It\u0027s more correct because if you write sphere,"},{"Start":"00:12.465 ","End":"00:15.975","Text":"sphere is just the outer shell of a ball."},{"Start":"00:15.975 ","End":"00:18.615","Text":"Ball is with the interior as well."},{"Start":"00:18.615 ","End":"00:22.410","Text":"Anyway, I found a nice picture on the Internet,"},{"Start":"00:22.410 ","End":"00:25.425","Text":"so I\u0027ll bring it up here."},{"Start":"00:25.425 ","End":"00:27.420","Text":"Here\u0027s the picture."},{"Start":"00:27.420 ","End":"00:33.795","Text":"There\u0027s only 1 parameter involved in determining a sphere and that is the radius."},{"Start":"00:33.795 ","End":"00:39.590","Text":"The volume is given by the well-known formula,"},{"Start":"00:39.590 ","End":"00:41.900","Text":"4/3 Pi r cubed."},{"Start":"00:41.900 ","End":"00:45.140","Text":"Now that\u0027s for the stating part."},{"Start":"00:45.140 ","End":"00:46.560","Text":"Now we have to prove it."},{"Start":"00:46.560 ","End":"00:53.605","Text":"We\u0027re going to prove it using definite integrals and solids of revolution."},{"Start":"00:53.605 ","End":"00:57.555","Text":"Well, I\u0027ll start off with a pair of axes."},{"Start":"00:57.555 ","End":"01:06.385","Text":"See, y-axis and x-axis, label them."},{"Start":"01:06.385 ","End":"01:11.855","Text":"Now I want to draw a semicircle of radius r, so let\u0027s see."},{"Start":"01:11.855 ","End":"01:14.000","Text":"Suppose this is r,"},{"Start":"01:14.000 ","End":"01:17.470","Text":"and then I want r over here."},{"Start":"01:17.470 ","End":"01:20.820","Text":"I want minus r,"},{"Start":"01:20.820 ","End":"01:25.490","Text":"and I want to just roughly draw a semicircle,"},{"Start":"01:25.490 ","End":"01:28.440","Text":"doesn\u0027t have to be precise."},{"Start":"01:31.670 ","End":"01:38.300","Text":"Now I\u0027m going to highlight it."},{"Start":"01:38.300 ","End":"01:43.250","Text":"You can easily imagine that if I revolve this semicircle about the x-axis,"},{"Start":"01:43.250 ","End":"01:45.155","Text":"then I\u0027ll get a sphere."},{"Start":"01:45.155 ","End":"01:49.190","Text":"At least if you could follow the example with the cylinder and with the cone,"},{"Start":"01:49.190 ","End":"01:51.740","Text":"should be no difficulty here."},{"Start":"01:51.740 ","End":"01:55.850","Text":"What I\u0027m missing at the moment is the equation"},{"Start":"01:55.850 ","End":"01:59.270","Text":"of the function which describes this semicircle."},{"Start":"01:59.270 ","End":"02:02.330","Text":"For the moment, I\u0027ll call it y equals f of x,"},{"Start":"02:02.330 ","End":"02:05.765","Text":"but I\u0027ll need to know what f of x is explicitly."},{"Start":"02:05.765 ","End":"02:07.625","Text":"The plan of action is,"},{"Start":"02:07.625 ","End":"02:11.585","Text":"then to use the formula."},{"Start":"02:11.585 ","End":"02:14.825","Text":"I borrowed this from the previous exercise."},{"Start":"02:14.825 ","End":"02:20.960","Text":"This is the solid of revolution obtained when you revolve"},{"Start":"02:20.960 ","End":"02:27.840","Text":"the curve f of x around the x-axis from a to b,"},{"Start":"02:27.840 ","End":"02:31.895","Text":"so this will be our a and this will be our b."},{"Start":"02:31.895 ","End":"02:35.390","Text":"All we\u0027re missing is the f of x,"},{"Start":"02:35.390 ","End":"02:39.529","Text":"and that\u0027s not really very difficult."},{"Start":"02:39.529 ","End":"02:41.570","Text":"We know the equation of a full circle."},{"Start":"02:41.570 ","End":"02:48.710","Text":"The equation of a full circle would be x squared plus y squared equals r squared."},{"Start":"02:48.710 ","End":"02:53.905","Text":"Well-known formula for a circle of the radius r centered at the origin."},{"Start":"02:53.905 ","End":"03:00.850","Text":"I can then say that y squared is equal to r squared minus x squared."},{"Start":"03:00.850 ","End":"03:08.390","Text":"Normally y would equal plus or minus the square root of r squared minus x squared,"},{"Start":"03:08.390 ","End":"03:10.010","Text":"I would say the plus or minus."},{"Start":"03:10.010 ","End":"03:13.250","Text":"But here we\u0027re obviously above the x-axis,"},{"Start":"03:13.250 ","End":"03:14.540","Text":"so y is positive,"},{"Start":"03:14.540 ","End":"03:16.310","Text":"so I don\u0027t need a plus or minus."},{"Start":"03:16.310 ","End":"03:22.610","Text":"I\u0027ll just put a plus to show that I deliberately did not include the minus, and so on."},{"Start":"03:22.610 ","End":"03:24.160","Text":"That will be my f of x,"},{"Start":"03:24.160 ","End":"03:32.475","Text":"so f of x is the square root of r squared minus x squared."},{"Start":"03:32.475 ","End":"03:34.370","Text":"We have everything we need."},{"Start":"03:34.370 ","End":"03:38.630","Text":"Now let\u0027s get over here and say that v, which is volume,"},{"Start":"03:38.630 ","End":"03:43.400","Text":"is equal to Pi times the integral from a to b,"},{"Start":"03:43.400 ","End":"03:48.985","Text":"which is minus r to r of f of x squared."},{"Start":"03:48.985 ","End":"03:51.780","Text":"Well, look at that. Aren\u0027t we lucky,"},{"Start":"03:51.780 ","End":"03:55.210","Text":"f of x is under a square root sign and if we square it,"},{"Start":"03:55.210 ","End":"03:57.980","Text":"it just means we throw out the square root."},{"Start":"03:57.980 ","End":"04:00.350","Text":"I\u0027m not even going to write that in 2 lines,"},{"Start":"04:00.350 ","End":"04:05.500","Text":"I\u0027m going to straight away write r squared minus x squared,"},{"Start":"04:05.500 ","End":"04:07.650","Text":"that is f of x squared,"},{"Start":"04:07.650 ","End":"04:10.345","Text":"and that is dx."},{"Start":"04:10.345 ","End":"04:16.200","Text":"That\u0027s a fairly straightforward integration. Let\u0027s see."},{"Start":"04:16.730 ","End":"04:21.505","Text":"We get that this is equal to Pi."},{"Start":"04:21.505 ","End":"04:23.900","Text":"Now we have a polynomial here."},{"Start":"04:23.900 ","End":"04:28.220","Text":"This polynomial, r squared is a constant as far as x goes,"},{"Start":"04:28.220 ","End":"04:30.125","Text":"so it\u0027s r squared x."},{"Start":"04:30.125 ","End":"04:35.815","Text":"Here integral of x squared is x cubed over 3 with a minus of course,"},{"Start":"04:35.815 ","End":"04:39.080","Text":"and we have to substitute upper limit and"},{"Start":"04:39.080 ","End":"04:44.475","Text":"lower limit r here minus r here this and then subtract this."},{"Start":"04:44.475 ","End":"04:46.965","Text":"What we get if we put in r,"},{"Start":"04:46.965 ","End":"04:50.790","Text":"we get Pi times,"},{"Start":"04:50.790 ","End":"04:55.995","Text":"r squared r is r cubed."},{"Start":"04:55.995 ","End":"05:04.370","Text":"Here, r cubed over 3 minus what I get if I substitute minus r,"},{"Start":"05:04.370 ","End":"05:06.985","Text":"which is Pi times"},{"Start":"05:06.985 ","End":"05:16.035","Text":"minus r here gives us r squared times minus r is minus r cubed."},{"Start":"05:16.035 ","End":"05:20.750","Text":"Here, minus cubed is minus."},{"Start":"05:20.750 ","End":"05:22.220","Text":"But because it\u0027s a minus,"},{"Start":"05:22.220 ","End":"05:23.525","Text":"then it\u0027s a plus,"},{"Start":"05:23.525 ","End":"05:28.765","Text":"plus r cubed over 3."},{"Start":"05:28.765 ","End":"05:34.680","Text":"If we take out Pi, let\u0027s see."},{"Start":"05:34.680 ","End":"05:36.945","Text":"I\u0027ll put the Pi at the end."},{"Start":"05:36.945 ","End":"05:38.865","Text":"What am I left with?"},{"Start":"05:38.865 ","End":"05:41.490","Text":"In fact, I\u0027ll take out Pi r cubed,"},{"Start":"05:41.490 ","End":"05:46.660","Text":"we\u0027re left with 1 minus 1/3,"},{"Start":"05:46.660 ","End":"05:51.005","Text":"minus, minus 1, which is plus 1,"},{"Start":"05:51.005 ","End":"05:54.230","Text":"and then minus 1/3."},{"Start":"05:54.230 ","End":"05:57.325","Text":"Let\u0027s see, what does this amount to?"},{"Start":"05:57.325 ","End":"05:59.900","Text":"1 minus 1/3 is 2/3,"},{"Start":"05:59.900 ","End":"06:01.220","Text":"and then there\u0027s another 1 of those,"},{"Start":"06:01.220 ","End":"06:09.665","Text":"so we get 4/3 so this is equal to 4/3 Pi r cubed."},{"Start":"06:09.665 ","End":"06:16.279","Text":"I would like to highlight that because that\u0027s the answer for"},{"Start":"06:16.279 ","End":"06:23.850","Text":"the V volume and it\u0027s the same as here and so we are done."}],"ID":4739},{"Watched":false,"Name":"Exercise 9","Duration":"8m 50s","ChapterTopicVideoID":4732,"CourseChapterTopicPlaylistID":3993,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this exercise, we have to state and prove"},{"Start":"00:02.760 ","End":"00:06.825","Text":"the formula for computing the volume of a cone."},{"Start":"00:06.825 ","End":"00:11.580","Text":"I found a nice diagram on the Internet and borrowed it."},{"Start":"00:11.580 ","End":"00:16.275","Text":"Here we have a picture of a cone and there\u0027s 2 main parameters,"},{"Start":"00:16.275 ","End":"00:18.765","Text":"the height and the radius of the base."},{"Start":"00:18.765 ","End":"00:22.530","Text":"Perhaps we should have added the words right circular cone,"},{"Start":"00:22.530 ","End":"00:25.935","Text":"but the default is this cone."},{"Start":"00:25.935 ","End":"00:27.629","Text":"Here\u0027s the volume."},{"Start":"00:27.629 ","End":"00:31.380","Text":"I\u0027ll just write it bigger."},{"Start":"00:31.380 ","End":"00:35.685","Text":"What we have is that V,"},{"Start":"00:35.685 ","End":"00:37.574","Text":"which is the volume,"},{"Start":"00:37.574 ","End":"00:45.000","Text":"is equal to 1/3 Pi times the radius squared,"},{"Start":"00:45.000 ","End":"00:47.925","Text":"that\u0027s the radius of the circular base,"},{"Start":"00:47.925 ","End":"00:50.160","Text":"and times the height."},{"Start":"00:50.160 ","End":"00:57.810","Text":"That answers the part about stating."},{"Start":"00:57.810 ","End":"01:01.565","Text":"Now, we\u0027re going to do the part about proving."},{"Start":"01:01.565 ","End":"01:06.360","Text":"Perhaps I should also extend it to here."},{"Start":"01:08.530 ","End":"01:10.760","Text":"Now when it says prove,"},{"Start":"01:10.760 ","End":"01:13.610","Text":"of course we\u0027re going to do it with integration and"},{"Start":"01:13.610 ","End":"01:16.835","Text":"solids of revolution because that\u0027s the chapter we\u0027re in."},{"Start":"01:16.835 ","End":"01:23.795","Text":"What I\u0027d like to do is describe the cone as a solid of revolution."},{"Start":"01:23.795 ","End":"01:26.315","Text":"Here\u0027s how I propose we do it."},{"Start":"01:26.315 ","End":"01:29.005","Text":"I\u0027ll draw some axes,"},{"Start":"01:29.005 ","End":"01:34.750","Text":"y-axis and there\u0027s an x-axis."},{"Start":"01:36.470 ","End":"01:39.240","Text":"Yeah, it\u0027s a bit straighter."