1 00:00:00,226 --> 00:00:04,926 [MUSIC]. 2 00:00:04,926 --> 00:00:08,769 Let's compute the volume of a bead but first we have to figure out how we're 3 00:00:08,769 --> 00:00:12,693 going to build the bead as a solid of revolution. 4 00:00:12,693 --> 00:00:18,450 So start with the X, Y plane and I'm going to draw a circle of radius two in 5 00:00:18,450 --> 00:00:23,878 the plane. So, its a circle of radius 2 if I take 6 00:00:23,878 --> 00:00:28,880 this and rotate around the Y axis and I got a sphere, but I want to bead, so I am 7 00:00:28,880 --> 00:00:35,687 going to imagine drilling out a hole around the Y axis. 8 00:00:35,687 --> 00:00:40,391 So to do that we can draw these vertical lines here. 9 00:00:40,391 --> 00:00:46,481 This is the line x equals 1. This is the line, x equals minus 1. 10 00:00:46,481 --> 00:00:53,449 [NOISE] Okay, now if I just take the the region in here, and rotate that region 11 00:00:53,449 --> 00:01:00,999 around the y axis, that gives me a bead. It's the same thing as giving me a 12 00:01:00,999 --> 00:01:05,023 sphere. With a hole drilled out around the y 13 00:01:05,023 --> 00:01:07,320 axis. We've got a choice. 14 00:01:07,320 --> 00:01:10,435 We can attack this problem with shells or with washers. 15 00:01:10,435 --> 00:01:14,827 If I want to do it with washers, since I'm rotating around the y axis, I'd be 16 00:01:14,827 --> 00:01:21,411 cutting this into horizontal strips. Because if I take one of these horizontal 17 00:01:21,411 --> 00:01:25,947 strips and rotate it around the y axis, that gives me a washer, and that looks 18 00:01:25,947 --> 00:01:32,810 like here is my here's my big washer. And that's, that works. 19 00:01:32,810 --> 00:01:35,834 There's nothing wrong with that, but it turns out that if you do it the way the 20 00:01:35,834 --> 00:01:41,250 interval that you get is kind of yucky. But let's try it instead with shells. 21 00:01:41,250 --> 00:01:45,940 If I go with shells, then I'd be cutting into vertical strips. 22 00:01:45,940 --> 00:01:50,924 Parallel to the axis of rotation. And when I take one of those little 23 00:01:50,924 --> 00:01:57,969 vertical strips and rotate it around. Well then exactly what I've got is, is 24 00:01:57,969 --> 00:02:03,515 one of these shell pieces. Now we invoke the formula for volume of a 25 00:02:03,515 --> 00:02:09,313 shell in terms of dx. So the formula is 2 pi, the radius of the 26 00:02:09,313 --> 00:02:13,061 shell. Which if I think of just one of these 27 00:02:13,061 --> 00:02:17,201 pieces say at x, the radius is x, since that's how far it is from the axis of 28 00:02:17,201 --> 00:02:22,715 rotation. The height of the shell, which I haven't 29 00:02:22,715 --> 00:02:27,760 figured out yet, times the thickness of the shell, which is then dx. 30 00:02:27,760 --> 00:02:30,355 Now I need to determine the height of the shell. 31 00:02:30,355 --> 00:02:37,006 So that big sphere has radius 2. And this orange curve is that circle with 32 00:02:37,006 --> 00:02:41,708 radius 2. So I can write down that that orange 33 00:02:41,708 --> 00:02:49,590 curve is y equals the square root of 2 squared minus x squared. 34 00:02:49,590 --> 00:02:53,278 And, that's almost enough information to tell me the height of the shell. 35 00:02:53,278 --> 00:02:56,983 Let me write down what the height of this shell is, then, the height of the shell 36 00:02:56,983 --> 00:03:01,724 is well, how tall is this thing. Well here, from here to here, is the 37 00:03:01,724 --> 00:03:06,146 square root of 2 squared minus x squared, and then it's the same distance, down to 38 00:03:06,146 --> 00:03:12,520 the other side here. So, the height, is twice this quantity. 39 00:03:12,520 --> 00:03:17,812 So, I can write that down; 2 Pi X times the height of the shell, which is 2 times 40 00:03:17,812 --> 00:03:24,590 the square root of 2 squared, which is 4 minus X squared. 41 00:03:24,590 --> 00:03:28,920 That's the height of the shell, and then the thickness of the shell dx. 42 00:03:28,920 --> 00:03:31,690 Let's think about the endpoints of integration. 43 00:03:31,690 --> 00:03:35,840 This purple line, right, was the line x equals 1. 44 00:03:35,840 --> 00:03:40,230 And the orange circle, right, has radius 2. 