1 00:00:00,264 --> 00:00:04,946 [MUSIC]. 2 00:00:04,946 --> 00:00:10,081 We can cut a three dimensional object up into shapes beyond just slabs, disks, 3 00:00:10,081 --> 00:00:13,643 washers. Specifically instead of building an 4 00:00:13,643 --> 00:00:18,625 object out of disks, instead of building it by stacking up disks of varying radii. 5 00:00:18,625 --> 00:00:22,260 We can build the object by wrapping it with these shells. 6 00:00:22,260 --> 00:00:25,970 This shell thing is supposed to be like a cardboard mailing tube. 7 00:00:25,970 --> 00:00:30,742 It's a cylinder sort of a thin wall. But how are shells different from 8 00:00:30,742 --> 00:00:33,800 washers? Well there are some similarities. 9 00:00:33,800 --> 00:00:37,686 Both shells and washers are things that you get by taking cylinders and drilling 10 00:00:37,686 --> 00:00:42,912 out other cylinders. But for washers, the thin dimension is 11 00:00:42,912 --> 00:00:46,422 its height. And for shells, the thin dimension is 12 00:00:46,422 --> 00:00:51,434 sort of the thickness of this wall. We need to write down a formula for the 13 00:00:51,434 --> 00:00:56,720 volume of a shell in terms of it's height, it's radius and that thin dr. 14 00:00:56,720 --> 00:00:59,660 Let me label those parts on the diagram here. 15 00:00:59,660 --> 00:01:04,460 So the height of the shell. I'll call h, the thickness of the shell. 16 00:01:04,460 --> 00:01:09,715 I'll call that dr, and the radius of the shell I'll just call r. 17 00:01:09,715 --> 00:01:14,732 Now, if I get out my scissors of mathematics here and cut the shell. 18 00:01:14,732 --> 00:01:19,282 You might imagine unrolling the shell, and then you just end up with a slab like 19 00:01:19,282 --> 00:01:22,590 this. How big is this slab? 20 00:01:23,990 --> 00:01:28,990 Well, the slab's height is still h. The thickness of this slab is dr. 21 00:01:28,990 --> 00:01:32,202 And how long is this slab? Well the slab's as long as the 22 00:01:32,202 --> 00:01:36,530 circumference of the circle , which is 2 pi r. 23 00:01:36,530 --> 00:01:41,250 And multiplying these three dimensions will give me the volume of the slab. 24 00:01:41,250 --> 00:01:50,330 So you're guessing that the volume of the shell or the slab is 2 pi rh dr. 25 00:01:50,330 --> 00:01:54,040 We can say something a bit more precise. Well here's the shell again. 26 00:01:54,040 --> 00:01:58,011 But instead of calling this dr. Let's think of this as a big cylinder, 27 00:01:58,011 --> 00:02:02,627 with a little cylinder drilled out. So both the cylinders have the same 28 00:02:02,627 --> 00:02:06,559 heighth, h. But let's say the big cylinder has a 29 00:02:06,559 --> 00:02:10,960 radius, r. And the little cylinder has a radius 30 00:02:10,960 --> 00:02:14,670 little r. Let's write down the formula for the 31 00:02:14,670 --> 00:02:20,372 volume of the shell in between. So now the volume of the shell, right, is 32 00:02:20,372 --> 00:02:25,332 the volume of the big cylinder pi big R squared h. 33 00:02:25,332 --> 00:02:30,727 Minus the volume of the little cylinder that I drilled out, pi little r squared 34 00:02:30,727 --> 00:02:33,708 h. In effect you'll recognize that this, 35 00:02:33,708 --> 00:02:38,354 this makes it look a lot like a washer. But here's the difference, little r is 36 00:02:38,354 --> 00:02:42,128 really close to big R. Right, I'm thinking of their difference 37 00:02:42,128 --> 00:02:44,923 as dr. Right, very thin walled cylinder. 38 00:02:44,923 --> 00:02:50,340 Well lets rewrite this volume anyhow. I'm going to factor this out pi. 39 00:02:50,340 --> 00:02:55,594 Big R squared, minus little r squared, h. Now let's rewrite this. 40 00:02:55,594 --> 00:03:01,930 This is a difference of squares, so I can write that as pie, big R plus little r. 41 00:03:03,010 --> 00:03:09,302 Big R minus little r, h. And thinking of the difference in these 42 00:03:09,302 --> 00:03:14,830 radii as being really small. So I'm going to rewrite this term as dr. 43 00:03:14,830 --> 00:03:20,642 So this is pi, big R plus little r, and I'll write h dr. 44 00:03:20,642 --> 00:03:25,800 Now what do I do with big R plus little r? 45 00:03:25,800 --> 00:03:29,886 Well this is the sum of these two radia. I mean, if you think about big R and 46 00:03:29,886 --> 00:03:35,520 little r as being really close together. I might as well just write this as 2 47 00:03:35,520 --> 00:03:39,630 little r. So I'll write this 2 pi little r. 48 00:03:39,630 --> 00:03:45,070 That's 2 little r's, if I'm imagining big R and little r to be close together, h 49 00:03:45,070 --> 00:03:48,470 dr. And you'll see this is the same as the 50 00:03:48,470 --> 00:03:51,810 formula that I got before for the volume of a shell. 