},{"Start":"01:39.240 ","End":"01:41.490","Text":"I\u0027ll label this x,"},{"Start":"01:41.490 ","End":"01:44.115","Text":"I\u0027ll label this y."},{"Start":"01:44.115 ","End":"01:49.535","Text":"What I claim is that if we draw a line through the origin,"},{"Start":"01:49.535 ","End":"01:55.485","Text":"a line segment and drop a perpendicular,"},{"Start":"01:55.485 ","End":"02:00.169","Text":"and in such a way that this distance is,"},{"Start":"02:00.169 ","End":"02:01.590","Text":"I\u0027ll choose a different color,"},{"Start":"02:01.590 ","End":"02:03.635","Text":"this distance is r,"},{"Start":"02:03.635 ","End":"02:08.080","Text":"and this distance is h,"},{"Start":"02:08.080 ","End":"02:15.180","Text":"then if I revolve this around the x-axis,"},{"Start":"02:15.500 ","End":"02:20.745","Text":"I will get exactly the cone."},{"Start":"02:20.745 ","End":"02:24.045","Text":"Let\u0027s say we just shade this area."},{"Start":"02:24.045 ","End":"02:28.295","Text":"Instead of shading I decided to highlight it, it looks nicer."},{"Start":"02:28.295 ","End":"02:32.075","Text":"If you revolve this shaded,"},{"Start":"02:32.075 ","End":"02:35.405","Text":"highlighted area around the x-axis,"},{"Start":"02:35.405 ","End":"02:37.145","Text":"you\u0027ll get the cone."},{"Start":"02:37.145 ","End":"02:39.589","Text":"If you\u0027re still not sure about that,"},{"Start":"02:39.589 ","End":"02:41.925","Text":"I have another picture for you."},{"Start":"02:41.925 ","End":"02:45.845","Text":"Here we are. I found this on the Internet also."},{"Start":"02:45.845 ","End":"02:52.330","Text":"This should explain how when we revolve this, we get this."},{"Start":"02:52.330 ","End":"03:01.490","Text":"Here we also have r and minus r. Why don\u0027t I just say that this point is the point h,"},{"Start":"03:01.650 ","End":"03:10.480","Text":"r. That explains that this is h and that this is 0."},{"Start":"03:12.620 ","End":"03:16.740","Text":"Maybe it\u0027s too much information never mind,"},{"Start":"03:16.740 ","End":"03:21.865","Text":"r and of course here we have minus r and so on."},{"Start":"03:21.865 ","End":"03:24.480","Text":"I think you get the idea."},{"Start":"03:24.480 ","End":"03:28.400","Text":"What we\u0027re missing is the equation of this line."},{"Start":"03:28.400 ","End":"03:33.775","Text":"This is like the curve we\u0027re going to revolve."},{"Start":"03:33.775 ","End":"03:37.415","Text":"This is a straight line through the origin."},{"Start":"03:37.415 ","End":"03:40.540","Text":"As a straight line through the origin,"},{"Start":"03:40.540 ","End":"03:46.290","Text":"they\u0027ve correctly noticed that we can write this"},{"Start":"03:46.290 ","End":"03:52.370","Text":"as f of x or y is m times x,"},{"Start":"03:52.370 ","End":"03:53.930","Text":"where m is the slope,"},{"Start":"03:53.930 ","End":"03:57.410","Text":"but this is the general form of a line through the origin,"},{"Start":"03:57.410 ","End":"04:07.110","Text":"so why don\u0027t I just write it on here that this line is part of y equals mx."},{"Start":"04:07.110 ","End":"04:10.450","Text":"We\u0027ll find what m is in a moment."},{"Start":"04:10.460 ","End":"04:17.690","Text":"It\u0027s just part of the line from x equals 0 to x equals,"},{"Start":"04:17.690 ","End":"04:22.535","Text":"that\u0027s not clear here this is an h. To find m,"},{"Start":"04:22.535 ","End":"04:28.350","Text":"the easiest thing to do is just to substitute the point h,"},{"Start":"04:28.350 ","End":"04:31.725","Text":"r into the equation of the line."},{"Start":"04:31.725 ","End":"04:35.840","Text":"In general, if I have y equals mx,"},{"Start":"04:35.840 ","End":"04:38.925","Text":"and I know that the point h,"},{"Start":"04:38.925 ","End":"04:42.890","Text":"r fits on this line,"},{"Start":"04:42.890 ","End":"04:44.180","Text":"the line passes through this point,"},{"Start":"04:44.180 ","End":"04:45.860","Text":"then I can substitute x is h,"},{"Start":"04:45.860 ","End":"04:53.000","Text":"y is r. So r equals m times h. That gives me that"},{"Start":"04:53.000 ","End":"05:00.500","Text":"m is equal to r over h. Once I have that,"},{"Start":"05:00.500 ","End":"05:05.105","Text":"then I have that the equation of this line is precisely,"},{"Start":"05:05.105 ","End":"05:06.995","Text":"I\u0027ll call it f of x."},{"Start":"05:06.995 ","End":"05:11.359","Text":"We have y, which equals f of x,"},{"Start":"05:11.359 ","End":"05:18.005","Text":"which equals m, which is r over h times x."},{"Start":"05:18.005 ","End":"05:20.910","Text":"This is this equation."},{"Start":"05:21.390 ","End":"05:26.010","Text":"The lower curve is the x-axis."},{"Start":"05:26.010 ","End":"05:33.670","Text":"We use the short formula with only 1 curve and let me copy it from the previous exercise."},{"Start":"05:33.670 ","End":"05:35.995","Text":"We have everything basically,"},{"Start":"05:35.995 ","End":"05:38.335","Text":"because this is my a,"},{"Start":"05:38.335 ","End":"05:45.170","Text":"this is my b and the f is this function here,"},{"Start":"05:45.170 ","End":"05:46.860","Text":"the f. I have a, b,"},{"Start":"05:46.860 ","End":"05:51.240","Text":"and f. All I have to do now is substitute."},{"Start":"05:51.240 ","End":"06:01.450","Text":"What I get is that the volume of the cone is Pi times the integral from 0 to h,"},{"Start":"06:01.450 ","End":"06:05.065","Text":"f of x is this bit here,"},{"Start":"06:05.065 ","End":"06:13.400","Text":"r over h x squared dx."},{"Start":"06:13.400 ","End":"06:15.050","Text":"Actually in this formula,"},{"Start":"06:15.050 ","End":"06:18.665","Text":"I meant to put the 2 outside,"},{"Start":"06:18.665 ","End":"06:20.680","Text":"just write that here."},{"Start":"06:20.680 ","End":"06:24.285","Text":"Otherwise, what\u0027s the point of the square brackets."},{"Start":"06:24.285 ","End":"06:30.870","Text":"Here we are and now let\u0027s continue just simplifying."},{"Start":"06:30.870 ","End":"06:34.290","Text":"V equals, r and h are"},{"Start":"06:34.290 ","End":"06:38.250","Text":"constants as far as x goes and I can take them outside the brackets."},{"Start":"06:38.250 ","End":"06:39.585","Text":"I have to first square them."},{"Start":"06:39.585 ","End":"06:46.650","Text":"So what I get is Pi times r over h squared,"},{"Start":"06:46.650 ","End":"06:53.135","Text":"then the integral from 0 to h of x squared dx."},{"Start":"06:53.135 ","End":"07:02.420","Text":"This is equal to Pi minus r squared over h squared times,"},{"Start":"07:02.420 ","End":"07:07.100","Text":"now the integral of x squared is x cubed over 3."},{"Start":"07:07.100 ","End":"07:12.050","Text":"I have to take this between 0 and"},{"Start":"07:12.050 ","End":"07:19.730","Text":"h. What I get is when I put x equals h,"},{"Start":"07:19.730 ","End":"07:23.290","Text":"I get h cubed over 3."},{"Start":"07:33.350 ","End":"07:36.810","Text":"If I put x equals 0,"},{"Start":"07:36.810 ","End":"07:40.430","Text":"0 cubed over 3 is 0 times all this, it\u0027s still 0."},{"Start":"07:40.430 ","End":"07:42.440","Text":"I like to write minus 0,"},{"Start":"07:42.440 ","End":"07:44.930","Text":"which shows that I haven\u0027t forgotten the lower limit."},{"Start":"07:44.930 ","End":"07:47.425","Text":"It just happens to be 0."},{"Start":"07:47.425 ","End":"07:50.775","Text":"As for this, we can cancel,"},{"Start":"07:50.775 ","End":"07:58.920","Text":"h squared goes into h cubed, just h times."},{"Start":"07:58.920 ","End":"08:02.730","Text":"What I\u0027m left with finally is"},{"Start":"08:02.730 ","End":"08:12.075","Text":"pi r squared times h over 3."},{"Start":"08:12.075 ","End":"08:20.890","Text":"It\u0027s 1/3 Pi r squared h. I think it deserves highlighting."},{"Start":"08:23.750 ","End":"08:26.220","Text":"That\u0027s the volume."},{"Start":"08:26.220 ","End":"08:30.770","Text":"I\u0027ll just write that again, the volume equals."},{"Start":"08:30.770 ","End":"08:34.895","Text":"Let\u0027s just check that we did get indeed the same answer."},{"Start":"08:34.895 ","End":"08:38.650","Text":"Remember, 1/3 Pi r squared h,"},{"Start":"08:38.650 ","End":"08:43.065","Text":"1/3 Pi r squared h,"},{"Start":"08:43.065 ","End":"08:45.360","Text":"looks the same to me."},{"Start":"08:45.360 ","End":"08:50.740","Text":"I believe that we are done. That\u0027s it."}],"ID":4740}],"Thumbnail":null,"ID":3993},{"Name":"Surface Area - Solids of Revolution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 (with Theory)","Duration":"3m 16s","ChapterTopicVideoID":4733,"CourseChapterTopicPlaylistID":3995,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we just have to write down a couple of formulas."},{"Start":"00:04.260 ","End":"00:07.080","Text":"1 is for computing"},{"Start":"00:07.080 ","End":"00:13.260","Text":"the surface area of the solid obtained by rotating a curve about the x-axis."},{"Start":"00:13.260 ","End":"00:15.825","Text":"That\u0027s diagram a."},{"Start":"00:15.825 ","End":"00:22.650","Text":"We rotate about the x-axis or revolve."},{"Start":"00:23.890 ","End":"00:28.080","Text":"You could consider it as the surface area of the solid or you"},{"Start":"00:28.080 ","End":"00:32.205","Text":"could just think of it as a shell so that the surface area,"},{"Start":"00:32.205 ","End":"00:40.830","Text":"it\u0027s just like taking this line and getting a hollow shell by rotating around the x-axis."},{"Start":"00:40.830 ","End":"00:42.090","Text":"Whatever way you look at it,"},{"Start":"00:42.090 ","End":"00:44.445","Text":"we\u0027re going to get the same formula."},{"Start":"00:44.445 ","End":"00:49.460","Text":"In part b, we\u0027re just going to do the same concept."},{"Start":"00:49.460 ","End":"00:57.740","Text":"We\u0027re going to have a curve and we\u0027re going to rotate it or revolve it around the y-axis."},{"Start":"00:57.740 ","End":"01:01.340","Text":"But you could also consider it as the surface of a solid."},{"Start":"01:01.340 ","End":"01:04.160","Text":"If you highlighted all this area,"},{"Start":"01:04.160 ","End":"01:09.345","Text":"you\u0027d get a solid and the surface of that would be the shell."},{"Start":"01:09.345 ","End":"01:13.845","Text":"Let\u0027s just write down the formulas."},{"Start":"01:13.845 ","End":"01:16.350","Text":"Off course I need the context."},{"Start":"01:16.350 ","End":"01:22.035","Text":"Here I have y as a function of x and I\u0027m taking it from a to b."},{"Start":"01:22.035 ","End":"01:23.970","Text":"So not the whole curve,"},{"Start":"01:23.970 ","End":"01:28.400","Text":"it\u0027s just the part that\u0027s between a and b and likewise here,"},{"Start":"01:28.400 ","End":"01:29.930","Text":"the curve may continue,"},{"Start":"01:29.930 ","End":"01:35.240","Text":"but we\u0027re just interested in the bit where y is between c and"},{"Start":"01:35.240 ","End":"01:41.725","Text":"d. Scroll down a bit and I\u0027ll write the formulas."