WEBVTT 00:00:00.000 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.950 Commons license. 00:00:03.950 --> 00:00:05.990 Your support will help MIT OpenCourseWare 00:00:05.990 --> 00:00:09.930 continue to offer high quality educational resources for free. 00:00:09.930 --> 00:00:12.550 To make a donation or to view additional materials 00:00:12.550 --> 00:00:16.150 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.150 --> 00:00:21.580 at ocw.mit.edu. 00:00:21.580 --> 00:00:23.630 PROFESSOR: OK. 00:00:23.630 --> 00:00:28.710 Now, today we get to move on from integral formulas 00:00:28.710 --> 00:00:34.620 and methods of integration back to some geometry. 00:00:34.620 --> 00:00:36.870 And this is more or less going to lead 00:00:36.870 --> 00:00:39.270 into the kinds of tools you'll be using 00:00:39.270 --> 00:00:43.120 in multivariable calculus. 00:00:43.120 --> 00:00:45.420 The first thing that we're going to do today 00:00:45.420 --> 00:00:58.220 is discuss arc length. 00:00:58.220 --> 00:01:02.940 Like all of the cumulative sums that we've worked on, 00:01:02.940 --> 00:01:06.710 this one has a storyline and a picture associated to it, 00:01:06.710 --> 00:01:09.360 which involves dividing things up. 00:01:09.360 --> 00:01:11.690 If you have a roadway, if you like, 00:01:11.690 --> 00:01:18.290 and you have mileage markers along the road, like this, 00:01:18.290 --> 00:01:22.490 all the way up to, say, s_n here, 00:01:22.490 --> 00:01:28.740 then the length along the road is described by this parameter, 00:01:28.740 --> 00:01:30.560 s. 00:01:30.560 --> 00:01:32.930 Which is arc length. 00:01:32.930 --> 00:01:37.680 And if we look at a graph of this sort of thing, 00:01:37.680 --> 00:01:42.160 if this is the last point b, and this is the first point a, 00:01:42.160 --> 00:01:47.120 then you can think in terms of having points above x_1, x_2, 00:01:47.120 --> 00:01:50.460 x_3, etc. 00:01:50.460 --> 00:01:53.940 The same as we did with Riemann sums. 00:01:53.940 --> 00:01:56.050 And then the way that we're going 00:01:56.050 --> 00:02:07.540 to approximate this is by taking the straight lines between each 00:02:07.540 --> 00:02:09.470 of these points. 00:02:09.470 --> 00:02:11.360 As things get smaller and smaller, 00:02:11.360 --> 00:02:15.210 the straight line is going to be fairly close to the curve. 00:02:15.210 --> 00:02:16.980 And that's the main idea. 00:02:16.980 --> 00:02:20.050 So let me just depict one little chunk of this. 00:02:20.050 --> 00:02:20.900 Which is like this. 00:02:20.900 --> 00:02:24.530 One straight line, and here's the curved surface there. 00:02:24.530 --> 00:02:26.830 And the distance along the curved surface is what 00:02:26.830 --> 00:02:31.690 I'm calling delta s, the change in the length between-- 00:02:31.690 --> 00:02:34.920 so this would be s_2 - s_1 if I depicted that one. 00:02:34.920 --> 00:02:42.490 So this would be delta s is, say s. s_i - s_(i-1), 00:02:42.490 --> 00:02:45.240 some increment there. 00:02:45.240 --> 00:02:49.670 And then I can figure out what the length 00:02:49.670 --> 00:02:51.750 of the orange segment is. 00:02:51.750 --> 00:02:54.630 Because the horizontal distance is delta x. 00:02:54.630 --> 00:02:57.290 And the vertical distance is delta y. 00:02:57.290 --> 00:03:02.950 And so the formula is that the hypotenuse is delta (delta x)^2 00:03:02.950 --> 00:03:06.470 + (delta y)^2. 00:03:06.470 --> 00:03:09.420 Square root. 00:03:09.420 --> 00:03:11.870 And delta s is approximately that. 00:03:11.870 --> 00:03:15.871 So what we're saying is that (delta s)^2 is approximately 00:03:15.871 --> 00:03:16.370 this. 00:03:16.370 --> 00:03:21.870 So this is the hypotenuse. 00:03:21.870 --> 00:03:23.500 Squared. 00:03:23.500 --> 00:03:29.520 And it's very close to the length of the curve. 00:03:29.520 --> 00:03:34.940 And the whole idea of calculus is that in the infinitesimal, 00:03:34.940 --> 00:03:44.150 this is exactly correct. 00:03:44.150 --> 00:03:48.010 So that's what's going to happen in the limit. 00:03:48.010 --> 00:03:52.000 And that is the basis for calculating arc length. 00:03:52.000 --> 00:03:56.060 I'm going to rewrite that formula on the next board. 00:03:56.060 --> 00:03:59.460 But I'm going to write it in the more customary fashion. 00:03:59.460 --> 00:04:02.200 We've done this before, a certain amount. 00:04:02.200 --> 00:04:03.970 But I just want to emphasize it here 00:04:03.970 --> 00:04:09.390 because this handwriting is a little bit peculiar. 00:04:09.390 --> 00:04:11.710 This ds is really all one thing. 00:04:11.710 --> 00:04:15.690 What I really mean is to put the parenthesis around it. 00:04:15.690 --> 00:04:16.660 It's one thing. 00:04:16.660 --> 00:04:19.070 It's not d times s, it's ds. 00:04:19.070 --> 00:04:20.020 It's one thing. 00:04:20.020 --> 00:04:20.940 And we square it. 00:04:20.940 --> 00:04:22.750 But for whatever reason people have 00:04:22.750 --> 00:04:26.112 gotten into the habit of omitting the parentheses. 00:04:26.112 --> 00:04:28.070 So you're just going to have to live with that. 00:04:28.070 --> 00:04:31.050 And realize that this is not d of s^2 or anything like that. 00:04:31.050 --> 00:04:32.755 And similarly, this is a single number, 00:04:32.755 --> 00:04:34.130 and this is a single number. 