1 00:00:00,025 --> 00:00:05,039 [MUSIC]. 2 00:00:05,039 --> 00:00:08,321 We're going to compute the volume of a sphere. 3 00:00:08,321 --> 00:00:15,229 Well here I got a clay sphere. And if I measure if I find that it's 4 00:00:15,229 --> 00:00:24,270 diameter its about 4.2. The diameter is about 4.2 centimeters. 5 00:00:24,270 --> 00:00:30,074 So its radius, say 2.1 centimeters. Now, we're going to take this sphere and 6 00:00:30,074 --> 00:00:35,255 crush it flat. And I want to make it so that it's about 7 00:00:35,255 --> 00:00:41,000 1 centimeter thick. Now I'll estimate the volume of that 8 00:00:41,000 --> 00:00:44,049 blob. So we measure this thing's maybe a bit 9 00:00:44,049 --> 00:00:51,150 under 7 centimeters by a bit over 5 centimeters, and it's a centimeter thick. 10 00:00:51,150 --> 00:00:56,875 So I'd guess that the volume is about 35 cubic centimeters. 11 00:00:56,875 --> 00:01:00,934 But that wasn't all that precise. But we can do the calculation precisely 12 00:01:00,934 --> 00:01:05,820 by taking the sphere, slicing it into discs thin cylinders. 13 00:01:05,820 --> 00:01:08,988 Integrating those thin cylinders right and that'll give us the exact volume of 14 00:01:08,988 --> 00:01:13,240 the sphere. So, we're going to take the sphere and 15 00:01:13,240 --> 00:01:20,670 I'm going to slice this thing up into lots and lots of little tiny cylinders. 16 00:01:20,670 --> 00:01:24,550 We have to figure out the radius of each of those thin cylinders. 17 00:01:24,550 --> 00:01:26,380 Well let me draw the picture again, alright? 18 00:01:26,380 --> 00:01:32,716 Here's the plane, this is the y axis and the x axis, and I've got my circle in the 19 00:01:32,716 --> 00:01:39,240 xy plane. And here is the sphere sitting in space. 20 00:01:39,240 --> 00:01:46,701 And in some height call that height y. I got a thin cylinder, what I want to do 21 00:01:46,701 --> 00:01:52,480 is to figure the radius of that thin cylinder. 22 00:01:53,550 --> 00:01:57,642 Really what that amounts is figuring out the coordinates of this point here. 23 00:01:57,642 --> 00:01:59,290 Right. I want to figure out what that x 24 00:01:59,290 --> 00:02:02,880 coordinate is, so that x, y lands on that circle. 25 00:02:02,880 --> 00:02:06,534 Well I know the equation for the circle in the xy plane. 26 00:02:06,534 --> 00:02:10,694 That's x squared plus y squared equals r squared, r is the radius of the sphere 27 00:02:10,694 --> 00:02:15,880 that I'm thinking about. And I can solve this for x. 28 00:02:15,880 --> 00:02:21,740 X equals the square root of r squared minus y squared. 29 00:02:21,740 --> 00:02:28,310 This is the radius of this little tiny thin blue cylinder here. 30 00:02:28,310 --> 00:02:30,942 And I'm going to cut up my sphere into a whole bunch of these little tiny 31 00:02:30,942 --> 00:02:33,743 cylinders. And add up the volume of those cylinders 32 00:02:33,743 --> 00:02:36,225 to give me the volume of the whole sphere. 33 00:02:36,225 --> 00:02:40,480 Now I want to figure out the volume of that thin cylinder. 34 00:02:40,480 --> 00:02:46,780 Now, what's the volume of this slice? What's the volume of that thin cylinder. 