1 00:00:00,247 --> 00:00:05,175 [MUSIC]. 2 00:00:05,175 --> 00:00:09,079 Let's figure out the volume of a certain solid of revolution. 3 00:00:09,079 --> 00:00:12,179 For example, let's think about this region inside here, the region that's 4 00:00:12,179 --> 00:00:15,288 below the graph of y equals the square root of x. 5 00:00:15,288 --> 00:00:19,174 And above the graph of y equals x squared, just this little wing shaped 6 00:00:19,174 --> 00:00:25,392 region inside here and let's take that region and rotate it around the x axis. 7 00:00:25,392 --> 00:00:29,174 When I take a region in the plane, and rotate it around an axis, I get a solid 8 00:00:29,174 --> 00:00:33,125 of revolution. And the question is what's the volume of 9 00:00:33,125 --> 00:00:36,719 that solid? Well let's cut this region up into little 10 00:00:36,719 --> 00:00:41,760 tiny thin rectangles, and then, instead of rotating the whole region, at once 11 00:00:41,760 --> 00:00:47,102 around the x axis. Let's just imagine rotating each one of 12 00:00:47,102 --> 00:00:51,433 these little rectangular pieces around the x axis, and then adding up the 13 00:00:51,433 --> 00:00:56,124 volumes of pieces. What do those little rectangles become 14 00:00:56,124 --> 00:01:00,116 when I rotate them around the x axis? I'm going to take one of these little 15 00:01:00,116 --> 00:01:03,180 thin rectangles, and rotate them around the x axis. 16 00:01:03,180 --> 00:01:06,280 Well, to just get a picture here let me just isolate a single rectangle, and 17 00:01:06,280 --> 00:01:10,499 imagine just taking this. Thin rectangle, and rotating it around 18 00:01:10,499 --> 00:01:13,541 the x axis. Well what would happen when I do that, 19 00:01:13,541 --> 00:01:17,079 alright, is that I'll end up getting a shape that looks a bit like that, 20 00:01:17,079 --> 00:01:22,498 alright? That shape we're going to call a washer. 21 00:01:22,498 --> 00:01:27,290 What's the volume of a washer? Let me make this picture a little bit 22 00:01:27,290 --> 00:01:30,760 bigger, alright? Here's a big picture of a washer. 23 00:01:30,760 --> 00:01:35,188 And what are the parts of a washer? Well, the washer is got a thickness which 24 00:01:35,188 --> 00:01:39,530 allows it to be thin. So, I'm going to call it dx. 25 00:01:39,530 --> 00:01:44,696 It's got a small radius which I'll call little r and then it's got a big radius 26 00:01:44,696 --> 00:01:51,378 which I'll call big R. Now, If I were jsut thinking about whole 27 00:01:51,378 --> 00:01:56,890 scylinder of radius big r. I know the volume of that. 28 00:01:56,890 --> 00:02:02,250 The area of a circle of radius big r is pi big r squared. 29 00:02:02,250 --> 00:02:06,570 So if I thicken that up, the volume of a cylinder of radius big r is pi, big r 30 00:02:06,570 --> 00:02:12,701 squared and the thickness here is dx. But then I'm drilling out this middle 31 00:02:12,701 --> 00:02:17,517 part here to make the washer. So I'm going to subtract from this the 32 00:02:17,517 --> 00:02:23,400 volume of this inside cylinder which is pi little r squared dx. 33 00:02:23,400 --> 00:02:27,690 So this is the volume of just the washer and the part that remains and I can write 34 00:02:27,690 --> 00:02:32,760 this may be a little bit more reasonably as pi. 35 00:02:32,760 --> 00:02:38,300 Times big r squared minus little r squared times the thickness, dx. 36 00:02:38,300 --> 00:02:42,021 Now I'll use that formula, the volume formula for washer to write down the 37 00:02:42,021 --> 00:02:46,920 integral that calculates the volume of solid of revolution. 38 00:02:46,920 --> 00:02:51,270 I'm going back to this picture here. I want to figure out the volume of say, 39 00:02:51,270 --> 00:02:56,020 the washer at x. So, first think about what's the big 40 00:02:56,020 --> 00:02:59,120 radius. Well the big radius here is on the red 41 00:02:59,120 --> 00:03:03,720 curve, right that's the outside of the washer. 42 00:03:03,720 --> 00:03:09,760 So big R is the square root of x. And then the inside radius is on the 43 00:03:09,760 --> 00:03:12,880 orange curve and that's given by x squared. 44 00:03:12,880 --> 00:03:17,494 so little r is x squared. And that's the outside radius and the 45 00:03:17,494 --> 00:03:21,815 inside radius in my in my washer picture here. 46 00:03:21,815 --> 00:03:28,535 Okay, now I just gotta use the formula for the volume of that washer, right. 47 00:03:28,535 --> 00:03:33,842 And the formula for the volume of the washer there is pi times big R squared 48 00:03:33,842 --> 00:03:41,470 minus little r squared times the thickness of the washer, which is dx. 49 00:03:41,470 --> 00:03:46,961 What about the endpoints of integration? Well in my picture, x could be as small 50 00:03:46,961 --> 00:03:52,659 as zero, or as big as one. Alright, I want to add up the volumes of 51 00:03:52,659 --> 00:03:58,496 washers for x between 0 and 1. So my integral is going to go from x 52 00:03:58,496 --> 00:04:02,606 equals 0 to 1. Now all that remains is to evaluate that 53 00:04:02,606 --> 00:04:06,272 integral. So to begin, I'll simplify the integral a 54 00:04:06,272 --> 00:04:10,488 bit, this is the integral from 0 to 1, π times square root of x square, that's 55 00:04:10,488 --> 00:04:15,843 just x, minus x squared squared, that's x to the 4th. 56 00:04:16,860 --> 00:04:19,970 I've got a constant pi. I'll pull that constant out. 57 00:04:19,970 --> 00:04:23,935 This is pi times the interval from zero to one of x minus x to the fourth, and 58 00:04:23,935 --> 00:04:29,200 I've just gotta write down an anti derivative for that. 59 00:04:29,200 --> 00:04:33,945 So that's pi times anti derivative of x is x squared over 2. 60 00:04:33,945 --> 00:04:38,390 Minus anti-derivative x to the 4th, x to the 5th over 5. 61 00:04:38,390 --> 00:04:41,990 I'm evaluating this at 1, and at 0, and taking the difference. 62 00:04:41,990 --> 00:04:46,958 when I plug in 0 I don't get anything, so the answer is just whatever happens when 63 00:04:46,958 --> 00:04:52,910 I plug in 1, which is pi times one half, minus one 5th. 64 00:04:52,910 --> 00:04:55,447 And, if I like, I could write this as a single fraction, this is pi times 5 65 00:04:55,447 --> 00:05:01,466 10ths, minus 2 10ths. Or, all together, pi times 3 10ths, so 66 00:05:01,466 --> 00:05:09,780 that's the volume of the solid of revolution. 67 00:05:09,780 --> 00:05:11,170 And we did it.