00:04:34.130 --> 00:04:36.060 Infinitesimal. 00:04:36.060 --> 00:04:39.400 So that's just the way that this idea here 00:04:39.400 --> 00:04:42.194 gets written in our notation. 00:04:42.194 --> 00:04:43.860 And this is the first time we're dealing 00:04:43.860 --> 00:04:46.070 with squares of infinitesimals. 00:04:46.070 --> 00:04:47.600 So it's just a little different. 00:04:47.600 --> 00:04:49.641 But immediately the first thing we're going to do 00:04:49.641 --> 00:04:51.750 is take the square root. 00:04:51.750 --> 00:04:56.660 If I take the square root, that's the square root of dx^2 00:04:56.660 --> 00:04:58.640 + dy^2. 00:04:58.640 --> 00:05:02.400 And this is the form in which I always remember this formula. 00:05:02.400 --> 00:05:07.030 Let's put it in some brightly decorated form. 00:05:07.030 --> 00:05:12.380 But there are about four, five, six other forms 00:05:12.380 --> 00:05:16.110 that you'll derive from this, which all mean the same thing. 00:05:16.110 --> 00:05:18.479 So this is, as I say, the way I remember it. 00:05:18.479 --> 00:05:20.270 But there are other ways of thinking of it. 00:05:20.270 --> 00:05:23.070 And let's just write a couple of them down. 00:05:23.070 --> 00:05:27.380 The first one is that you can factor out the dx. 00:05:27.380 --> 00:05:29.480 So that looks like this. 00:05:29.480 --> 00:05:34.680 1 + (dy / dx)^2. 00:05:34.680 --> 00:05:37.630 And then I factored out the dx. 00:05:37.630 --> 00:05:39.820 So this is a variant. 00:05:39.820 --> 00:05:43.190 And this is the one which actually we'll 00:05:43.190 --> 00:05:47.490 be using in practice right now on our examples. 00:05:47.490 --> 00:05:55.600 So the conclusion is that the arc length, 00:05:55.600 --> 00:06:05.320 which if you like is this total s_n - s_0, if you like, 00:06:05.320 --> 00:06:09.390 is going to be equal to the integral from a to b 00:06:09.390 --> 00:06:16.430 of the square root of 1 + (dy/dx)^2, dx. 00:06:20.490 --> 00:06:27.500 In practice, it's also very often written informally 00:06:27.500 --> 00:06:29.090 as this. 00:06:29.090 --> 00:06:30.750 The integral ds. 00:06:30.750 --> 00:06:32.830 So the change in this little variable 00:06:32.830 --> 00:06:40.009 s, and this is what you'll see notationally in many textbooks. 00:06:40.009 --> 00:06:42.550 So that's one way of writing it, and of course the second way 00:06:42.550 --> 00:06:45.950 of writing it which is practically the same thing is 00:06:45.950 --> 00:06:50.040 square root of 1 + f'(x)^2, dx. 00:06:50.040 --> 00:06:52.330 Mixing in a little bit of Newton's notation. 00:06:52.330 --> 00:06:57.250 And this is with y = f(x). 00:06:57.250 --> 00:07:03.310 So this is the formula for arc length. 00:07:03.310 --> 00:07:05.770 And as I say, I remember it this way. 00:07:05.770 --> 00:07:09.060 But you're going to have to derive various variants of it. 00:07:09.060 --> 00:07:11.010 And you'll have to use some arithmetic 00:07:11.010 --> 00:07:12.650 to get to various formulas. 00:07:12.650 --> 00:07:15.190 And there will be more later. 00:07:15.190 --> 00:07:16.030 Yeah, question. 00:07:16.030 --> 00:07:20.820 STUDENT: [INAUDIBLE] 00:07:20.820 --> 00:07:27.350 PROFESSOR: OK, the question is, is f'(x)^2 equal to f''(x). 00:07:27.350 --> 00:07:31.020 And the answer is no. 00:07:31.020 --> 00:07:33.090 And let's just see what it is. 00:07:33.090 --> 00:07:39.490 So, for example, if f(x) = x^2, which is an example which will 00:07:39.490 --> 00:07:47.910 come up in a few minutes, then f'(x) = 2x and f'(x)^2 = 00:07:47.910 --> 00:07:52.380 (2x)^2, which is 4x^2. 00:07:52.380 --> 00:07:56.490 Whereas f''(x) is equal to, if I differentiate this another 00:07:56.490 --> 00:07:58.750 time, it's equal to 2. 00:07:58.750 --> 00:08:03.440 So they don't mean the same thing. 00:08:03.440 --> 00:08:04.700 The same thing over here. 00:08:04.700 --> 00:08:07.200 You can see this dy / dx, this is the rate of change 00:08:07.200 --> 00:08:08.190 of y with respect to x. 00:08:08.190 --> 00:08:09.632 The quantity squared. 00:08:09.632 --> 00:08:11.340 So in other words, this thing is supposed 00:08:11.340 --> 00:08:13.300 to mean the same as that. 00:08:13.300 --> 00:08:13.800 Yeah. 00:08:13.800 --> 00:08:19.580 Another question. 00:08:19.580 --> 00:08:25.660 STUDENT: [INAUDIBLE] 00:08:25.660 --> 00:08:27.970 PROFESSOR: So the question is, you 00:08:27.970 --> 00:08:30.430 got a little nervous because I left out these limits. 00:08:30.430 --> 00:08:32.120 And indeed, I did that on purpose 00:08:32.120 --> 00:08:34.530 because I didn't want to specify what was going on. 00:08:34.530 --> 00:08:36.590 Really, if you wrote it in terms of ds, 00:08:36.590 --> 00:08:38.710 you'd have to write it as starting at s_0 00:08:38.710 --> 00:08:42.180 and ending at s_n to be consistent with the variable s. 00:08:42.180 --> 00:08:45.730 But of course, if you write it in terms of another variable, 00:08:45.730 --> 00:08:46.940 you put that variable in. 00:08:46.940 --> 00:08:49.190 So this is what we do when we change variables, right? 