45 00:03:40,230 --> 00:03:43,646 So, x can take values between 1 and 2, and that tells me what my end points of 46 00:03:43,646 --> 00:03:47,739 integration should be. So, this is the volume of just one of 47 00:03:47,739 --> 00:03:51,399 those shells, and I'm going to integrate that x goes from 1 to 2, to get all of 48 00:03:51,399 --> 00:03:58,169 the shells, and add them all up. Finally, we integrate it's 2 pi times the 49 00:03:58,169 --> 00:04:06,050 integral from 1 to 2, and I'll write 2x, the square root of 4 minus x squared dx. 50 00:04:06,050 --> 00:04:10,886 And this is really where shells shine, I set this up now, and you can see I can 51 00:04:10,886 --> 00:04:17,606 make a u substitution here. So I'll set u equal 4 minus x squared, du 52 00:04:17,606 --> 00:04:24,730 in that case is minus 2x dx, put a minus sign there and a minus sign there. 53 00:04:24,730 --> 00:04:32,220 Now I've got du square root of u, so this integral becomes minus 2 pi. 54 00:04:33,270 --> 00:04:37,852 Square of u d u and I can change the endpoints, u goes from when x is 1, u is 55 00:04:37,852 --> 00:04:42,836 3. When x is 2, u is 0, and this is negative 56 00:04:42,836 --> 00:04:49,058 from 3 to 0, so I can rewrite this as 2 pi the integral from 0 to 3 of the square 57 00:04:49,058 --> 00:05:02,780 root of u du. Now, that's not hard to do. 58 00:05:02,780 --> 00:05:08,430 I can rewrite that integrand as 2 pi the interval of u to the 1/2 du from 0 to 3. 59 00:05:08,430 --> 00:05:11,480 And I can easily anti differentiate that using the power rule. 60 00:05:11,480 --> 00:05:17,550 So this is 2 pi, u to the 3 halves over 3 halves evaluated at 0 and 3. 61 00:05:17,550 --> 00:05:21,942 Now, when I plug in zero I just get zero, so my answer is whatever I get when I 62 00:05:21,942 --> 00:05:28,736 plug in three. So the volume is 2 pi times 3 to the 3 63 00:05:28,736 --> 00:05:36,386 halves power over 3 halves. Now, instead of writing 3 to the 3 halves 64 00:05:36,386 --> 00:05:40,922 power, I could write this as 2 pi over 3 halves times 3 times the square root of 65 00:05:40,922 --> 00:05:44,472 3. Right, that's the same as 3 to the 3 66 00:05:44,472 --> 00:05:48,870 halves power. But look, this 3 and this 3 cancel and 67 00:05:48,870 --> 00:05:58,130 I'm dividing by a half here, so this ends up being 4 pi times the square root of 3. 68 00:05:58,130 --> 00:06:03,800 This is the volume of our bead. We did it, but is the answer reasonable? 69 00:06:03,800 --> 00:06:06,680 Well to think about how reasonable our calculation for the volume of the bead 70 00:06:06,680 --> 00:06:08,890 was. We could think about how we build the 71 00:06:08,890 --> 00:06:11,742 bead, alright we build this thing by starting with the sphere of radius two 72 00:06:11,742 --> 00:06:15,680 and then drilling out a tube of radius one. 73 00:06:15,680 --> 00:06:19,100 So if I start with the sphere of radius two and I subtract even more volume 74 00:06:19,100 --> 00:06:22,691 right, this cylinder of height four extends above the top of this sphere and 75 00:06:22,691 --> 00:06:28,680 below the bottom of the sphere. So this is even more material than I 76 00:06:28,680 --> 00:06:32,544 removed to get the bead. Well that means that the volume of this 77 00:06:32,544 --> 00:06:35,693 sphere minus the volume of the cylinder's got to be even a little bit less than the 78 00:06:35,693 --> 00:06:39,843 volume of the bead, right? Because this is a little bit more, 79 00:06:39,843 --> 00:06:44,430 material than the material that we removed to get the bead. 80 00:06:44,430 --> 00:06:48,490 Well how big are these three things? This thing here is a 4 3rds pie radius 81 00:06:48,490 --> 00:06:54,186 cubed thats the volume of a sphere minus this thing here is pie r squared times 82 00:06:54,186 --> 00:07:02,175 height is the volume of a cylinder. I could simplify this a bit this is a 4 83 00:07:02,175 --> 00:07:07,131 times 8 so that's 32 3rd pi minus 4 pi, and I can simplify that a little bit 84 00:07:07,131 --> 00:07:15,450 further even, that's 20 3rd pi. And what do we get to the volume of the 85 00:07:15,450 --> 00:07:22,630 big bead we calculated that to be 4 pi square root of 3. 86 00:07:22,630 --> 00:07:27,950 And yeah, this quanitiy here is just a bit smaller than this quantity here. 87 00:07:27,950 --> 00:07:29,920 So, it seems like our answer's reasonable.