},{"Start":"01:41.725 ","End":"01:46.310","Text":"Formulas are in many ways similar to the formula for arc length,"},{"Start":"01:46.310 ","End":"01:48.350","Text":"but I\u0027ll just write down."},{"Start":"01:48.350 ","End":"01:52.205","Text":"Here we have S. S for surface area,"},{"Start":"01:52.205 ","End":"02:03.220","Text":"is going to equal 2 Pi times the integral from a"},{"Start":"02:03.220 ","End":"02:10.530","Text":"to b of f of x times"},{"Start":"02:10.530 ","End":"02:21.490","Text":"the square root of 1 plus f prime of x squared dx."},{"Start":"02:22.100 ","End":"02:30.055","Text":"Let\u0027s go for the other case where we are revolving around the y-axis."},{"Start":"02:30.055 ","End":"02:37.900","Text":"In this case, the surface area obtained is also equal to 2 Pi."},{"Start":"02:37.900 ","End":"02:42.660","Text":"It\u0027s completely analogous, the integral from c to d,"},{"Start":"02:42.660 ","End":"02:49.840","Text":"but it\u0027s y that\u0027s going from c to d. We have g of y"},{"Start":"02:50.030 ","End":"03:01.020","Text":"times the square root of 1 plus g prime of y,"},{"Start":"03:01.020 ","End":"03:05.445","Text":"I better put the brackets,"},{"Start":"03:05.445 ","End":"03:09.475","Text":"squared, and this time it\u0027s dy."},{"Start":"03:09.475 ","End":"03:11.240","Text":"In the remaining exercises,"},{"Start":"03:11.240 ","End":"03:14.615","Text":"we\u0027ll be giving some examples of how to use these."},{"Start":"03:14.615 ","End":"03:17.640","Text":"We are done. These are the formulas."}],"ID":4741},{"Watched":false,"Name":"Exercise 2","Duration":"5m 52s","ChapterTopicVideoID":4734,"CourseChapterTopicPlaylistID":3995,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"Here we have another one of these surface area questions"},{"Start":"00:04.545 ","End":"00:09.680","Text":"for a solid of revolution about the x-axis."},{"Start":"00:09.680 ","End":"00:14.280","Text":"Let\u0027s first of all sketch the graph of the curve,"},{"Start":"00:14.280 ","End":"00:19.905","Text":"which is this in the interval from minus 1 to 1."},{"Start":"00:19.905 ","End":"00:22.665","Text":"Here are some axes."},{"Start":"00:22.665 ","End":"00:24.810","Text":"We don\u0027t have to make a very accurate sketch."},{"Start":"00:24.810 ","End":"00:27.360","Text":"I see it\u0027s defined from minus 1 to 1."},{"Start":"00:27.360 ","End":"00:31.680","Text":"Let\u0027s say that this here is minus 1,"},{"Start":"00:31.680 ","End":"00:34.560","Text":"and that this here is 1."},{"Start":"00:34.560 ","End":"00:36.750","Text":"When x is 0,"},{"Start":"00:36.750 ","End":"00:38.970","Text":"I see that y is 2."},{"Start":"00:38.970 ","End":"00:42.885","Text":"If that\u0027s 1 this must be 2,"},{"Start":"00:42.885 ","End":"00:45.670","Text":"doesn\u0027t have to be precise."},{"Start":"00:45.680 ","End":"00:52.890","Text":"We get something like 1,"},{"Start":"00:52.890 ","End":"00:55.169","Text":"it\u0027s equal to square root of 3,"},{"Start":"00:55.169 ","End":"00:56.415","Text":"which is less than 2."},{"Start":"00:56.415 ","End":"01:01.475","Text":"Something like this, and I\u0027ll just emphasize it by these points."},{"Start":"01:01.475 ","End":"01:05.740","Text":"This is the point above minus 1 and above 1."},{"Start":"01:05.740 ","End":"01:09.650","Text":"This is the bit of curve that I\u0027m going to rotate around"},{"Start":"01:09.650 ","End":"01:14.265","Text":"the x-axis and then I\u0027ll get a surface area."},{"Start":"01:14.265 ","End":"01:18.360","Text":"I\u0027ll immediately write down the formula."},{"Start":"01:18.360 ","End":"01:21.980","Text":"The surface area, which I call S,"},{"Start":"01:21.980 ","End":"01:27.820","Text":"is equal to 2Pi times the integral from a to b,"},{"Start":"01:27.820 ","End":"01:30.910","Text":"and all will be explained in due course,"},{"Start":"01:30.910 ","End":"01:36.020","Text":"of f of x times the square root of"},{"Start":"01:36.020 ","End":"01:42.835","Text":"1 plus f prime of x squared dx."},{"Start":"01:42.835 ","End":"01:46.294","Text":"I really should have labeled this curve,"},{"Start":"01:46.294 ","End":"01:50.270","Text":"and the curve is that y equals,"},{"Start":"01:50.270 ","End":"01:52.955","Text":"I\u0027ll call it also f of x,"},{"Start":"01:52.955 ","End":"01:58.295","Text":"which is the square root of 4 minus x squared."},{"Start":"01:58.295 ","End":"02:00.380","Text":"Point is that this is the curve,"},{"Start":"02:00.380 ","End":"02:02.495","Text":"this is what f of x is."},{"Start":"02:02.495 ","End":"02:06.990","Text":"Now I have f of x and I have a and b."},{"Start":"02:06.990 ","End":"02:12.030","Text":"This is going to be a and this is going to be b. I have a, b and f of x."},{"Start":"02:12.030 ","End":"02:16.735","Text":"The only thing I\u0027m missing really is f prime of x."},{"Start":"02:16.735 ","End":"02:19.510","Text":"Let\u0027s do that differentiation."},{"Start":"02:19.510 ","End":"02:26.314","Text":"If f of x is equal to the square root of 4 minus x squared,"},{"Start":"02:26.314 ","End":"02:29.000","Text":"then f prime of x equals,"},{"Start":"02:29.000 ","End":"02:32.540","Text":"there is that template for square root of something,"},{"Start":"02:32.540 ","End":"02:37.790","Text":"it\u0027s 1 over twice the square root of the same thing,"},{"Start":"02:37.790 ","End":"02:41.390","Text":"which is 4 minus x squared."},{"Start":"02:41.390 ","End":"02:44.420","Text":"Actually, I usually don\u0027t put the 1"},{"Start":"02:44.420 ","End":"02:48.710","Text":"because then I have to multiply by the inner derivative,"},{"Start":"02:48.710 ","End":"02:52.205","Text":"which in this case is minus 2x."},{"Start":"02:52.205 ","End":"03:01.109","Text":"This is the derivative and of course I\u0027ll cancel the 2 with the 2,"},{"Start":"03:01.109 ","End":"03:03.015","Text":"and now I have everything."},{"Start":"03:03.015 ","End":"03:12.245","Text":"Let\u0027s just substitute into the formula and get that S equals 2Pi times the integral,"},{"Start":"03:12.245 ","End":"03:14.825","Text":"a and b as we said are minus 1 and 1,"},{"Start":"03:14.825 ","End":"03:16.805","Text":"so from minus 1 to 1."},{"Start":"03:16.805 ","End":"03:25.745","Text":"F of x, the square root of 4 minus x squared times the square root,"},{"Start":"03:25.745 ","End":"03:30.455","Text":"that\u0027s this square root of 1 plus f prime of x squared."},{"Start":"03:30.455 ","End":"03:32.920","Text":"I need the square of this thing."},{"Start":"03:32.920 ","End":"03:36.315","Text":"The minus doesn\u0027t matter if it\u0027s a square."},{"Start":"03:36.315 ","End":"03:39.110","Text":"Let\u0027s see square top and bottom of the fraction."},{"Start":"03:39.110 ","End":"03:42.295","Text":"On the top of the fraction I have x squared,"},{"Start":"03:42.295 ","End":"03:44.445","Text":"and on the bottom of the fraction,"},{"Start":"03:44.445 ","End":"03:46.880","Text":"if you have the square root of something and you square it,"},{"Start":"03:46.880 ","End":"03:49.490","Text":"then you just throw out the square root sign."},{"Start":"03:49.490 ","End":"03:52.070","Text":"That\u0027s the way it works because the square root of something"},{"Start":"03:52.070 ","End":"03:54.800","Text":"squared is the thing itself and dx,"},{"Start":"03:54.800 ","End":"03:59.795","Text":"of course. Let\u0027s continue."},{"Start":"03:59.795 ","End":"04:01.955","Text":"Use a bit of algebra here."},{"Start":"04:01.955 ","End":"04:07.400","Text":"I have the product of two square roots under the integral sign,"},{"Start":"04:07.400 ","End":"04:10.400","Text":"and the square root of 8 times the square root of b is the square root of a,"},{"Start":"04:10.400 ","End":"04:14.560","Text":"b. I can put one big square root sign and a dx."},{"Start":"04:14.560 ","End":"04:16.835","Text":"Now I just multiply this by this."},{"Start":"04:16.835 ","End":"04:20.840","Text":"So 4 minus x squared times 1 is just 4 minus x"},{"Start":"04:20.840 ","End":"04:27.710","Text":"squared and 4 minus x squared times something over 4 minus x squared,"},{"Start":"04:27.710 ","End":"04:30.380","Text":"I just get rid of the denominator because this would cancel"},{"Start":"04:30.380 ","End":"04:33.890","Text":"with this so I just get plus x squared."},{"Start":"04:33.890 ","End":"04:36.450","Text":"How convenient."},{"Start":"04:36.450 ","End":"04:41.025","Text":"The x squared and the x squared cancel,"},{"Start":"04:41.025 ","End":"04:44.594","Text":"the square root of 4 is 2,"},{"Start":"04:44.594 ","End":"04:49.040","Text":"and so I end up with"},{"Start":"04:49.040 ","End":"04:57.945","Text":"2Pi integral from minus 1 to 1 of 2 dx."},{"Start":"04:57.945 ","End":"05:01.694","Text":"That is equal to,"},{"Start":"05:01.694 ","End":"05:05.620","Text":"2Pi times 2 is 4Pi,"},{"Start":"05:06.110 ","End":"05:12.515","Text":"4Pi times the integral of 1 is just x,"},{"Start":"05:12.515 ","End":"05:20.100","Text":"and this x has to be evaluated between minus 1 and 1."},{"Start":"05:21.610 ","End":"05:24.815","Text":"This is equal to 4Pi,"},{"Start":"05:24.815 ","End":"05:28.700","Text":"if I substitute x equals 1,"},{"Start":"05:28.700 ","End":"05:31.730","Text":"and x equals 1, sounds silly, but that\u0027s yeah."},{"Start":"05:31.730 ","End":"05:34.220","Text":"If I substitute x equals minus 1,"},{"Start":"05:34.220 ","End":"05:36.005","Text":"then x equals minus 1."},{"Start":"05:36.005 ","End":"05:38.070","Text":"I subtract those."},{"Start":"05:38.070 ","End":"05:41.220","Text":"So 1 minus minus 1 is 2,"},{"Start":"05:41.220 ","End":"05:45.580","Text":"2 times 4Pi gives me 8Pi,"},{"Start":"05:47.360 ","End":"05:53.410","Text":"that is what the surface area is and we are done."}],"ID":4742},{"Watched":false,"Name":"Exercise 3","Duration":"10m 16s","ChapterTopicVideoID":4735,"CourseChapterTopicPlaylistID":3995,"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":"Here we have to state and prove the formula for computing the surface area of a cone."},{"Start":"00:06.135 ","End":"00:09.990","Text":"Now, I want to save time and effort because I\u0027ve"},{"Start":"00:09.990 ","End":"00:16.335","Text":"already done a similar question with the volume of a cone."},{"Start":"00:16.335 ","End":"00:22.215","Text":"I am going to copy some of the pictures and text from that exercise."},{"Start":"00:22.215 ","End":"00:30.390","Text":"Hang on, I\u0027ve brought all this stuff over from the volume of a cone."},{"Start":"00:30.390 ","End":"00:40.580","Text":"However, there\u0027s something not quite right here because here that talks about volume."},{"Start":"00:40.580 ","End":"00:45.780","Text":"What I\u0027ll do is I\u0027ll replace this picture."