00:08:49.190 --> 00:08:51.570 We have many different choices for these limits. 00:08:51.570 --> 00:08:54.530 And this is the clue as to which variable we use. 00:08:54.530 --> 00:08:59.060 STUDENT: [INAUDIBLE] 00:08:59.060 --> 00:09:01.720 PROFESSOR: Correct. s_0 and s_n are not 00:09:01.720 --> 00:09:03.310 the same thing as a and b. 00:09:03.310 --> 00:09:05.290 In fact, this is x_n. 00:09:05.290 --> 00:09:07.740 And this x_0, over here. 00:09:07.740 --> 00:09:08.990 That's what a and b are. 00:09:08.990 --> 00:09:13.242 But s_0 and s_n are mileage markers on the road. 00:09:13.242 --> 00:09:15.450 They're not the same thing as keeping track of what's 00:09:15.450 --> 00:09:16.950 happening on the x axis. 00:09:16.950 --> 00:09:18.710 So when we measure arc length, remember 00:09:18.710 --> 00:09:27.980 it's mileage along the curved path. 00:09:27.980 --> 00:09:32.410 So now, I need to give you some examples. 00:09:32.410 --> 00:09:40.820 And my first example is going to be really basic. 00:09:40.820 --> 00:09:45.630 But I hope that it helps to give some perspective here. 00:09:45.630 --> 00:09:48.830 So I'm going to take the example y = m 00:09:48.830 --> 00:09:52.550 x, which is a linear function, a straight line. 00:09:52.550 --> 00:09:58.630 And then y' would be m, and so ds is going to be the square 00:09:58.630 --> 00:10:02.650 root of 1 + (y')^2, dx. 00:10:02.650 --> 00:10:10.840 Which is the square root of 1 + m^2, dx. 00:10:10.840 --> 00:10:17.700 And now, the length, say, if we go from, I don't know, 00:10:17.700 --> 00:10:27.300 let's say 0 to 10, let's say, of the graph is going to be 00:10:27.300 --> 00:10:33.990 the integral from 0 to 10 of the square root of 1 + m^2, dx. 00:10:33.990 --> 00:10:39.910 Which of course is just 10 square root of 1 + m^2. 00:10:39.910 --> 00:10:41.810 Not very surprising. 00:10:41.810 --> 00:10:43.360 This is a constant. 00:10:43.360 --> 00:10:46.830 It just factors out and the integral from 0 to 10 of dx 00:10:46.830 --> 00:10:51.210 is 10. 00:10:51.210 --> 00:10:54.340 Let's just draw a picture of this. 00:10:54.340 --> 00:10:57.530 This is something which has slope m here. 00:10:57.530 --> 00:10:59.020 And it's going to 10. 00:10:59.020 --> 00:11:02.340 So this horizontal is 10. 00:11:02.340 --> 00:11:05.930 And the vertical is 10m. 00:11:05.930 --> 00:11:08.030 Those are the dimensions of this. 00:11:08.030 --> 00:11:11.810 And the Pythagorean theorem says that the hypotenuse, 00:11:11.810 --> 00:11:15.457 not surprisingly, let's draw it in here in orange to remind 00:11:15.457 --> 00:11:18.040 ourselves that it was the same type of orange that we had over 00:11:18.040 --> 00:11:27.160 there, this length here is the square root of 10^2 + (10m)^2. 00:11:27.160 --> 00:11:31.400 That's the formula for the hypotenuse. 00:11:31.400 --> 00:11:38.520 And that's exactly the same as this. 00:11:38.520 --> 00:11:40.540 Maybe you're saying duh, this is obvious. 00:11:40.540 --> 00:11:44.160 But the point that I'm trying to make is this. 00:11:44.160 --> 00:11:48.220 If you can figure out these formulas for linear functions, 00:11:48.220 --> 00:11:52.510 calculus tells you how to do it for every function. 00:11:52.510 --> 00:11:56.370 The idea of calculus is that this easy calculation here, 00:11:56.370 --> 00:11:58.530 which you can do without any calculus 00:11:58.530 --> 00:12:04.360 at all, all of the tools, the notations of differentials 00:12:04.360 --> 00:12:06.115 and limits and integrals, is going 00:12:06.115 --> 00:12:10.040 to make you be able to do it for any curve. 00:12:10.040 --> 00:12:12.760 Because we can break things up into these little infinitesimal 00:12:12.760 --> 00:12:13.260 bits. 00:12:13.260 --> 00:12:15.820 This is the whole idea of all of the methods 00:12:15.820 --> 00:12:18.620 that we had to set up integrals here. 00:12:18.620 --> 00:12:25.770 This is the main point of these integrals. 00:12:25.770 --> 00:12:32.480 Now, so let's do something slightly more interesting. 00:12:32.480 --> 00:12:39.970 Our next example is going to be the circle, 00:12:39.970 --> 00:12:41.540 so y = square root of 1-x^2. 00:12:48.590 --> 00:12:51.820 If you like, that's the graph of a semicircle. 00:12:51.820 --> 00:12:57.040 And maybe we'll set it up here this way. 00:12:57.040 --> 00:13:00.030 So that the semicircle goes around like this. 00:13:00.030 --> 00:13:02.310 And we'll start it here at x = 0. 00:13:02.310 --> 00:13:04.250 And we'll go over to a. 00:13:04.250 --> 00:13:06.560 And we'll take this little piece of the circle. 00:13:06.560 --> 00:13:08.150 So down to here. 00:13:08.150 --> 00:13:11.850 If you like. 00:13:11.850 --> 00:13:14.970 So here's the portion of the circle 00:13:14.970 --> 00:13:17.660 that I'm going to measure the length of. 00:13:17.660 --> 00:13:19.110 Now, we know that length. 00:13:19.110 --> 00:13:20.160 It's called arc length. 00:13:20.160 --> 00:13:21.659 And I'm going to give it a name, I'm 00:13:21.659 --> 00:13:23.400 going to call it alpha here. 00:13:23.400 --> 00:13:39.640 So alpha's the arc length along the circle. 