35 00:02:46,780 --> 00:02:51,310 Well, it's pi times the radius of that cylinider, which is given by this. 36 00:02:52,860 --> 00:02:56,820 So the radius is the square root of r squared minus y squared and it's that 37 00:02:56,820 --> 00:03:00,955 radius squared. Right, this is pi radius of cylinder 38 00:03:00,955 --> 00:03:04,707 squared, times the height of that cylinder, well the height of that 39 00:03:04,707 --> 00:03:09,913 cylinder is dy. So this is giving me the volume of this 40 00:03:09,913 --> 00:03:13,994 thin cylinder. We have to think about the end points of 41 00:03:13,994 --> 00:03:18,720 integration. Well how big and how small can y be? 42 00:03:18,720 --> 00:03:23,676 Well y goes all the way down here to minus r and y goes all the way up here to 43 00:03:23,676 --> 00:03:28,870 r. So when I set up my integral and I want 44 00:03:28,870 --> 00:03:33,400 to integrate this, y goes from minus r to r. 45 00:03:34,900 --> 00:03:38,780 Now we can integrate. So first I'll simplify the integrand. 46 00:03:38,780 --> 00:03:43,720 This is the integral y goes from minus r to r of pi times the square root squared, 47 00:03:43,720 --> 00:03:50,458 just r squared minus y squared dy. Now I can simplify this a bit more. 48 00:03:50,458 --> 00:03:54,420 I've got a constant, always pull that constant out. 49 00:03:54,420 --> 00:04:00,660 This is pi times the integral from minus r to r of r squared minus y squared dy. 50 00:04:00,660 --> 00:04:02,625 Now I've gotta cook up an anti-derivative of this. 51 00:04:02,625 --> 00:04:05,570 Alright, to use a fundamental term of calculus. 52 00:04:05,570 --> 00:04:09,104 So this'll be pi times, so what's an anti-derivative of just r squared, r is 53 00:04:09,104 --> 00:04:13,933 just a constant here. So an anti-derivative of that will just 54 00:04:13,933 --> 00:04:18,581 be r squared times y minus an anti-derivative of y squared is y cubed 55 00:04:18,581 --> 00:04:25,544 over three. I'll be ecaluating that from r and minus 56 00:04:25,544 --> 00:04:28,596 r. But if you think about it a bit I can get 57 00:04:28,596 --> 00:04:32,151 away with using from 0 to r if I multiply this by 2. 58 00:04:32,151 --> 00:04:34,740 And that'll simplify the calculation somewhat. 59 00:04:35,880 --> 00:04:42,080 Alright, now I was going to plug in r and plug in 0. 60 00:04:42,080 --> 00:04:50,960 So, when I plug in r, I get r squared times r minus r cubed over 3. 61 00:04:50,960 --> 00:04:55,040 And when I plug in 0 I just get 0. So I don't have to subtract that. 62 00:04:55,040 --> 00:04:56,850 And I can keep working on this. What do I do? 63 00:04:56,850 --> 00:05:07,653 I got 2 pi r cubed minus 1 3rd r cubed. Let's call this 3 3rd's cubed. 64 00:05:07,653 --> 00:05:14,013 Alright, so this is now 2 3rd r cubed times 2 pi or this ends up being 4 3rd's 65 00:05:14,013 --> 00:05:20,640 pi r cubed. We did it, yeah the volume of a sphere 66 00:05:20,640 --> 00:05:27,233 here of a radius r. We just calculated it, its 4 3rds pi r 67 00:05:27,233 --> 00:05:34,390 cubed and we could reach into even higher dimensions. 68 00:05:34,390 --> 00:05:38,062 For example, you could try to write down an integral that calculates the volume of 69 00:05:38,062 --> 00:05:41,100 a hyper sphere. What's a hyper sphere? 70 00:05:41,100 --> 00:05:44,700 Well, that's the higher dimensional analog of the regular old sphere, a 71 00:05:44,700 --> 00:05:53,890 sphere is sort of a stack of circles. So a hypersphere will be a stack of 72 00:05:53,890 --> 00:06:03,631 spheres but in the fourth dimension.