},{"Start":"00:46.070 ","End":"00:52.935","Text":"Here\u0027s a more generic picture and there is a formula."},{"Start":"00:52.935 ","End":"00:59.869","Text":"The formula for the surface area of the cone is simply that S,"},{"Start":"00:59.869 ","End":"01:09.920","Text":"S for surface area is equal to pi times r times, this letter is an l,"},{"Start":"01:09.920 ","End":"01:13.820","Text":"it looks like a capital I or something, but it\u0027s an l,"},{"Start":"01:13.820 ","End":"01:22.500","Text":"is pi rl. L If we want it in terms of h and r,"},{"Start":"01:22.500 ","End":"01:26.600","Text":"we can say that l is equal to the square root"},{"Start":"01:26.600 ","End":"01:31.430","Text":"of h plus r squared by Pythagoras\u0027s theorem,"},{"Start":"01:31.430 ","End":"01:38.315","Text":"since this is a right angle and we\u0027re talking about a right circular cone, of course."},{"Start":"01:38.315 ","End":"01:44.540","Text":"This is the formula S equals pi rl. Another cautionary note,"},{"Start":"01:44.540 ","End":"01:47.750","Text":"in some books or in some situations,"},{"Start":"01:47.750 ","End":"01:53.270","Text":"the cone is not hollow like an ice cream cone or a clown\u0027s hat,"},{"Start":"01:53.270 ","End":"01:57.950","Text":"the base is included as part of the area and"},{"Start":"01:57.950 ","End":"02:04.250","Text":"so there is an extra bit, which is the area of the base,"},{"Start":"02:04.250 ","End":"02:07.110","Text":"which is Pi r squared."},{"Start":"02:07.220 ","End":"02:11.960","Text":"I\u0027m going to erase that because our cones are going to be hollow,"},{"Start":"02:11.960 ","End":"02:15.660","Text":"how are you going to get the ice cream in otherwise?"},{"Start":"02:16.100 ","End":"02:23.750","Text":"Our task is to prove this using surfaces of revolution and definite integrals and so on."},{"Start":"02:23.750 ","End":"02:25.910","Text":"We\u0027ve stated the formula,"},{"Start":"02:25.910 ","End":"02:28.055","Text":"now we just have to prove it."},{"Start":"02:28.055 ","End":"02:33.620","Text":"Basically, if we take a straight line as a graph"},{"Start":"02:33.620 ","End":"02:39.560","Text":"through the origin and in such a way that here it\u0027s h and here it\u0027s r. In other words,"},{"Start":"02:39.560 ","End":"02:45.570","Text":"the point h,r is on the straight line,"},{"Start":"02:45.570 ","End":"02:48.965","Text":"then and I highlighted,"},{"Start":"02:48.965 ","End":"02:51.860","Text":"this area if rotated gives me the solid cone,"},{"Start":"02:51.860 ","End":"02:57.650","Text":"but just this line gives me the surface of the cone."},{"Start":"02:57.650 ","End":"03:01.340","Text":"That\u0027s the red line here. That\u0027s this."},{"Start":"03:01.340 ","End":"03:05.180","Text":"Of course, I have put this on its side and this figure"},{"Start":"03:05.180 ","End":"03:09.785","Text":"helps show how when we rotate this around the x-axis,"},{"Start":"03:09.785 ","End":"03:12.890","Text":"of course, it\u0027s rotated around the x-axis,"},{"Start":"03:12.890 ","End":"03:16.740","Text":"then this is what we get as the cone."},{"Start":"03:17.480 ","End":"03:23.330","Text":"What I said before was that this straight line through the origin,"},{"Start":"03:23.330 ","End":"03:27.205","Text":"in this case, y equals f of x in general."},{"Start":"03:27.205 ","End":"03:32.360","Text":"I deduced that it was equal to this formula, r/h times x."},{"Start":"03:32.360 ","End":"03:33.695","Text":"How did I do this?"},{"Start":"03:33.695 ","End":"03:39.275","Text":"In general, the equation of a line through the origin and in fact,"},{"Start":"03:39.275 ","End":"03:42.029","Text":"they even say it here,"},{"Start":"03:42.190 ","End":"03:46.820","Text":"is that y is some number m times the x,"},{"Start":"03:46.820 ","End":"03:51.785","Text":"where m is the slope and I have that here."},{"Start":"03:51.785 ","End":"03:59.750","Text":"In our case, I can figure out what m is because this line passes through h,r."},{"Start":"03:59.750 ","End":"04:06.195","Text":"If I substitute h,r in this should work so y is r,"},{"Start":"04:06.195 ","End":"04:08.715","Text":"x is h. We get r equals m,h,"},{"Start":"04:08.715 ","End":"04:10.965","Text":"m is r /h."},{"Start":"04:10.965 ","End":"04:16.490","Text":"Then I replace the m. This bit was the m,"},{"Start":"04:16.490 ","End":"04:18.845","Text":"I replace it by r/h."},{"Start":"04:18.845 ","End":"04:22.165","Text":"This becomes my function."},{"Start":"04:22.165 ","End":"04:30.790","Text":"Now, I\u0027m going to write the formula for the area of the surface of revolution."},{"Start":"04:30.790 ","End":"04:41.630","Text":"The general formula is that S is equal to 2 pi times the integral from"},{"Start":"04:41.630 ","End":"04:45.890","Text":"a to b of f of x times"},{"Start":"04:45.890 ","End":"04:53.370","Text":"the square root of 1 plus f prime of x squared dx."},{"Start":"04:53.370 ","End":"04:56.855","Text":"I have to tell you what all these things mean."},{"Start":"04:56.855 ","End":"05:02.670","Text":"Now, our a and v are just 0"},{"Start":"05:02.670 ","End":"05:07.985","Text":"and h. A is equal to 0 and this of course,"},{"Start":"05:07.985 ","End":"05:18.779","Text":"is h so that b is equal to h. The f of x is written here."},{"Start":"05:18.779 ","End":"05:27.005","Text":"All I\u0027m missing now is f prime of x but that\u0027s easy enough to compute."},{"Start":"05:27.005 ","End":"05:28.685","Text":"If this is f of x,"},{"Start":"05:28.685 ","End":"05:30.590","Text":"then all computed here,"},{"Start":"05:30.590 ","End":"05:33.275","Text":"then f prime of x,"},{"Start":"05:33.275 ","End":"05:35.555","Text":"this is a constant times x."},{"Start":"05:35.555 ","End":"05:36.800","Text":"When you differentiate it,"},{"Start":"05:36.800 ","End":"05:38.270","Text":"you\u0027re just left with the constant,"},{"Start":"05:38.270 ","End":"05:43.445","Text":"in our case, r/h."},{"Start":"05:43.445 ","End":"05:47.640","Text":"Now, I\u0027m going to substitute everything in here."},{"Start":"05:47.750 ","End":"05:54.014","Text":"S equals 2 pi times the integral."},{"Start":"05:54.014 ","End":"05:55.800","Text":"Now, this is a and this is b,"},{"Start":"05:55.800 ","End":"06:04.210","Text":"it\u0027s from 0 to h. F of x is r/h times x."},{"Start":"06:05.210 ","End":"06:08.145","Text":"Next, the square root,"},{"Start":"06:08.145 ","End":"06:15.610","Text":"1 plus f prime of x, r/h constant squared."},{"Start":"06:17.090 ","End":"06:20.490","Text":"I\u0027ll just write it as r squared over h squared."},{"Start":"06:20.490 ","End":"06:27.380","Text":"I don\u0027t think anyone has a problem with that if I square it right away and dx."},{"Start":"06:27.380 ","End":"06:32.580","Text":"Now, look, all these bits without x,"},{"Start":"06:32.580 ","End":"06:34.470","Text":"I mean the r/h doesn\u0027t have x in it,"},{"Start":"06:34.470 ","End":"06:35.870","Text":"the whole square root doesn\u0027t have x in it."},{"Start":"06:35.870 ","End":"06:37.660","Text":"They\u0027re all constants as far as x goes."},{"Start":"06:37.660 ","End":"06:41.815","Text":"Let me take all the constants out in front and have less clutter."},{"Start":"06:41.815 ","End":"06:48.960","Text":"What I get is 2 pi and then I take out the r/h,"},{"Start":"06:48.960 ","End":"06:55.650","Text":"take out the square root of 1 plus r squared over h squared."},{"Start":"06:55.650 ","End":"07:03.500","Text":"All I\u0027m left with is the integral from 0 to h of x dx."},{"Start":"07:03.500 ","End":"07:06.485","Text":"I decided to just compute this bit at the side."},{"Start":"07:06.485 ","End":"07:15.860","Text":"The integral from 0 to h of x dx is equal to the integral of x is 1/2x squared."},{"Start":"07:15.860 ","End":"07:22.505","Text":"I have 1/2 of x squared and the x squared is taken from 0 to h,"},{"Start":"07:22.505 ","End":"07:29.435","Text":"which gives me 1/2 of H squared minus 0 squared,"},{"Start":"07:29.435 ","End":"07:32.180","Text":"which is just 1/2h squared."},{"Start":"07:32.180 ","End":"07:36.350","Text":"I mean, I\u0027ll write the 0 squared just so you\u0027ll see I haven\u0027t forgotten it."},{"Start":"07:36.350 ","End":"07:38.915","Text":"When I get back over here,"},{"Start":"07:38.915 ","End":"07:44.240","Text":"I already have the integral and so altogether I just have a"},{"Start":"07:44.240 ","End":"07:50.620","Text":"constant which is 2 pi, and then r/h."},{"Start":"07:50.620 ","End":"07:55.220","Text":"The square root, let me just write the square root differently with a common denominator,"},{"Start":"07:55.220 ","End":"08:00.900","Text":"which is h squared plus r squared over h squared,"},{"Start":"08:00.900 ","End":"08:02.655","Text":"I have a reason for doing this."},{"Start":"08:02.655 ","End":"08:05.855","Text":"Then times the integral,"},{"Start":"08:05.855 ","End":"08:09.400","Text":"which is 1/2h squared."},{"Start":"08:10.150 ","End":"08:13.580","Text":"Now, I\u0027m claiming that a lot of stuff will cancel,"},{"Start":"08:13.580 ","End":"08:19.950","Text":"that this will boil down ultimately to what we needed to get so let\u0027s see."},{"Start":"08:19.950 ","End":"08:26.910","Text":"The 2 with the 2 here cancels and 1 of"},{"Start":"08:26.910 ","End":"08:33.120","Text":"the h\u0027s cancels with this h. I want"},{"Start":"08:33.120 ","End":"08:39.530","Text":"to remind you that h squared plus r squared is equal to l squared."},{"Start":"08:39.530 ","End":"08:43.595","Text":"Let me scroll back up and see."},{"Start":"08:43.595 ","End":"08:45.935","Text":"Yes, here it is."},{"Start":"08:45.935 ","End":"08:53.780","Text":"Directly from Pythagoras, h squared plus r squared is l squared."},{"Start":"08:53.780 ","End":"08:56.240","Text":"The other thing is that while we\u0027re up here,"},{"Start":"08:56.240 ","End":"09:04.290","Text":"why don\u0027t I just copy and paste that down there so I can remember it."},{"Start":"09:05.180 ","End":"09:08.060","Text":"As they say Eyes on the Prize,"},{"Start":"09:08.060 ","End":"09:09.840","Text":"this is what we\u0027re aiming for."},{"Start":"09:09.840 ","End":"09:16.265","Text":"Now we\u0027re continuing here but we have that h squared plus r squared is l squared."},{"Start":"09:16.265 ","End":"09:25.385","Text":"Now I\u0027m continuing with the simplification and I get pi times r times the square root"},{"Start":"09:25.385 ","End":"09:35.570","Text":"of l squared over h squared and then times h. This is what we\u0027re finally left with."},{"Start":"09:35.570 ","End":"09:41.070","Text":"Now the square root of l squared over h squared is just l/h."},{"Start":"09:41.070 ","End":"09:44.050","Text":"I have pi r,"},{"Start":"09:44.640 ","End":"09:51.035","Text":"l/h times h and once again, the h\u0027s cancel."