00:13:39.640 --> 00:13:42.660 Now, let's figure out what it is. 00:13:42.660 --> 00:13:45.970 First, in order to do this, I have to figure out what y' is. 00:13:45.970 --> 00:13:47.960 Or, if you like, dy/dx. 00:13:47.960 --> 00:13:50.680 Now, that's a calculation that we've done a number of times. 00:13:50.680 --> 00:13:52.830 And I'm going to do it slightly faster. 00:13:52.830 --> 00:13:57.160 But you remember it gives you a square root in the denominator. 00:13:57.160 --> 00:13:59.500 And then you have the derivative of what's 00:13:59.500 --> 00:14:01.180 inside the square root. 00:14:01.180 --> 00:14:02.260 Which is -2x. 00:14:02.260 --> 00:14:05.590 But then there's also 1/2, because in disguise it's really 00:14:05.590 --> 00:14:07.840 (1 - x^2)^(1/2). 00:14:07.840 --> 00:14:10.584 So we've done this calculation enough times 00:14:10.584 --> 00:14:12.500 that I'm not going to carry it out completely. 00:14:12.500 --> 00:14:14.290 I want you to think about what it is. 00:14:14.290 --> 00:14:17.690 It turns out to -x up here, because the 1/2 and the 2 00:14:17.690 --> 00:14:22.890 cancel. 00:14:22.890 --> 00:14:25.870 And now the thing that we have to integrate 00:14:25.870 --> 00:14:30.190 is this arc length element, as it's called, ds. 00:14:30.190 --> 00:14:38.350 And that's going to be the square root of 1 + (y')^2, dx. 00:14:38.350 --> 00:14:41.060 And so I'm going to have to carry out the calculation, 00:14:41.060 --> 00:14:42.970 some messy calculation here. 00:14:42.970 --> 00:14:47.020 Which is that this is 1 plus the quantity -x over square root 00:14:47.020 --> 00:14:49.274 of 1 - x^2, squared. 00:14:49.274 --> 00:14:51.440 So I have to figure out what's under the square root 00:14:51.440 --> 00:14:55.740 sign over here in order to carry out this calculation. 00:14:55.740 --> 00:14:58.570 Now let's do that. 00:14:58.570 --> 00:15:03.310 This is 1 + x^2 / (1 - x^2). 00:15:03.310 --> 00:15:06.140 That's what this simplifies to. 00:15:06.140 --> 00:15:11.190 And then that's equal to, over a common denominator, 1 - x^2. 00:15:11.190 --> 00:15:13.580 1 - x^2 + x^2. 00:15:13.580 --> 00:15:16.500 And there is a little bit of simplification now. 00:15:16.500 --> 00:15:19.190 Because the two x^2's cancel. 00:15:19.190 --> 00:15:20.390 And we get 1/(1-x^2). 00:15:28.640 --> 00:15:35.750 So now I get to finish off the calculation 00:15:35.750 --> 00:15:40.580 by actually figuring out what the arc length is. 00:15:40.580 --> 00:15:42.460 And what is it? 00:15:42.460 --> 00:15:51.834 Well, this alpha is equal to the integral from 0 to a of ds. 00:15:51.834 --> 00:15:53.750 Well, it's going to be the square root of what 00:15:53.750 --> 00:15:55.110 I have here. 00:15:55.110 --> 00:15:57.210 This was a square, this is just what 00:15:57.210 --> 00:15:58.710 was underneath the square root sign. 00:15:58.710 --> 00:16:01.500 This is 1 + (y')^2. 00:16:01.500 --> 00:16:03.270 Have to take the square root of that. 00:16:03.270 --> 00:16:08.380 So what I get here is dx over the square root of 1 - x^2. 00:16:13.350 --> 00:16:18.590 And now, we recognize this. 00:16:18.590 --> 00:16:21.100 The antiderivative of this is something that we know. 00:16:21.100 --> 00:16:23.790 This is the inverse sine. 00:16:23.790 --> 00:16:25.830 Evaluated at 0 and a. 00:16:25.830 --> 00:16:29.490 Which is just giving us the inverse sine 00:16:29.490 --> 00:16:35.860 of a, because the inverse sine of 0 is equal to 0. 00:16:35.860 --> 00:16:43.980 So alpha is equal to the inverse sine of a. 00:16:43.980 --> 00:16:51.320 That's a very fancy way of saying that sin(alpha) = a. 00:16:51.320 --> 00:16:54.870 That's the equivalent statement here. 00:16:54.870 --> 00:16:59.830 And what's going on here is something that's just 00:16:59.830 --> 00:17:01.950 a little deeper than it looks. 00:17:01.950 --> 00:17:03.120 Which is this. 00:17:03.120 --> 00:17:08.030 We've just figured out a geometric interpretation 00:17:08.030 --> 00:17:09.350 of what's going on here. 00:17:09.350 --> 00:17:13.680 That is, that we went a distance alpha along this arc. 00:17:13.680 --> 00:17:28.570 And now remember that the radius here is 1. 00:17:28.570 --> 00:17:34.430 And this horizontal distance here is a. 00:17:34.430 --> 00:17:37.450 This distance here is a. 00:17:37.450 --> 00:17:40.600 And so the geometric interpretation of this 00:17:40.600 --> 00:17:51.000 is that this angle is alpha radians. 00:17:51.000 --> 00:17:54.950 And sin(alpha) = a. 00:17:54.950 --> 00:17:57.570 So this is consistent with our definition 00:17:57.570 --> 00:18:00.430 previously, our previous geometric definition 00:18:00.430 --> 00:18:02.530 of radians. 00:18:02.530 --> 00:18:07.190 But this is really your first true definition of radians. 00:18:07.190 --> 00:18:09.850 We never actually-- People told you 00:18:09.850 --> 00:18:12.570 that radians were the arc length along this curve. 00:18:12.570 --> 00:18:14.770 This is the first time you're deriving it. 00:18:14.770 --> 00:18:18.420 This is the first time you're seeing it correctly done. 00:18:18.420 --> 00:18:20.134 And furthermore, this is the first time 00:18:20.134 --> 00:18:21.550 you're seeing a correct definition 00:18:21.550 --> 00:18:24.100 of the sine function. 