51 00:03:51,810 --> 00:03:55,202 This formula, the formula for the volume of a shell, to write down an integral 52 00:03:55,202 --> 00:03:59,420 that calculates the volume of a solid of revolution. 53 00:03:59,420 --> 00:04:04,146 As an example, let's take this region. This is the graph y equals 1 minus x 54 00:04:04,146 --> 00:04:07,511 squared. And I just want to consider y greater and 55 00:04:07,511 --> 00:04:11,360 equal to 0, so consider this region inside here. 56 00:04:11,360 --> 00:04:14,160 So let me write y less than or equal to 1 minus x. 57 00:04:14,160 --> 00:04:17,744 So just to emphasize this, it's this region inside here, and let's take that, 58 00:04:17,744 --> 00:04:21,520 and let's rotate that region around the y axis. 59 00:04:21,520 --> 00:04:25,414 To give myself this solid of revolution, I'm going to calculate the volume of that 60 00:04:25,414 --> 00:04:29,193 solid. Admittedly, we could do this problem 61 00:04:29,193 --> 00:04:34,787 without using shells, but using washers. In that case, I'd be chopping the region 62 00:04:34,787 --> 00:04:39,800 up into horizontal slices. And each of these horizontal slices, when 63 00:04:39,800 --> 00:04:43,690 I rotate it around the y axis would give me a disc. 64 00:04:43,690 --> 00:04:47,265 And then I just integrate the volumes of those discs to give me the volume of the 65 00:04:47,265 --> 00:04:51,810 solid of revolution. But let's try it with shells. 66 00:04:51,810 --> 00:04:56,334 In this case, I'll cut it up into vertical slices and I'll rotate these 67 00:04:56,334 --> 00:05:01,540 vertical slices around the y axes to produce shells. 68 00:05:01,540 --> 00:05:04,170 And I'll integrate the volumes of those shells. 69 00:05:04,170 --> 00:05:07,580 To give me the volume of the whole solid of revolution. 70 00:05:07,580 --> 00:05:10,660 Now, we're going to use the formula, for the volume of a shell. 71 00:05:10,660 --> 00:05:17,686 Let's just look at one of those shells. So, here's one of these thin rectangles, 72 00:05:17,686 --> 00:05:24,527 I imagine, rotating this thin rectangle, around. 73 00:05:24,527 --> 00:05:33,770 The the y axis to produce this shell. Now, how how tall is this shell? 74 00:05:33,770 --> 00:05:39,400 Well, the height of that shell is given right here. 75 00:05:39,400 --> 00:05:45,646 It's 1 minus x squared. The radius of the shell is x and the 76 00:05:45,646 --> 00:05:51,480 thickness of that shell is dx. So, I put all those pieces together. 77 00:05:51,480 --> 00:05:57,295 I can write down the volume of just that one, one shell. 78 00:05:57,295 --> 00:06:03,310 And it'll be 2 pi, the radius of that shell. 79 00:06:03,310 --> 00:06:09,030 The height of that shell, the thickness of that shell. 80 00:06:09,030 --> 00:06:12,060 Now I have to worry about the end points of integration. 81 00:06:12,060 --> 00:06:16,745 You might think that I'd be integrating from x equals minus 1 to 1. 82 00:06:16,745 --> 00:06:21,590 But actually I only have to integrate from x equals 0 to 1. 83 00:06:21,590 --> 00:06:26,012 Because rotating a thin rectangle over here also accounts for stuff happening 84 00:06:26,012 --> 00:06:29,970 over here. So it's enough just to integrate from x 85 00:06:29,970 --> 00:06:34,712 goes from 0 to 1. So I'll write that here, x goes from 0 to 86 00:06:34,712 --> 00:06:37,316 1. This integral will calculate the volume 87 00:06:37,316 --> 00:06:41,240 of my solid of revolution. Finally, it's just a matter of evaluating 88 00:06:41,240 --> 00:06:44,350 the integral with the fundamental theorem of calculus. 89 00:06:44,350 --> 00:06:48,605 And to do this integral. Well I'll pull out these constants, 2 pi, 90 00:06:48,605 --> 00:06:54,848 the integral from 0 to 1 of x minus x cubed, when I combine these two terms. 91 00:06:54,848 --> 00:06:58,877 Now I just have to write down an antiderivative here. 92 00:06:58,877 --> 00:07:03,728 2 pi, antiderivative of x is x squared over 2, anti-derivative x cubed is x to 93 00:07:03,728 --> 00:07:09,280 the 4th over 4. And I'm evaluating this at x equals, 1, 94 00:07:09,280 --> 00:07:13,210 and x equals 0. And I'm subtracting, alright? 95 00:07:13,210 --> 00:07:15,410 That's the fundamental theorem of calculus. 96 00:07:15,410 --> 00:07:19,094 But when I plug in 0, I just get 0. So the answer is just whatever I get when 97 00:07:19,094 --> 00:07:24,650 I plug in 1. And, that's 2 pi, a half minus a 4th, and 98 00:07:24,650 --> 00:07:32,085 a half minus a 4th is a 4th. So, 2 pi times a 4th, and 2 times a 4th 99 00:07:32,085 --> 00:07:37,221 is a half. So, pi over 2, is the volume of my solid 100 00:07:37,221 --> 00:07:40,760 of revolution. We did it.