},{"Start":"09:51.035 ","End":"10:00.110","Text":"Isn\u0027t this just equal to pi r l? Which is this."},{"Start":"10:00.110 ","End":"10:03.710","Text":"I mean, I\u0027m going to just highlight it so you\u0027ll see better,"},{"Start":"10:03.710 ","End":"10:09.600","Text":"s this is pi r l and this is s equals pi r l,"},{"Start":"10:09.600 ","End":"10:12.435","Text":"which is a copy-paste from what I wrote above."},{"Start":"10:12.435 ","End":"10:16.960","Text":"Looks like we\u0027ve proven it so we are done."}],"ID":4743},{"Watched":false,"Name":"Exercise 4","Duration":"7m 28s","ChapterTopicVideoID":4736,"CourseChapterTopicPlaylistID":3995,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.120","Text":"I would just like to begin with a word on notation and terminology."},{"Start":"00:06.120 ","End":"00:10.950","Text":"We\u0027ve been talking about solids of rotation or evolution,"},{"Start":"00:10.950 ","End":"00:15.239","Text":"and then discussing the surface area."},{"Start":"00:15.239 ","End":"00:19.905","Text":"We could actually just consider the surface as like a shell,"},{"Start":"00:19.905 ","End":"00:24.330","Text":"and there is a concept called surface of revolution,"},{"Start":"00:24.330 ","End":"00:28.980","Text":"which you\u0027ll see in the literature and they\u0027ll often say a question like"},{"Start":"00:28.980 ","End":"00:34.620","Text":"find the area of the surface of revolution of the curve so and so."},{"Start":"00:34.620 ","End":"00:40.615","Text":"I\u0027m also not quite sure what the difference is between revolution and rotation,"},{"Start":"00:40.615 ","End":"00:44.000","Text":"because I see them used pretty interchangeably in the literature."},{"Start":"00:44.000 ","End":"00:46.705","Text":"I use them synonymously."},{"Start":"00:46.705 ","End":"00:49.130","Text":"I could say in this question,"},{"Start":"00:49.130 ","End":"00:56.285","Text":"determine the surface of revolution of the graph of so and so about the y-axis?"},{"Start":"00:56.285 ","End":"00:57.875","Text":"It\u0027s the same thing."},{"Start":"00:57.875 ","End":"01:00.780","Text":"Let\u0027s get to the exercise."},{"Start":"01:01.190 ","End":"01:05.855","Text":"I\u0027m starting with a pair of axes and we\u0027ll make a diagram."},{"Start":"01:05.855 ","End":"01:12.800","Text":"The important thing to notice is that we are revolving or rotating about the y-axis,"},{"Start":"01:12.800 ","End":"01:16.190","Text":"and I\u0027ll mark that here already to remind ourselves"},{"Start":"01:16.190 ","End":"01:20.760","Text":"because up to now we\u0027ve been mostly revolving around the x axis."},{"Start":"01:21.170 ","End":"01:27.725","Text":"We are in luck because x is given explicitly in terms of y."},{"Start":"01:27.725 ","End":"01:28.820","Text":"If this was not so,"},{"Start":"01:28.820 ","End":"01:32.720","Text":"we might have to do some algebra to extract x in terms of y."},{"Start":"01:32.720 ","End":"01:33.920","Text":"Since this is the case,"},{"Start":"01:33.920 ","End":"01:36.390","Text":"we don\u0027t need to worry. Now let\u0027s see."},{"Start":"01:36.390 ","End":"01:39.185","Text":"The domain is from minus 2-2."},{"Start":"01:39.185 ","End":"01:41.255","Text":"Let\u0027s mark on the graph."},{"Start":"01:41.255 ","End":"01:51.695","Text":"Let\u0027s say that this is y equals 2 and that on the other side we have y equals minus 2,"},{"Start":"01:51.695 ","End":"01:58.280","Text":"and this here is the origin and I don\u0027t need an exact sketch,"},{"Start":"01:58.280 ","End":"02:01.205","Text":"but just roughly, let\u0027s see if y is 0,"},{"Start":"02:01.205 ","End":"02:03.890","Text":"then x is square root of 9, which is 3."},{"Start":"02:03.890 ","End":"02:06.110","Text":"If this is 2, this is around 3,"},{"Start":"02:06.110 ","End":"02:07.730","Text":"so it passes through here,"},{"Start":"02:07.730 ","End":"02:10.295","Text":"and if y is plus or minus 2,"},{"Start":"02:10.295 ","End":"02:12.110","Text":"I get the square root of 5,"},{"Start":"02:12.110 ","End":"02:13.970","Text":"which is 2. something."},{"Start":"02:13.970 ","End":"02:19.810","Text":"So let\u0027s just say it\u0027s somewhere here and here."},{"Start":"02:19.810 ","End":"02:23.400","Text":"I\u0027ll just sketch a bit of a curve here."},{"Start":"02:24.700 ","End":"02:27.410","Text":"Let\u0027s give this function a name."},{"Start":"02:27.410 ","End":"02:29.555","Text":"Let\u0027s call it g(y)."},{"Start":"02:29.555 ","End":"02:33.320","Text":"I prefer a g then f when x is a function of y."},{"Start":"02:33.320 ","End":"02:39.200","Text":"So that here also we have that x is equal to g(y),"},{"Start":"02:39.200 ","End":"02:47.035","Text":"where g(y) is square root of 9 minus y squared."},{"Start":"02:47.035 ","End":"02:52.430","Text":"Now I\u0027m going to write down the formula for the area of surface of revolution as"},{"Start":"02:52.430 ","End":"02:58.665","Text":"the surface area is equal to 2 Pi times the integral,"},{"Start":"02:58.665 ","End":"03:01.160","Text":"but this time the limit\u0027s around y."},{"Start":"03:01.160 ","End":"03:03.230","Text":"But it\u0027s still from,"},{"Start":"03:03.230 ","End":"03:09.095","Text":"let\u0027s say from c to d of instead f(x),"},{"Start":"03:09.095 ","End":"03:11.895","Text":"we now have g(y)."},{"Start":"03:11.895 ","End":"03:16.370","Text":"We still have the square root 1 plus and then we have"},{"Start":"03:16.370 ","End":"03:21.570","Text":"a derivative squared only this time it\u0027s g\u0027(y),"},{"Start":"03:21.570 ","End":"03:24.180","Text":"that is squared and it\u0027s dy."},{"Start":"03:24.180 ","End":"03:28.155","Text":"So our c is this,"},{"Start":"03:28.155 ","End":"03:30.345","Text":"our d is this,"},{"Start":"03:30.345 ","End":"03:34.110","Text":"our g is what we have here."},{"Start":"03:34.110 ","End":"03:38.890","Text":"This is g. All we\u0027re missing now is g\u0027."},{"Start":"03:39.280 ","End":"03:42.080","Text":"I\u0027ll compute that at the side."},{"Start":"03:42.080 ","End":"03:45.330","Text":"Let me just scroll a bit,"},{"Start":"03:46.000 ","End":"03:48.180","Text":"g(y) is equal to,"},{"Start":"03:48.180 ","End":"03:49.295","Text":"I have it written here,"},{"Start":"03:49.295 ","End":"03:54.270","Text":"square root of 9 minus y squared,"},{"Start":"03:54.760 ","End":"03:59.990","Text":"and so g\u0027(y) is equal to,"},{"Start":"03:59.990 ","End":"04:01.280","Text":"whenever I have a square root,"},{"Start":"04:01.280 ","End":"04:07.160","Text":"it\u0027s a template, I put 1 over 2 the square root of the same thing."},{"Start":"04:07.160 ","End":"04:11.149","Text":"I don\u0027t put the 1 here because I already put the inner derivative,"},{"Start":"04:11.149 ","End":"04:13.940","Text":"which is minus 2y here,"},{"Start":"04:13.940 ","End":"04:18.030","Text":"and I cancel the 2."},{"Start":"04:18.030 ","End":"04:19.530","Text":"So now I have g\u0027(y)."},{"Start":"04:19.530 ","End":"04:23.660","Text":"Let me go back up."},{"Start":"04:23.660 ","End":"04:27.930","Text":"So we have everything we need to substitute in here."},{"Start":"04:28.040 ","End":"04:40.020","Text":"I\u0027m going to continue and say that S is equal to 2 Pi times the integral from minus 2-2,"},{"Start":"04:41.510 ","End":"04:48.600","Text":"g(y) is the square root of 9 minus y squared,"},{"Start":"04:48.600 ","End":"04:57.990","Text":"and here I have the square root of 1 plus g\u0027(y squared)."},{"Start":"04:57.990 ","End":"05:01.460","Text":"Let me square it and then put it inside here."},{"Start":"05:01.460 ","End":"05:02.480","Text":"Now if I square this,"},{"Start":"05:02.480 ","End":"05:05.850","Text":"I square the numerator and the denominator."},{"Start":"05:06.130 ","End":"05:10.940","Text":"The minus, of course squared is just a plus so I throw it out."},{"Start":"05:10.940 ","End":"05:13.015","Text":"So I\u0027ve got y squared,"},{"Start":"05:13.015 ","End":"05:15.260","Text":"and on the denominator,"},{"Start":"05:15.260 ","End":"05:18.365","Text":"the square root squared is just the thing itself."},{"Start":"05:18.365 ","End":"05:23.280","Text":"We\u0027ve seen this many times and finally, dy."},{"Start":"05:24.070 ","End":"05:26.840","Text":"S is equal to 2 Pi,"},{"Start":"05:26.840 ","End":"05:29.510","Text":"the integral from minus 2-2."},{"Start":"05:29.510 ","End":"05:33.430","Text":"I\u0027m going to use a bit of algebra that the square root of a times"},{"Start":"05:33.430 ","End":"05:37.390","Text":"the square root of b is the square root of ab."},{"Start":"05:37.390 ","End":"05:40.015","Text":"In other words, I\u0027m going to multiply these things out,"},{"Start":"05:40.015 ","End":"05:46.735","Text":"so I get one big square root and I\u0027ll put brackets around here just to help us."},{"Start":"05:46.735 ","End":"05:51.760","Text":"9 minus y squared times 1 is 9 minus y"},{"Start":"05:51.760 ","End":"05:58.615","Text":"squared and 9 minus y squared times a fraction where this is on the bottom,"},{"Start":"05:58.615 ","End":"06:01.180","Text":"the 9 minus y squared will cancel,"},{"Start":"06:01.180 ","End":"06:06.680","Text":"so the second term will be just y squared."},{"Start":"06:07.060 ","End":"06:11.195","Text":"That\u0027s looking very simple because we\u0027re in luck."},{"Start":"06:11.195 ","End":"06:14.255","Text":"More cancel out, this thing cancels with this."},{"Start":"06:14.255 ","End":"06:17.795","Text":"The square root of 9 is 3."},{"Start":"06:17.795 ","End":"06:20.375","Text":"So if I put this in front,"},{"Start":"06:20.375 ","End":"06:24.890","Text":"then all I\u0027m going to end up with is that S is equal to 3 times"},{"Start":"06:24.890 ","End":"06:34.210","Text":"2 is 6 Pi times the integral of just dy."},{"Start":"06:35.630 ","End":"06:40.940","Text":"I sometimes I like to put it as 1dy so we can see what it is."},{"Start":"06:40.940 ","End":"06:43.660","Text":"So it\u0027s equal to 6 Pi,"},{"Start":"06:43.660 ","End":"06:47.920","Text":"I forgot to write minus 2-2 of course."},{"Start":"06:48.260 ","End":"06:58.485","Text":"This is equal to 6 Pi and the integral of 1 is just y taken from minus 2-2,"},{"Start":"06:58.485 ","End":"07:04.080","Text":"and this is equal to 6 Pi."},{"Start":"07:04.080 ","End":"07:06.090","Text":"Now, when y is 2,"},{"Start":"07:06.090 ","End":"07:08.480","Text":"then y is 2, and when y is minus 2,"},{"Start":"07:08.480 ","End":"07:14.830","Text":"y is minus 2, and I subtract 2 minus minus 2, this is just 4,"},{"Start":"07:14.830 ","End":"07:24.015","Text":"and so what I end up with is 24 Pi and I\u0027m going to highlight this,"},{"Start":"07:24.015 ","End":"07:29.230","Text":"that our surface area is 24 Pi and we are done."