00:18:24.100 --> 00:18:26.156 Remember we had this crazy way, we 00:18:26.156 --> 00:18:27.530 defined the exponential function, 00:18:27.530 --> 00:18:29.661 then we had another way of defining the log 00:18:29.661 --> 00:18:30.660 function as an integral. 00:18:30.660 --> 00:18:32.900 Then we defined the exponential in terms of it. 00:18:32.900 --> 00:18:34.690 Well, this is the same sort of thing. 00:18:34.690 --> 00:18:36.700 What's really happening here is that if you 00:18:36.700 --> 00:18:38.190 want to know what radians are, you 00:18:38.190 --> 00:18:40.847 have to calculate this number. 00:18:40.847 --> 00:18:42.430 If you've calculated this number, then 00:18:42.430 --> 00:18:47.930 by definition if sine is the thing whose alpha radian 00:18:47.930 --> 00:18:49.890 amount gives you a, then it must be 00:18:49.890 --> 00:18:52.520 that this is sine inverse of a. 00:18:52.520 --> 00:18:55.430 And so the first thing that gets defined is the arcsine. 00:18:55.430 --> 00:18:56.930 And the next thing that gets defined 00:18:56.930 --> 00:19:00.250 is the sine, afterwards. 00:19:00.250 --> 00:19:04.500 This is the way the foundational approach actually 00:19:04.500 --> 00:19:06.730 works when you start from first principles. 00:19:06.730 --> 00:19:10.200 This arc length being one of the first principles. 00:19:10.200 --> 00:19:13.720 So now we have a solid foundation for trig functions. 00:19:13.720 --> 00:19:15.760 And this is giving that connection. 00:19:15.760 --> 00:19:18.260 Of course, it's consistent with everything you already knew, 00:19:18.260 --> 00:19:22.079 so you don't have to make any transitional thinking here. 00:19:22.079 --> 00:19:23.620 It's just that this is the first time 00:19:23.620 --> 00:19:25.570 it's being done rigorously. 00:19:25.570 --> 00:19:36.380 Because you only now have arc length. 00:19:36.380 --> 00:19:41.220 So these are examples, as I say, that maybe you already know. 00:19:41.220 --> 00:19:44.820 And maybe we'll do one that we don't know quite as well. 00:19:44.820 --> 00:19:49.300 Let's find the length of a parabola. 00:19:49.300 --> 00:19:59.180 This is Example 3. 00:19:59.180 --> 00:20:00.660 Now, that was what I was suggesting 00:20:00.660 --> 00:20:03.140 we were going to do earlier. 00:20:03.140 --> 00:20:06.730 So this is the function y x^2. 00:20:06.730 --> 00:20:09.800 y' = 2x. 00:20:09.800 --> 00:20:20.120 And so ds is equal to the square root of 1 + (2x)^2, dx. 00:20:20.120 --> 00:20:24.600 And now I can figure out what a piece of a parabola is. 00:20:24.600 --> 00:20:28.220 So I'll draw the piece of parabola up to a, 00:20:28.220 --> 00:20:30.840 let's say, starting from 0. 00:20:30.840 --> 00:20:32.680 So that's the chunk. 00:20:32.680 --> 00:20:45.450 And then its arc length, between 0 and a of this curve, 00:20:45.450 --> 00:21:02.400 is the integral from 0 to a of square root of 1 + 4x^2, dx. 00:21:02.400 --> 00:21:08.490 OK, now if you like, this is the answer to the question. 00:21:08.490 --> 00:21:11.040 But people hate looking at answers 00:21:11.040 --> 00:21:13.800 when they're integrals if they can be evaluated. 00:21:13.800 --> 00:21:16.510 So one of the reasons why we went through all this rigmarole 00:21:16.510 --> 00:21:18.800 of calculating these things is to show you 00:21:18.800 --> 00:21:22.120 that we can actually evaluate a bunch of these functions 00:21:22.120 --> 00:21:23.340 here more explicitly. 00:21:23.340 --> 00:21:28.180 It doesn't help a lot, but there is an explicit calculation 00:21:28.180 --> 00:21:28.680 of this. 00:21:28.680 --> 00:21:30.560 So remember how you would do this. 00:21:30.560 --> 00:21:33.040 So this is just a little bit of review. 00:21:33.040 --> 00:21:35.420 What we did in techniques of integration. 00:21:35.420 --> 00:21:39.000 The first step is what? 00:21:39.000 --> 00:21:40.700 A substitution. 00:21:40.700 --> 00:21:43.950 It's a trig substitution. 00:21:43.950 --> 00:21:45.030 And what is it? 00:21:45.030 --> 00:21:47.250 STUDENT: [INAUDIBLE] 00:21:47.250 --> 00:21:50.270 PROFESSOR: So x equals something tan(theta). 00:21:50.270 --> 00:21:54.587 I claim that it's 1/2 tan, and I'm going to call it u. 00:21:54.587 --> 00:21:56.420 Because I'm going to use theta for something 00:21:56.420 --> 00:21:58.161 else in a couple of days. 00:21:58.161 --> 00:21:58.660 OK? 00:21:58.660 --> 00:22:01.420 So this is the substitution. 00:22:01.420 --> 00:22:10.620 And then of course dx = 1/2 sec^2 u du, etc. 00:22:10.620 --> 00:22:12.750 So what happens if you do this? 00:22:12.750 --> 00:22:15.160 I'll write down the answer, but I'm not 00:22:15.160 --> 00:22:16.330 going to carry this out. 00:22:16.330 --> 00:22:19.090 Because every one of these is horrendous. 00:22:19.090 --> 00:22:22.190 But I think I worked it out. 00:22:22.190 --> 00:22:23.370 Let's see if I'm lucky. 00:22:23.370 --> 00:22:24.300 Oh yeah. 00:22:24.300 --> 00:22:26.380 I think this is what it is. 00:22:26.380 --> 00:22:40.979 It's a 1/4 ln(2x + square root of (1+4x^2) + 1/2 x square root 00:22:40.979 --> 00:22:41.520 of (1+4x^2)). 00:22:46.540 --> 00:22:52.190 Evaluated at a and 0. 