1 00:00:00,242 --> 00:00:05,006 [MUSIC]. 2 00:00:05,006 --> 00:00:08,605 We've done a bunch of area calculations at this point, so let's move up a 3 00:00:08,605 --> 00:00:13,394 dimension and think about volume. Let me ask really loudly. 4 00:00:13,394 --> 00:00:18,122 What is volume? Of course, what is area? 5 00:00:18,122 --> 00:00:20,091 Right? Area is the thing that integral's 6 00:00:20,091 --> 00:00:22,176 calculate. So instead of trying to give a definition 7 00:00:22,176 --> 00:00:25,738 of volume, lets just focus on how were going to compute volume, whatever it is. 8 00:00:25,738 --> 00:00:30,130 And, were going to comput volume by integrating the volume of little tiny 9 00:00:30,130 --> 00:00:35,323 pieces that we cut up our objects into. Let's work through an example. 10 00:00:35,323 --> 00:00:38,426 Well, here's a cone, I want to compute the volume of this cone. 11 00:00:38,426 --> 00:00:43,236 And just for concreteness, let's suppose that the cone has a height of 2 units, 12 00:00:43,236 --> 00:00:48,485 and a radius of 1 unit, and I want to compute its volume. 13 00:00:48,485 --> 00:00:52,069 I want to set this thing up as an integration problem, I want to cut this 14 00:00:52,069 --> 00:00:57,294 thing up into slices, and add up the volumes of the little slices. 15 00:00:57,294 --> 00:01:03,216 So, imagine that I take this, and I just slice it up into a bunch of really thin 16 00:01:03,216 --> 00:01:07,555 pieces. Now, what's the volume of a slice? 17 00:01:07,555 --> 00:01:10,977 Well, that slice, is practically a little tiny cylinder. 18 00:01:10,977 --> 00:01:15,410 So what's the volume of one of these very thin cylinders? 19 00:01:15,410 --> 00:01:18,950 Well let's suppose the cylinder has some radius r, and I'm going to call its 20 00:01:18,950 --> 00:01:24,129 height, an its height should be really thin, so I'll call its height dx. 21 00:01:24,129 --> 00:01:29,455 And in that case the volume of the cylinder is, what's the area of the top? 22 00:01:29,455 --> 00:01:32,991 It's phi r squared times its height, which is dx. 23 00:01:32,991 --> 00:01:38,275 So this here will be our formula for the volume of one of these thin cylinders. 24 00:01:38,275 --> 00:01:42,652 And once again, we're seeing the importance of the differentials. 25 00:01:42,652 --> 00:01:45,030 Right? We're using the differentials to write 26 00:01:45,030 --> 00:01:48,275 down an equation for the volume of the thin cylinder. 27 00:01:48,275 --> 00:01:51,489 Well any how, lets put this together in interval. 28 00:01:51,489 --> 00:01:54,561 But to set up the integral to calculate the volume of this cone, I first have to 29 00:01:54,561 --> 00:01:58,635 figure out the volume of one of the slices that make up this cone. 30 00:01:58,635 --> 00:02:02,990 To do that, let's turn this cone on it's side, and let's imagine that this is the 31 00:02:02,990 --> 00:02:07,196 point 0, 0. So let's make this the Y axis, and the X 32 00:02:07,196 --> 00:02:13,645 axis will head in that direction. Then, what do I have here? 33 00:02:13,645 --> 00:02:19,537 Well, this line, is the line y equals x over 2. 34 00:02:19,537 --> 00:02:25,326 Now, I'll use that to figure out the radius of one of those slices. 35 00:02:25,326 --> 00:02:29,706 Pick some value of X, and let's think about one of the slices, one of these 36 00:02:29,706 --> 00:02:34,924 thin cylinders that I'm using to build up the big cone. 37 00:02:34,924 --> 00:02:40,483 I don't know the radius of that that particular cylinder. 38 00:02:40,483 --> 00:02:45,163 Well, it's this distance here, and I know the Y coordinate up there, the Y 39 00:02:45,163 --> 00:02:51,613 coordinate is x over 2, so the radius of this cylinder is x over 2. 40 00:02:51,613 --> 00:02:59,550 I can write down the volume of that disk. The volume's Phi times the radius, which 41 00:02:59,550 --> 00:03:06,157 is x over 2 square, times the thickness of that disk, which is DX. 42 00:03:06,157 --> 00:03:11,191 And now I'm integrating this from where to where? 43 00:03:11,191 --> 00:03:14,196 Well, X goes from 0 all the way down to 2. 44 00:03:14,196 --> 00:03:18,923 So this'll be the integral from x equals 0 to 2. 45 00:03:18,923 --> 00:03:22,509 And this integral will calculate the volume of my cone. 46 00:03:22,509 --> 00:03:25,419 And now we use the fundamental theorem of calculus. 47 00:03:25,419 --> 00:03:29,939 I can simplify this a bit, I've got, some constants here I can pull out. 48 00:03:29,939 --> 00:03:32,647 Right? I can pull out the phi, and I can pull 49 00:03:32,647 --> 00:03:37,822 out 1/2 squared. So this is phi over 4 times the integral 50 00:03:37,822 --> 00:03:44,210 from 0 to 2, of just x squared dx. Now what's an anti-derivative for x 51 00:03:44,210 --> 00:03:47,264 squared? Well, x to the 3rd over 3, and then I'm 52 00:03:47,264 --> 00:03:53,101 integrating that, I'm evaluating that at 0 and at 2, and taking the difference. 53 00:03:53,101 --> 00:03:56,629 So when I plug in zero don't get anything, but when I plug in 2, I get 8, 54 00:03:56,629 --> 00:04:02,521 that's 2 cubed over 3, and then minus 0, so this is the volume of the cone. 55 00:04:02,521 --> 00:04:12,076 Which, I can simplified somewhat, I can write this as 2 times phi over 3. 56 00:04:12,076 --> 00:04:15,521 We did it! Well, I mean, you could have just looked 57 00:04:15,521 --> 00:04:18,440 up this formula at the back of some calculus textbook. 58 00:04:18,440 --> 00:04:20,758 But that's beside the point. Right? 59 00:04:20,758 --> 00:04:24,245 The point here, is that we're teaching you the secrets of volume. 60 00:04:24,245 --> 00:04:28,007 You can now verify that the formulas in the back of the textbook are correct, and 61 00:04:28,007 --> 00:04:33,118 that's part of the fun of mathematics. We're not just giving you information, 62 00:04:33,118 --> 00:04:36,590 we're giving you information, and the tools with which to verify that that 63 00:04:36,590 --> 00:04:38,953 information is correct.