}],"ID":4744}],"Thumbnail":null,"ID":3995},{"Name":"Volume by Integrating Cross Sections","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 (with Theory)","Duration":"13m 10s","ChapterTopicVideoID":4737,"CourseChapterTopicPlaylistID":3994,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this exercise, we have to find a formula for computing"},{"Start":"00:04.410 ","End":"00:08.370","Text":"the volume of a right pyramid with height H and"},{"Start":"00:08.370 ","End":"00:12.600","Text":"a square base with side of length A. I\u0027d"},{"Start":"00:12.600 ","End":"00:17.830","Text":"like to illustrate this with a picture I found on the Internet."},{"Start":"00:17.840 ","End":"00:20.325","Text":"Here\u0027s the picture."},{"Start":"00:20.325 ","End":"00:21.960","Text":"That\u0027s what a pyramid looked like."},{"Start":"00:21.960 ","End":"00:23.535","Text":"Its base is square,"},{"Start":"00:23.535 ","End":"00:29.740","Text":"and the right pyramid means that the apex is directly above the center of the base."},{"Start":"00:29.740 ","End":"00:32.900","Text":"Now, in general, for all pyramids,"},{"Start":"00:32.900 ","End":"00:38.790","Text":"the volume is the area of the base times the height divided by 3."},{"Start":"00:38.930 ","End":"00:42.170","Text":"In our case, well, in this diagram,"},{"Start":"00:42.170 ","End":"00:43.520","Text":"the side of the base is B,"},{"Start":"00:43.520 ","End":"00:46.040","Text":"in our case it\u0027s A, no big difference."},{"Start":"00:46.040 ","End":"00:50.210","Text":"Their formula comes out to be as follows."},{"Start":"00:50.210 ","End":"00:52.550","Text":"The area of the base b times b times h,"},{"Start":"00:52.550 ","End":"00:55.720","Text":"which is b squared h, and divide it by 3."},{"Start":"00:55.720 ","End":"01:00.000","Text":"I just wanted to keep the results so I\u0027ll know what I\u0027m aiming for."},{"Start":"01:00.000 ","End":"01:02.210","Text":"Let me write that down that in our case,"},{"Start":"01:02.210 ","End":"01:06.680","Text":"we are expecting to get that v is equal"},{"Start":"01:06.680 ","End":"01:13.380","Text":"to a squared times h over 3,"},{"Start":"01:13.380 ","End":"01:17.320","Text":"and I\u0027m going to get rid of the picture now."},{"Start":"01:17.320 ","End":"01:22.825","Text":"But I found another picture that\u0027s going to be more useful for the continuation,"},{"Start":"01:22.825 ","End":"01:25.085","Text":"and here it is,"},{"Start":"01:25.085 ","End":"01:31.040","Text":"and here\u0027s the diagram that is useful for us."},{"Start":"01:31.040 ","End":"01:36.500","Text":"In general, the method I\u0027m going to use is called the method of"},{"Start":"01:36.500 ","End":"01:41.975","Text":"cross-sections or volumes by integrating cross-sections,"},{"Start":"01:41.975 ","End":"01:48.645","Text":"and we can use this for more general volumes other than solids"},{"Start":"01:48.645 ","End":"01:55.820","Text":"of revolution as long as we have a body that\u0027s sandwiched between 2 planes."},{"Start":"01:55.820 ","End":"01:58.130","Text":"In this case, in this diagram,"},{"Start":"01:58.130 ","End":"02:05.630","Text":"it\u0027s between x equals 0 and x equals h. But in general,"},{"Start":"02:05.630 ","End":"02:08.690","Text":"it will be sandwiched between 2 planes,"},{"Start":"02:08.690 ","End":"02:15.170","Text":"x equals a and x equals b. I say planes because there\u0027s also a z-axis,"},{"Start":"02:15.170 ","End":"02:18.590","Text":"and as long as we know the cross-section a of"},{"Start":"02:18.590 ","End":"02:22.160","Text":"x is a function of x for each point between a and b,"},{"Start":"02:22.160 ","End":"02:24.275","Text":"or in our case, 0 and h,"},{"Start":"02:24.275 ","End":"02:32.900","Text":"then the formula will be that the volume is the integral from a to b,"},{"Start":"02:32.900 ","End":"02:42.620","Text":"and if I call the area as a function of xa of x, dx."},{"Start":"02:42.620 ","End":"02:46.880","Text":"Oops, I\u0027ve used the little a twice."},{"Start":"02:46.880 ","End":"02:53.095","Text":"One is for the lower limit of the integration and one is for the side of the square base."},{"Start":"02:53.095 ","End":"02:57.560","Text":"So what I\u0027ll do is I\u0027ll just highlight the a from the lower limit"},{"Start":"02:57.560 ","End":"03:01.670","Text":"of the integration and hope there\u0027s no confusion."},{"Start":"03:01.670 ","End":"03:04.039","Text":"This is the formula that works in general."},{"Start":"03:04.039 ","End":"03:06.320","Text":"If we know that, again,"},{"Start":"03:06.320 ","End":"03:13.055","Text":"there\u0027s a solid shape sandwiched between the planes x equals a and x equals b,"},{"Start":"03:13.055 ","End":"03:17.510","Text":"and the cross-section which is perpendicular to the x-axis,"},{"Start":"03:17.510 ","End":"03:21.635","Text":"of course, is a of x at each point x between a and b,"},{"Start":"03:21.635 ","End":"03:23.720","Text":"then this is the volume."},{"Start":"03:23.720 ","End":"03:25.460","Text":"Now, in our case,"},{"Start":"03:25.460 ","End":"03:33.965","Text":"we have a pyramid and the a in our case is equal to 0,"},{"Start":"03:33.965 ","End":"03:39.400","Text":"and b in our case is equal to h,"},{"Start":"03:39.400 ","End":"03:41.540","Text":"and the question is,"},{"Start":"03:41.540 ","End":"03:45.150","Text":"what is a of x equal to?"},{"Start":"03:46.940 ","End":"03:50.520","Text":"I\u0027d like to redraw this not in 3D,"},{"Start":"03:50.520 ","End":"03:53.040","Text":"but in 2D just with the x-y plane."},{"Start":"03:53.040 ","End":"03:57.344","Text":"I\u0027ll squash it down or project it onto the x-y plane."},{"Start":"03:57.344 ","End":"03:59.625","Text":"Here are some coordinates."},{"Start":"03:59.625 ","End":"04:02.255","Text":"Oh, there\u0027s more thing I need to mark here,"},{"Start":"04:02.255 ","End":"04:06.654","Text":"because we said that the side is of length a,"},{"Start":"04:06.654 ","End":"04:09.440","Text":"then from here to here,"},{"Start":"04:09.440 ","End":"04:16.695","Text":"this is the point where y is a over 2 because it\u0027s altogether a,"},{"Start":"04:16.695 ","End":"04:20.480","Text":"so this point would be the point where it\u0027s minus"},{"Start":"04:20.480 ","End":"04:24.335","Text":"a over 2 altogether a but divided into 2 bits,"},{"Start":"04:24.335 ","End":"04:26.840","Text":"and the height, as we said, is h. Now,"},{"Start":"04:26.840 ","End":"04:29.165","Text":"if I write all this stuff here,"},{"Start":"04:29.165 ","End":"04:33.890","Text":"what I need is for here to write a over 2,"},{"Start":"04:33.890 ","End":"04:35.345","Text":"this is the origin of course,"},{"Start":"04:35.345 ","End":"04:38.210","Text":"and for here to write h,"},{"Start":"04:38.210 ","End":"04:41.245","Text":"and then I need to join these with a line."},{"Start":"04:41.245 ","End":"04:43.290","Text":"This is a diagram,"},{"Start":"04:43.290 ","End":"04:46.760","Text":"not a function, so I\u0027m going to write,"},{"Start":"04:46.760 ","End":"04:48.670","Text":"show the other side too."},{"Start":"04:48.670 ","End":"04:55.260","Text":"On this side, I have minus a over 2,"},{"Start":"04:55.260 ","End":"04:58.210","Text":"and I\u0027ll join that up also."},{"Start":"04:58.210 ","End":"05:02.495","Text":"There we are. At a general point,"},{"Start":"05:02.495 ","End":"05:05.900","Text":"let\u0027s call it x. I need the cross-section."},{"Start":"05:05.900 ","End":"05:07.970","Text":"I\u0027ll do it in a different color."},{"Start":"05:07.970 ","End":"05:09.770","Text":"Why not go with red?"},{"Start":"05:09.770 ","End":"05:14.100","Text":"Yeah. This will be the cross-section."},{"Start":"05:14.100 ","End":"05:20.360","Text":"This line here actually represents this square here,"},{"Start":"05:20.360 ","End":"05:26.760","Text":"just viewed from the scene from the side."},{"Start":"05:27.290 ","End":"05:33.020","Text":"The a of x will be the area of this cross-section,"},{"Start":"05:33.020 ","End":"05:35.090","Text":"which is the area of this,"},{"Start":"05:35.090 ","End":"05:39.670","Text":"and that\u0027s what we have to figure out now."},{"Start":"05:39.680 ","End":"05:45.030","Text":"I\u0027d like to label this point x,y."},{"Start":"05:45.030 ","End":"05:49.475","Text":"This point is going to be x, minus y."},{"Start":"05:49.475 ","End":"05:56.550","Text":"The length of this red thing will then be 2y,"},{"Start":"05:56.660 ","End":"05:59.305","Text":"then on this diagram,"},{"Start":"05:59.305 ","End":"06:04.945","Text":"this is a square where this is 2y and this is 2y,"},{"Start":"06:04.945 ","End":"06:11.600","Text":"and so a of x is equal to 4y squared."},{"Start":"06:11.730 ","End":"06:16.570","Text":"All I need is to find y in terms of x,"},{"Start":"06:16.570 ","End":"06:21.460","Text":"which means finding the equation of this line here,"},{"Start":"06:21.460 ","End":"06:25.150","Text":"which I will highlight."},{"Start":"06:25.150 ","End":"06:28.944","Text":"What do I know about this line?"},{"Start":"06:28.944 ","End":"06:34.750","Text":"Main thing is that it passes through the point h,0 and 0,a over 2."},{"Start":"06:34.750 ","End":"06:38.125","Text":"I\u0027ll start with the general equation of a line,"},{"Start":"06:38.125 ","End":"06:44.764","Text":"y equals mx plus B. I know 2 points on the line."},{"Start":"06:44.764 ","End":"06:54.840","Text":"I know that the line passes through 0,a over 2 and it passes through h,0."},{"Start":"06:54.840 ","End":"06:56.370","Text":"Okay. Trying the first one,"},{"Start":"06:56.370 ","End":"06:58.230","Text":"x is 0, y is a over 2,"},{"Start":"06:58.230 ","End":"07:05.955","Text":"I get a over 2 equals m times 0 plus b,"},{"Start":"07:05.955 ","End":"07:10.850","Text":"and the other equation I get is that y is 0,"},{"Start":"07:10.850 ","End":"07:16.434","Text":"x is h, 0 equals mh plus b."},{"Start":"07:16.434 ","End":"07:22.000","Text":"What we\u0027re looking for, m and b."},{"Start":"07:24.050 ","End":"07:27.680","Text":"From the first one,"},{"Start":"07:27.680 ","End":"07:29.720","Text":"because m times 0 is 0,"},{"Start":"07:29.720 ","End":"07:37.540","Text":"we immediately get that b is equal to a over 2,"},{"Start":"07:38.330 ","End":"07:44.465","Text":"and if I put that into the second equation,"},{"Start":"07:44.465 ","End":"07:54.569","Text":"then I get that 0 equals mh plus a over 2,"},{"Start":"07:54.569 ","End":"07:57.554","Text":"and what I want to extract is m,"},{"Start":"07:57.554 ","End":"08:03.510","Text":"so I get that m is equal to a. I bring the a over 2 to the other side,"},{"Start":"08:03.510 ","End":"08:06.030","Text":"so that\u0027s minus a over 2."},{"Start":"08:06.030 ","End":"08:08.430","Text":"I also divide by h,"},{"Start":"08:08.430 ","End":"08:13.245","Text":"so I get that m is minus a over 2h."