00:22:52.190 --> 00:22:53.090 So yick. 00:22:53.090 --> 00:22:53.810 I mean, you know. 00:22:53.810 --> 00:22:55.720 STUDENT: [INAUDIBLE] 00:22:55.720 --> 00:22:58.850 PROFESSOR: Why I did I make it 1/2? 00:22:58.850 --> 00:23:00.989 Because it turns out that when you differentiate. 00:23:00.989 --> 00:23:02.780 So the question is, why is there 1/2 there? 00:23:02.780 --> 00:23:05.734 If you differentiate it without the 1/2, you get this term 00:23:05.734 --> 00:23:07.650 and it looks like it's going to be just right. 00:23:07.650 --> 00:23:10.191 But then if you differentiate this one you get another thing. 00:23:10.191 --> 00:23:12.170 And it all mixes together. 00:23:12.170 --> 00:23:13.660 And it turns out that there's more. 00:23:13.660 --> 00:23:15.280 So it turns out that it's 1/2. 00:23:15.280 --> 00:23:18.750 Differentiate it and check. 00:23:18.750 --> 00:23:21.530 So this just an incredibly long calculation. 00:23:21.530 --> 00:23:24.220 It would take fifteen minutes or something like that. 00:23:24.220 --> 00:23:26.220 But the point is, you do know in principle 00:23:26.220 --> 00:23:27.840 how to do these things. 00:23:27.840 --> 00:23:43.455 STUDENT: [INAUDIBLE] 00:23:43.455 --> 00:23:45.330 PROFESSOR: Oh, he was talking about this 1/2, 00:23:45.330 --> 00:23:47.100 not this crazy 1/2 here. 00:23:47.100 --> 00:23:47.600 Sorry. 00:23:47.600 --> 00:23:48.637 STUDENT: [INAUDIBLE] 00:23:48.637 --> 00:23:49.470 PROFESSOR: Yeah, OK. 00:23:49.470 --> 00:23:50.740 So sorry about that. 00:23:50.740 --> 00:23:53.080 Thank you for helping. 00:23:53.080 --> 00:23:56.270 This factor of 1/2 here comes about because when you square 00:23:56.270 --> 00:23:58.690 x, you don't get tan^2. 00:23:58.690 --> 00:24:02.510 When you square 2x, you get (4x)^2 and that matches 00:24:02.510 --> 00:24:03.930 perfectly with this thing. 00:24:03.930 --> 00:24:07.300 And that's why you need this factor here. 00:24:07.300 --> 00:24:07.800 Yeah. 00:24:07.800 --> 00:24:09.216 Another question, way in the back. 00:24:09.216 --> 00:24:18.190 STUDENT: [INAUDIBLE] 00:24:18.190 --> 00:24:20.780 PROFESSOR: The question is, when you do this substitution, 00:24:20.780 --> 00:24:25.040 doesn't the limit from 0 to a change. 00:24:25.040 --> 00:24:27.020 And the answer is, absolutely yes. 00:24:27.020 --> 00:24:30.230 The limits in terms of u are not the same 00:24:30.230 --> 00:24:31.800 as the limits in terms of a. 00:24:31.800 --> 00:24:34.870 But if I then translate back to the x variables, which 00:24:34.870 --> 00:24:40.820 I've done here in this bottom formula, of x = 0 and x = a, 00:24:40.820 --> 00:24:44.740 it goes back to those in the original variables. 00:24:44.740 --> 00:24:46.840 So if I write things in the original variables, 00:24:46.840 --> 00:24:48.980 I have the original limits. 00:24:48.980 --> 00:24:51.962 If I use the u variables, I would have to change limits. 00:24:51.962 --> 00:24:53.670 But I'm not carrying out the integration, 00:24:53.670 --> 00:24:55.060 because I don't want to. 00:24:55.060 --> 00:25:00.630 So I brought it back to the x formula. 00:25:00.630 --> 00:25:07.080 Other questions. 00:25:07.080 --> 00:25:11.580 OK, so now we're ready to launch into three-space a little bit 00:25:11.580 --> 00:25:14.250 here. 00:25:14.250 --> 00:25:41.310 We're going to talk about surface area. 00:25:41.310 --> 00:25:43.750 You're going to be doing a lot with surface 00:25:43.750 --> 00:25:48.550 area in multivariable calculus. 00:25:48.550 --> 00:25:50.990 It's one of the really fun things. 00:25:50.990 --> 00:25:55.430 And just remember, when it gets complicated, 00:25:55.430 --> 00:25:57.926 that the simplest things are the most important. 00:25:57.926 --> 00:25:59.300 And the simple things are, if you 00:25:59.300 --> 00:26:01.820 can handle things for linear functions, 00:26:01.820 --> 00:26:02.920 you know all the rest. 00:26:02.920 --> 00:26:04.544 So there's going to be some complicated 00:26:04.544 --> 00:26:06.390 stuff but it'll really only involve 00:26:06.390 --> 00:26:09.480 what's happening on planes. 00:26:09.480 --> 00:26:11.680 So let's start with surface area. 00:26:11.680 --> 00:26:15.615 And the example that I'd like to give 00:26:15.615 --> 00:26:20.110 - this is the only type of example that we'll have - 00:26:20.110 --> 00:26:28.560 is the surface of rotation. 00:26:28.560 --> 00:26:31.990 And as long as we have our parabola there, 00:26:31.990 --> 00:26:33.300 we'll use that one. 00:26:33.300 --> 00:26:51.920 So we have y = x^2, rotated around the x-axis. 00:26:51.920 --> 00:26:54.480 So let's take a look at what this looks like. 00:26:54.480 --> 00:26:57.910 It's the parabola, which is going like that. 00:26:57.910 --> 00:27:01.850 And then it's being spun around the x-axis. 00:27:01.850 --> 00:27:08.390 So some kind of shape like this with little circles. 00:27:08.390 --> 00:27:17.880 It's some kind of trumpet shape, right? 00:27:17.880 --> 00:27:20.170 And that's the shape that we're-- Now, again, 00:27:20.170 --> 00:27:20.980 it's the surface. 00:27:20.980 --> 00:27:27.770 It's just the metal of the trumpet, not the insides. 00:27:27.770 --> 00:27:33.280 Now, the principle for figuring out what the formula for area 00:27:33.280 --> 00:27:36.470 is, is not that different from what we 00:27:36.470 --> 00:27:38.530 did for surfaces of revolution. 00:27:38.530 --> 00:27:42.610 But it just requires a little bit of thought and imagination. 00:27:42.610 --> 00:27:50.360 We have a little chunk of arc length along here. 00:27:50.360 --> 00:27:55.580 And we're going to spin that around this axis. 00:27:55.580 --> 00:28:01.610 Now, if this were a horizontal piece of arc length, 00:28:01.610 --> 00:28:04.500 then it would spin around just like a shell. 00:28:04.500 --> 00:28:07.550 It would just be a surface. 00:28:07.550 --> 00:28:11.670 But if it's tilted, if it's tilted, 00:28:11.670 --> 00:28:15.030 then there's more surface area proportional to the amount 00:28:15.030 --> 00:28:17.040 that it's tilted. 00:28:17.040 --> 00:28:19.310 So it's proportional to the length of the segment 00:28:19.310 --> 00:28:22.890 that you spin around. 00:28:22.890 --> 00:28:29.430 So the total is going to be ds, that's one of the factors here. 00:28:29.430 --> 00:28:32.220 Maybe I'll write that second. 00:28:32.220 --> 00:28:33.560 That's one of the dimensions. 00:28:33.560 --> 00:28:36.660 And then the other dimension is the circumference. 00:28:36.660 --> 00:28:43.180 Which is 2 pi, in this case, y. 00:28:43.180 --> 00:28:46.300 So that's the end of the calculation. 00:28:46.300 --> 00:28:55.990 This is the area element of surface area. 00:28:55.990 --> 00:28:59.860 Now, when you get to 18.02, and maybe even before that, 00:28:59.860 --> 00:29:03.000 you'll also see some people referring to this area element 00:29:03.000 --> 00:29:09.110 when it's a curvy surface like this with a notation dS. 00:29:09.110 --> 00:29:10.610 That's a little confusing because we 00:29:10.610 --> 00:29:12.070 have a lower case s here. 00:29:12.070 --> 00:29:15.450 We're not going to use it right now. 00:29:15.450 --> 00:29:17.620 But the lower case s is usually arc length. 00:29:17.620 --> 00:29:23.950 The upper case S is usually surface area. 00:29:23.950 --> 00:29:25.650 So. 00:29:25.650 --> 00:29:32.510 Also used for dA. 00:29:32.510 --> 00:29:33.730 The area element. 00:29:33.730 --> 00:29:39.780 Because this is a curved area element. 00:29:39.780 --> 00:29:47.630 So let's figure out this example. 00:29:47.630 --> 00:29:54.090 So in the example-- ...is equal to x ^2 then the situation is, 00:29:54.090 --> 00:30:01.050 we have the surface area is equal to the integral from, 00:30:01.050 --> 00:30:03.770 I don't know, 0 to a if those are the limits that we wanted 00:30:03.770 --> 00:30:05.040 to choose. 00:30:05.040 --> 00:30:10.470 Of 2 pi x^2, right? 00:30:10.470 --> 00:30:11.160 Because y = x^2. 00:30:11.160 --> 00:30:17.960 Times the square root of 1 + 4x^2, dx. 00:30:17.960 --> 00:30:20.300 Remember we had this from our previous example. 00:30:20.300 --> 00:30:32.850 This was ds from previous. 00:30:32.850 --> 00:30:41.930 And this, of course, is 2 pi y. 00:30:41.930 --> 00:30:49.150 Now again, the calculation of this integral is kind of long. 00:30:49.150 --> 00:30:52.060 And I'm going to omit it. 00:30:52.060 --> 00:30:54.300 But let me just point out that it follows 00:30:54.300 --> 00:30:56.350 from the same substitution. 00:30:56.350 --> 00:31:05.150 Namely, x = 1/2 tan u is going to work for this integral. 00:31:05.150 --> 00:31:06.200 It's kind of a mess. 00:31:06.200 --> 00:31:08.590 There's a tan squared here and the secant squared. 00:31:08.590 --> 00:31:10.170 There's another secant and so on. 00:31:10.170 --> 00:31:12.640 So it's one of these trig integrals 00:31:12.640 --> 00:31:19.670 that then takes a while to do. 00:31:19.670 --> 00:31:22.940 So that just is going to trail off into nothing. 00:31:22.940 --> 00:31:25.540 And the reason is that what's important here 00:31:25.540 --> 00:31:27.250 is more the method. 00:31:27.250 --> 00:31:29.650 And the setup of the integrals. 00:31:29.650 --> 00:31:32.902 The actual computation, in fact, you could go to a program 00:31:32.902 --> 00:31:34.610 and you could plug in something like this 00:31:34.610 --> 00:31:37.300 and you would spit out an answer immediately. 00:31:37.300 --> 00:31:41.100 So really what we just want is for you to have enough control 00:31:41.100 --> 00:31:43.640 to see that it's an integral that's a manageable one. 00:31:43.640 --> 00:31:45.630 And also to know that if you plugged it in, 00:31:45.630 --> 00:31:50.970 you would get an answer. 00:31:50.970 --> 00:31:53.320 When I actually do carry out a calculation, though, 00:31:53.320 --> 00:31:57.740 what I want to do is to do something 00:31:57.740 --> 00:32:00.190 that has an answer that you can remember. 00:32:00.190 --> 00:32:02.090 And that's a nice answer. 00:32:02.090 --> 00:32:05.940 So that turns out to be the example of the surface 00:32:05.940 --> 00:32:07.580 area of a sphere. 00:32:07.580 --> 00:32:10.110 So it's analogous to this 2 here. 00:32:10.110 --> 00:32:15.170 And maybe I should remember this result here. 00:32:15.170 --> 00:32:24.390 Which was that the arc length element was given by this. 00:32:24.390 --> 00:32:38.700 So we'll save that for a second. 