},{"Start":"08:13.245 ","End":"08:19.819","Text":"Now, I have b and I have m. I can now rewrite"},{"Start":"08:19.819 ","End":"08:26.540","Text":"this equation as y equals m,"},{"Start":"08:26.540 ","End":"08:31.760","Text":"which is minus a over 2h times"},{"Start":"08:31.760 ","End":"08:38.320","Text":"x plus a over 2."},{"Start":"08:38.320 ","End":"08:44.290","Text":"I can take a over 2 outside the brackets and get that this is equal"},{"Start":"08:44.290 ","End":"08:54.485","Text":"to a over 2 times minus x over h plus 1,"},{"Start":"08:54.485 ","End":"08:56.345","Text":"and I\u0027ll rewrite the y."},{"Start":"08:56.345 ","End":"09:00.385","Text":"This is the equation of this line."},{"Start":"09:00.385 ","End":"09:05.790","Text":"For any given x,"},{"Start":"09:05.790 ","End":"09:09.235","Text":"we have that the area is 4y squared,"},{"Start":"09:09.235 ","End":"09:12.930","Text":"so the area in terms of x,"},{"Start":"09:12.930 ","End":"09:15.915","Text":"which is 4y squared,"},{"Start":"09:15.915 ","End":"09:19.950","Text":"is equal to a squared over"},{"Start":"09:19.950 ","End":"09:26.870","Text":"4 times this thing squared."},{"Start":"09:26.870 ","End":"09:33.930","Text":"I\u0027ll just write it as minus x over h plus 1 all squared,"},{"Start":"09:33.930 ","End":"09:36.270","Text":"and then I\u0027ll do it on the next line,"},{"Start":"09:36.270 ","End":"09:43.530","Text":"which equals a squared over 4 times,"},{"Start":"09:43.530 ","End":"09:46.910","Text":"and using the formula for the sum of 2 terms squared,"},{"Start":"09:46.910 ","End":"09:48.745","Text":"it\u0027s the first 1 squared,"},{"Start":"09:48.745 ","End":"09:55.570","Text":"which is x squared over h squared plus twice the product, in this case,"},{"Start":"09:55.570 ","End":"10:00.975","Text":"minus 2x over h,"},{"Start":"10:00.975 ","End":"10:03.420","Text":"and the last 1 squared,"},{"Start":"10:03.420 ","End":"10:05.830","Text":"1 squared is 1."},{"Start":"10:06.170 ","End":"10:08.370","Text":"To compute the volume,"},{"Start":"10:08.370 ","End":"10:15.650","Text":"I\u0027m going to need the integral from 0 to h of this a of x,"},{"Start":"10:15.650 ","End":"10:23.420","Text":"so what we get is that v is equal to,"},{"Start":"10:23.420 ","End":"10:28.429","Text":"I can take the a squared over 4 outside the integral sign."},{"Start":"10:28.429 ","End":"10:33.140","Text":"So from 0 to h of this thing here,"},{"Start":"10:33.140 ","End":"10:40.470","Text":"x squared over h squared minus 2x over h plus 1dx,"},{"Start":"10:44.290 ","End":"10:47.510","Text":"and this is equal to."},{"Start":"10:47.510 ","End":"10:53.660","Text":"Let\u0027s see. The integral of x squared over h squared is"},{"Start":"10:53.660 ","End":"10:59.595","Text":"just x cubed over 3h squared,"},{"Start":"10:59.595 ","End":"11:06.360","Text":"and here, I have minus the integral of 2x is just x squared over h,"},{"Start":"11:06.360 ","End":"11:13.025","Text":"and the integral of 1 is x evaluated between 0 and h,"},{"Start":"11:13.025 ","End":"11:18.860","Text":"and this is equal to a squared over 4."},{"Start":"11:18.860 ","End":"11:22.160","Text":"When I substitute 0, I get nothing."},{"Start":"11:22.160 ","End":"11:26.000","Text":"So all I have to do is substitute the h in here."},{"Start":"11:26.000 ","End":"11:33.605","Text":"What I get is h cubed over 3h squared"},{"Start":"11:33.605 ","End":"11:41.890","Text":"minus h squared over h plus h. Let\u0027s see if we can simplify this."},{"Start":"11:41.890 ","End":"11:45.070","Text":"This is a squared over 4."},{"Start":"11:45.070 ","End":"11:49.520","Text":"Now, h cubed over h squared is just h,"},{"Start":"11:49.520 ","End":"11:54.650","Text":"so this is 1/3 h. H squared over h is"},{"Start":"11:54.650 ","End":"12:02.940","Text":"h plus h. This cancels with this."},{"Start":"12:02.960 ","End":"12:08.030","Text":"This is one of those oops moment where I realized I\u0027m not going to get the right answer."},{"Start":"12:08.030 ","End":"12:15.125","Text":"I looked back and I noticed that the problem is that I forgot to carry this 4 over,"},{"Start":"12:15.125 ","End":"12:18.810","Text":"and if I had, I wouldn\u0027t have had this 4."},{"Start":"12:18.810 ","End":"12:22.930","Text":"This 4 basically just disappears all the way through,"},{"Start":"12:22.930 ","End":"12:25.510","Text":"and I\u0027m sorry about that."},{"Start":"12:25.510 ","End":"12:28.650","Text":"Really cross it out."},{"Start":"12:30.760 ","End":"12:34.530","Text":"Now, we\u0027re in good shape."},{"Start":"12:35.060 ","End":"12:39.260","Text":"What we get is that if we multiply out,"},{"Start":"12:39.260 ","End":"12:42.035","Text":"we get a squared times 1/3 h,"},{"Start":"12:42.035 ","End":"12:49.710","Text":"which equals a squared h over 3."},{"Start":"12:49.820 ","End":"12:56.670","Text":"That is exactly the answer that we were supposed to get. I\u0027ll show you."},{"Start":"12:58.130 ","End":"13:02.700","Text":"There we are. A squared h over 3,"},{"Start":"13:02.700 ","End":"13:06.780","Text":"a squared h over 3 for the volume."},{"Start":"13:06.780 ","End":"13:10.270","Text":"So we are done."}],"ID":4745},{"Watched":false,"Name":"Exercise 2","Duration":"13m 10s","ChapterTopicVideoID":4738,"CourseChapterTopicPlaylistID":3994,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"What we have here is a triangular pyramid."},{"Start":"00:03.780 ","End":"00:06.420","Text":"The base is a right triangle,"},{"Start":"00:06.420 ","End":"00:11.025","Text":"so here\u0027s the 90 degrees and the lengths are of lengths a and b."},{"Start":"00:11.025 ","End":"00:15.074","Text":"Let\u0027s call this one a and this one b,"},{"Start":"00:15.074 ","End":"00:18.570","Text":"and the height is c. What I\u0027ve done is,"},{"Start":"00:18.570 ","End":"00:21.555","Text":"I\u0027ve put the height along the x-axis."},{"Start":"00:21.555 ","End":"00:24.510","Text":"This is the x-axis going upwards."},{"Start":"00:24.510 ","End":"00:28.005","Text":"That means that this point here, the apex,"},{"Start":"00:28.005 ","End":"00:33.580","Text":"is the point where x is equal to c and this is the origin."},{"Start":"00:33.920 ","End":"00:41.405","Text":"Now, I allowed us to assume that the apex is directly above."},{"Start":"00:41.405 ","End":"00:43.160","Text":"Everything\u0027s a right angle here.,"},{"Start":"00:43.160 ","End":"00:46.370","Text":"though the same answer would be obtained even if"},{"Start":"00:46.370 ","End":"00:49.620","Text":"it wasn\u0027t directly above the right angle,"},{"Start":"00:49.620 ","End":"00:51.725","Text":"if this c was anywhere above"},{"Start":"00:51.725 ","End":"00:58.205","Text":"the base at height c. We\u0027re going to take everything as right angles."},{"Start":"00:58.205 ","End":"01:02.660","Text":"Now, we\u0027re going to use the method which last"},{"Start":"01:02.660 ","End":"01:07.325","Text":"time I called volumes by integrating cross-sections."},{"Start":"01:07.325 ","End":"01:10.625","Text":"As you can see, I\u0027ve drawn a cross-section."},{"Start":"01:10.625 ","End":"01:16.380","Text":"Now, what we\u0027re going to do is we\u0027re going to let this bit here,"},{"Start":"01:16.400 ","End":"01:24.330","Text":"be of length x in general and x is going to go from 0 to c. For each x,"},{"Start":"01:24.330 ","End":"01:29.325","Text":"we\u0027re going to get the cross-section and I\u0027ll highlight it."},{"Start":"01:29.325 ","End":"01:34.820","Text":"Note that this cross-section is parallel to the base because"},{"Start":"01:34.820 ","End":"01:39.950","Text":"all the cross-sections are perpendicular to the x-axis."},{"Start":"01:39.950 ","End":"01:42.455","Text":"That\u0027s how we require it."},{"Start":"01:42.455 ","End":"01:47.020","Text":"This is also parallel to the base and this is a right angle too."},{"Start":"01:47.020 ","End":"01:51.410","Text":"In fact, this triangle is a similar triangle to the base."},{"Start":"01:51.410 ","End":"01:54.364","Text":"Now, let\u0027s mark some more things."},{"Start":"01:54.364 ","End":"01:56.780","Text":"The area of this,"},{"Start":"01:56.780 ","End":"02:00.340","Text":"I\u0027ll call A of x."},{"Start":"02:00.340 ","End":"02:06.810","Text":"The area of this is a function of x. I also want to"},{"Start":"02:06.810 ","End":"02:14.555","Text":"note that if this is x from here to here and from here to here is c,"},{"Start":"02:14.555 ","End":"02:17.105","Text":"then this bit, the remaining bit,"},{"Start":"02:17.105 ","End":"02:18.980","Text":"say from here to here,"},{"Start":"02:18.980 ","End":"02:23.130","Text":"is going to be c minus x."},{"Start":"02:23.130 ","End":"02:28.385","Text":"That completes x to c. Now,"},{"Start":"02:28.385 ","End":"02:32.000","Text":"the way we\u0027re going to do this is we\u0027re going to, for each x,"},{"Start":"02:32.000 ","End":"02:36.230","Text":"compute what A of x is and that will be our next task because"},{"Start":"02:36.230 ","End":"02:40.910","Text":"then we\u0027ll be able to write the formula that the volume of"},{"Start":"02:40.910 ","End":"02:45.800","Text":"the pyramid V will be equal to the integral as x"},{"Start":"02:45.800 ","End":"02:53.385","Text":"goes from 0 to c of A of x, dx."},{"Start":"02:53.385 ","End":"02:57.180","Text":"As I said, our challenge is to find now,"},{"Start":"02:57.180 ","End":"02:58.830","Text":"this A of x."},{"Start":"02:58.830 ","End":"03:02.540","Text":"It\u0027s a right angle triangle and if I just knew the 2 sides,"},{"Start":"03:02.540 ","End":"03:03.890","Text":"then I could compute the area."},{"Start":"03:03.890 ","End":"03:05.495","Text":"Let\u0027s do them one at a time."},{"Start":"03:05.495 ","End":"03:14.620","Text":"This side here, we\u0027ll call it y. I\u0027m going to find y by the method of similar triangles."},{"Start":"03:14.620 ","End":"03:17.270","Text":"We have 2 similar triangles here."},{"Start":"03:17.270 ","End":"03:20.375","Text":"Let me highlight the smaller one,"},{"Start":"03:20.375 ","End":"03:26.355","Text":"which is this, this, and this."},{"Start":"03:26.355 ","End":"03:30.135","Text":"This is similar to the larger one,"},{"Start":"03:30.135 ","End":"03:39.900","Text":"which is this, this, and this."},{"Start":"03:39.900 ","End":"03:42.500","Text":"They\u0027re similar because this is 90 degrees and this is"},{"Start":"03:42.500 ","End":"03:45.650","Text":"90 degrees, and this angle is common."},{"Start":"03:45.650 ","End":"03:50.325","Text":"Once you have 2 angles the same, then they\u0027re similar."},{"Start":"03:50.325 ","End":"03:58.340","Text":"For similar triangles, then you have that the ratios of respective sides are equal."},{"Start":"03:58.340 ","End":"04:00.650","Text":"In our particular case,"},{"Start":"04:00.650 ","End":"04:05.510","Text":"what I\u0027m saying is that the base of this over the base of this,"},{"Start":"04:05.510 ","End":"04:10.025","Text":"which means y over b,"},{"Start":"04:10.025 ","End":"04:14.585","Text":"will equal to the height of this little triangle which is c minus"},{"Start":"04:14.585 ","End":"04:21.720","Text":"x over the height of the larger triangle which is this,"},{"Start":"04:21.720 ","End":"04:25.440","Text":"which is c. From this,"},{"Start":"04:25.440 ","End":"04:29.400","Text":"I can extract y and I can conclude,"},{"Start":"04:29.400 ","End":"04:30.890","Text":"and I\u0027ll write it over here,"},{"Start":"04:30.890 ","End":"04:35.330","Text":"that y is equal to just multiplying by b,"},{"Start":"04:35.330 ","End":"04:41.600","Text":"b over c times c minus x."},{"Start":"04:41.600 ","End":"04:46.490","Text":"This is going to be 1 of 2 equations because the other one,"},{"Start":"04:46.490 ","End":"04:48.935","Text":"I\u0027m going to give another name too."},{"Start":"04:48.935 ","End":"04:56.045","Text":"Let\u0027s say, we\u0027ll call that one z or zed if you\u0027re in England."},{"Start":"04:56.045 ","End":"05:00.920","Text":"Once again, we\u0027re going to have 2 similar triangles."},{"Start":"05:00.920 ","End":"05:08.850","Text":"The smaller one is going to be here, here, and here."},{"Start":"05:08.850 ","End":"05:12.900","Text":"I\u0027m going to try again."},{"Start":"05:13.760 ","End":"05:20.730","Text":"The larger triangle will be this,"},{"Start":"05:20.730 ","End":"05:25.920","Text":"this, not doing very well with this but you get the idea,"},{"Start":"05:25.920 ","End":"05:31.755","Text":"and this, so this triangle and this triangle."},{"Start":"05:31.755 ","End":"05:37.650","Text":"Once again, we can use similar triangles."},{"Start":"05:37.650 ","End":"05:40.945","Text":"If I say that the base over the base,"},{"Start":"05:40.945 ","End":"05:47.430","Text":"I\u0027ll get z over a."},{"Start":"05:47.430 ","End":"05:50.170","Text":"This will equal to the height of the small triangle,"},{"Start":"05:50.170 ","End":"05:52.210","Text":"which once again, is c minus x."},{"Start":"05:52.210 ","End":"05:54.070","Text":"It\u0027s got a bit obliterated here,"},{"Start":"05:54.070 ","End":"05:55.915","Text":"but that\u0027s c minus x,"},{"Start":"05:55.915 ","End":"06:01.590","Text":"over the height of this which is once again,"},{"Start":"06:01.590 ","End":"06:07.410","Text":"c. From here, if I extract z,"},{"Start":"06:07.410 ","End":"06:14.490","Text":"I get that z is equal to a over c,"},{"Start":"06:14.490 ","End":"06:20.535","Text":"bringing the a over, times c minus x."},{"Start":"06:20.535 ","End":"06:25.860","Text":"Now, what I want to do is compute this A of x."},{"Start":"06:25.860 ","End":"06:30.900","Text":"Now, notice it\u0027s a right angle triangle with sides y and z."},{"Start":"06:30.900 ","End":"06:37.670","Text":"The formula for the area of a right triangle is the product of the sides over 2."},{"Start":"06:37.670 ","End":"06:42.875","Text":"1/2 y times z. I\u0027ll do it as a side exercise over here."},{"Start":"06:42.875 ","End":"06:51.525","Text":"I have that A of x is equal to 1/2 times y times z."},{"Start":"06:51.525 ","End":"06:54.065","Text":"Let\u0027s see what that comes out to."},{"Start":"06:54.065 ","End":"06:58.685","Text":"This comes out to 1/2."},{"Start":"06:58.685 ","End":"07:03.850","Text":"Now, y times z, let\u0027s see."},{"Start":"07:03.850 ","End":"07:06.854","Text":"If I just multiply this by this,"},{"Start":"07:06.854 ","End":"07:16.990","Text":"I get ab over c squared and then this times this is c minus x squared."},{"Start":"07:18.800 ","End":"07:25.100","Text":"This is a formula for A of x just in terms of x and the parameters a,"},{"Start":"07:25.100 ","End":"07:29.890","Text":"b and c, which are like constants and I can plug this into here."},{"Start":"07:29.890 ","End":"07:34.594","Text":"But constants come out of the integral."},{"Start":"07:34.594 ","End":"07:37.220","Text":"What I can say is that V is equal to,"},{"Start":"07:37.220 ","End":"07:39.695","Text":"instead of just putting A of x here like this,"},{"Start":"07:39.695 ","End":"07:47.330","Text":"all this bit here is a constant and I can put it in front of the integral sign."},{"Start":"07:47.330 ","End":"07:53.295","Text":"Let me write it as ab over 2c squared."},{"Start":"07:53.295 ","End":"07:57.845","Text":"Then the integral from 0 to c of this bit,"},{"Start":"07:57.845 ","End":"08:03.690","Text":"c minus x squared dx."},{"Start":"08:06.190 ","End":"08:14.915","Text":"V is equal to ab over 2c squared and the integral of this,"},{"Start":"08:14.915 ","End":"08:16.685","Text":"well, if it was just x squared,"},{"Start":"08:16.685 ","End":"08:20.430","Text":"it would be x cubed over 3."},{"Start":"08:21.410 ","End":"08:23.910","Text":"But it\u0027s not x squared."},{"Start":"08:23.910 ","End":"08:26.210","Text":"It\u0027s c minus x instead of x,"},{"Start":"08:26.210 ","End":"08:27.890","Text":"which is a linear expression."},{"Start":"08:27.890 ","End":"08:30.680","Text":"The same thing works except that we have to"},{"Start":"08:30.680 ","End":"08:34.740","Text":"divide by the inner derivative or the coefficient of x,"},{"Start":"08:34.740 ","End":"08:38.660","Text":"so I have to divide also by minus 1,"},{"Start":"08:38.660 ","End":"08:41.465","Text":"which is the inner derivative."},{"Start":"08:41.465 ","End":"08:43.160","Text":"That\u0027s the integral."},{"Start":"08:43.160 ","End":"08:47.784","Text":"I have to take this between the limits"},{"Start":"08:47.784 ","End":"08:54.280","Text":"of 0 and c. Now,"},{"Start":"08:54.280 ","End":"08:56.480","Text":"there\u0027s a trick I\u0027d like to use."},{"Start":"08:56.480 ","End":"08:57.635","Text":"I use it often."},{"Start":"08:57.635 ","End":"09:00.245","Text":"It\u0027s that when I have a negative,"},{"Start":"09:00.245 ","End":"09:03.830","Text":"then I get rid of it by reversing the upper and"},{"Start":"09:03.830 ","End":"09:07.055","Text":"lower limits because it just changes the order of subtraction."},{"Start":"09:07.055 ","End":"09:11.855","Text":"What I\u0027m saying is that we have ab over 2c cubed."},{"Start":"09:11.855 ","End":"09:16.230","Text":"You know what, I can even take the 3 out,"},{"Start":"09:16.230 ","End":"09:22.005","Text":"so allow me to take out the 3 and put it in with the 2."},{"Start":"09:22.005 ","End":"09:24.630","Text":"I\u0027ll just put it, 3 times 2,"},{"Start":"09:24.630 ","End":"09:32.755","Text":"and what I\u0027m left with is c minus x cubed."},{"Start":"09:32.755 ","End":"09:36.890","Text":"I\u0027m reversing the limits so it\u0027s 0 at the top,"},{"Start":"09:36.890 ","End":"09:40.095","Text":"c at the bottom, and the brackets here."},{"Start":"09:40.095 ","End":"09:46.940","Text":"Yes. Once again, because of the minus I reversed the order of upper and lower limit."},{"Start":"09:46.940 ","End":"09:53.290","Text":"The 3 I just took outside of here and placed it here."},{"Start":"09:53.290 ","End":"09:55.965","Text":"Now, let\u0027s continue."},{"Start":"09:55.965 ","End":"10:06.000","Text":"What we get is that V is equal to ab over 6c cubed times."},{"Start":"10:06.000 ","End":"10:07.820","Text":"Now, let\u0027s make the substitution."},{"Start":"10:07.820 ","End":"10:14.015","Text":"First, let\u0027s substitute 0 and then we get if x is 0,"},{"Start":"10:14.015 ","End":"10:18.460","Text":"c minus 0 cubed is just c cubed."},{"Start":"10:18.460 ","End":"10:21.585","Text":"If I substitute c,"},{"Start":"10:21.585 ","End":"10:27.085","Text":"what I get is c minus c cubed which is 0 cubed."},{"Start":"10:27.085 ","End":"10:29.690","Text":"0 cubed is not necessary,"},{"Start":"10:29.690 ","End":"10:35.400","Text":"but I included it just to show we haven\u0027t forgotten because it\u0027s just 0."},{"Start":"10:41.540 ","End":"10:43.980","Text":"This shouldn\u0027t be a 3."},{"Start":"10:43.980 ","End":"10:46.755","Text":"It should be a 2. I\u0027ll correct it at once."},{"Start":"10:46.755 ","End":"10:51.420","Text":"Continuing, what I get is that V is equal"},{"Start":"10:51.420 ","End":"11:00.495","Text":"to abc cubed over 6c squared."},{"Start":"11:00.495 ","End":"11:07.680","Text":"Now, c squared goes into c cubed exactly c times."},{"Start":"11:07.680 ","End":"11:19.230","Text":"I end up with the formula that V is equal to abc over 6."},{"Start":"11:19.230 ","End":"11:22.850","Text":"That is the answer."},{"Start":"11:22.850 ","End":"11:26.830","Text":"However, I\u0027m not going to stop here because I\u0027d like to"},{"Start":"11:26.830 ","End":"11:31.010","Text":"verify this using regular geometry."},{"Start":"11:31.010 ","End":"11:33.400","Text":"Let\u0027s see if we can get the same answer another way,"},{"Start":"11:33.400 ","End":"11:37.250","Text":"then I\u0027ll be happier knowing that we haven\u0027t made a mistake."},{"Start":"11:37.590 ","End":"11:40.775","Text":"Let me do it another way."},{"Start":"11:40.775 ","End":"11:48.730","Text":"Let me get back to this diagram here and if I can fit everything in."},{"Start":"11:49.910 ","End":"11:55.170","Text":"The volume of a pyramid in general,"},{"Start":"11:55.170 ","End":"12:00.515","Text":"whatever its base is, is 1/3."},{"Start":"12:00.515 ","End":"12:02.140","Text":"I\u0027ll just write it in words,"},{"Start":"12:02.140 ","End":"12:08.950","Text":"area of base times height."},{"Start":"12:08.950 ","End":"12:11.380","Text":"That\u0027s true for all pyramids."},{"Start":"12:11.380 ","End":"12:17.270","Text":"Now, in our particular case what we have is 1/2."},{"Start":"12:17.870 ","End":"12:20.985","Text":"There again, I go with the 2s and the 3s."},{"Start":"12:20.985 ","End":"12:23.400","Text":"I meant 1/3, sorry."},{"Start":"12:23.400 ","End":"12:26.190","Text":"Apologies. First, I change a 2 into a 3,"},{"Start":"12:26.190 ","End":"12:28.140","Text":"then I change a 3 into a 2."},{"Start":"12:28.140 ","End":"12:31.515","Text":"Yes, it\u0027s 1/3 of the base times the height."},{"Start":"12:31.515 ","End":"12:39.000","Text":"The area of the base because it\u0027s a right triangle is ab over 2."},{"Start":"12:39.000 ","End":"12:43.260","Text":"This is a times b over 2,"},{"Start":"12:43.260 ","End":"12:49.110","Text":"and the height is c. If I"},{"Start":"12:49.110 ","End":"12:55.620","Text":"just write this slightly differently then the 3 with the 2 is 6, the abc."},{"Start":"12:55.620 ","End":"13:00.780","Text":"This is equal to abc over 6."},{"Start":"13:00.780 ","End":"13:05.000","Text":"This looks remarkably like what we got using integration."},{"Start":"13:05.000 ","End":"13:10.320","Text":"I\u0027m confident that this is our answer and we are done."}],"ID":4746}],"Thumbnail":null,"ID":3994}]

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