00:32:38.700 --> 00:32:41.710 So we're going to do this surface area now. 00:32:41.710 --> 00:32:43.930 So if you like, this is another example. 00:32:43.930 --> 00:32:54.820 The surface area of a sphere. 00:32:54.820 --> 00:32:59.670 This is a good example, and one, as I say, 00:32:59.670 --> 00:33:01.160 that has a really nice answer. 00:33:01.160 --> 00:33:07.130 So it's worth doing. 00:33:07.130 --> 00:33:09.770 So first of all, I'm not going to set it up quite the way 00:33:09.770 --> 00:33:11.794 I did in Example 2. 00:33:11.794 --> 00:33:13.710 Instead, I'm going to take the general sphere, 00:33:13.710 --> 00:33:18.790 because I'd like to watch the dependence on the radius. 00:33:18.790 --> 00:33:22.160 So here this is going to be the radius. 00:33:22.160 --> 00:33:27.250 It's going to be radius a. 00:33:27.250 --> 00:33:30.440 And now, if I carry out the same calculations 00:33:30.440 --> 00:33:33.840 as before, if you think about it for a second, 00:33:33.840 --> 00:33:42.322 you're going to get this result. And then, the rest 00:33:42.322 --> 00:33:43.780 of the arithmetic, which is sitting 00:33:43.780 --> 00:33:47.750 up there in the case, a = 1, will give us 00:33:47.750 --> 00:33:53.120 that ds is equal to what? 00:33:53.120 --> 00:33:56.670 Well, maybe I'll just carry it out. 00:33:56.670 --> 00:33:58.510 Because that's always nice. 00:33:58.510 --> 00:34:03.630 So we have 1 + x^2 / (a^2 - x^2). 00:34:03.630 --> 00:34:07.060 That's 1 + (y')^2. 00:34:07.060 --> 00:34:09.630 And now I put this over a common denominator. 00:34:09.630 --> 00:34:11.800 And I get a^2 - x^2. 00:34:11.800 --> 00:34:15.030 And I have in the numerator a^2 - x^2 + x^2. 00:34:15.030 --> 00:34:17.590 So the same cancellation occurs. 00:34:17.590 --> 00:34:25.390 But now we get an a^2 in the numerator. 00:34:25.390 --> 00:34:28.070 So now I can set up the ds. 00:34:28.070 --> 00:34:30.200 And so here's what happens. 00:34:30.200 --> 00:34:35.370 The area of a section of the sphere, so let's see. 00:34:35.370 --> 00:34:38.920 We're going to start at some starting place x_1, 00:34:38.920 --> 00:34:40.760 and end at some place x_2. 00:34:40.760 --> 00:34:43.110 So what does that look like? 00:34:43.110 --> 00:34:45.140 Here's the sphere. 00:34:45.140 --> 00:34:47.490 And we're starting at a place x_1. 00:34:47.490 --> 00:34:49.810 And we're ending at a place x_2. 00:34:49.810 --> 00:34:53.830 And we're taking more or less the slice here, if you like. 00:34:53.830 --> 00:34:59.480 The section of this sphere. 00:34:59.480 --> 00:35:02.150 So the area's going to equal this. 00:35:02.150 --> 00:35:06.730 And what is it going to be? 00:35:06.730 --> 00:35:12.820 Well, so I have here 2 pi y. 00:35:12.820 --> 00:35:15.310 I'll write it out, just leave it as y for now. 00:35:15.310 --> 00:35:19.190 And then I have ds. 00:35:19.190 --> 00:35:22.110 So that's always what the formula is when you're 00:35:22.110 --> 00:35:25.880 revolving around the x-axis. 00:35:25.880 --> 00:35:29.920 And then I'll plug in for those things. 00:35:29.920 --> 00:35:35.990 So 2 pi, the formula for y is square root a^2 - x^2. 00:35:38.700 --> 00:35:42.220 And the formula for ds, well, it's 00:35:42.220 --> 00:35:44.970 the square root of this times dx. 00:35:44.970 --> 00:35:48.920 So it's the square root of a^2 / (a^2 - x^2), dx. 00:35:51.760 --> 00:35:54.880 So this part is ds. 00:35:54.880 --> 00:36:02.170 And this part is y. 00:36:02.170 --> 00:36:07.590 And now, I claim we have a nice cancellation that takes place. 00:36:07.590 --> 00:36:09.770 Square root of a^2 is a. 00:36:09.770 --> 00:36:12.740 And then there's another good cancellation. 00:36:12.740 --> 00:36:13.990 As you can see. 00:36:13.990 --> 00:36:17.140 Now, what we get here is the integral from x_1 to x_2, 00:36:17.140 --> 00:36:21.510 of 2 pi a dx, which is about the easiest integral 00:36:21.510 --> 00:36:23.340 you can imagine. 00:36:23.340 --> 00:36:24.940 It's just the integral of a constant. 00:36:24.940 --> 00:36:27.980 So it's 2 pi a (x_2 - x_1). 00:36:36.810 --> 00:36:40.780 Let's check this in a couple of examples. 00:36:40.780 --> 00:36:48.390 And then see what it's saying geometrically, a little bit. 00:36:48.390 --> 00:36:53.220 So what this is saying-- So special cases 00:36:53.220 --> 00:36:56.560 that you should always check, when 00:36:56.560 --> 00:36:59.220 you have a nice formula like this, at least. 00:36:59.220 --> 00:37:00.730 But really with anything in order 00:37:00.730 --> 00:37:03.250 to make sure that you've got the right answer. 00:37:03.250 --> 00:37:05.710 If you take, for example, the hemisphere. 00:37:05.710 --> 00:37:08.920 So you take 1/2 of this sphere. 00:37:08.920 --> 00:37:11.090 So that would be starting at 0, sorry. 00:37:11.090 --> 00:37:14.300 And ending at a. 00:37:14.300 --> 00:37:17.270 So that's the integral from 0 to a. 00:37:17.270 --> 00:37:21.890 So this is the case x_1 = 0. x_2 = a. 00:37:21.890 --> 00:37:29.190 And what you're going to get is a hemisphere. 00:37:29.190 --> 00:37:36.270 And the area is 2 pi a times a. 00:37:36.270 --> 00:37:38.150 Or in other words, 2 pi a^2. 00:37:42.410 --> 00:37:48.920 And if you take the whole sphere, 00:37:48.920 --> 00:37:55.900 that's starting at x_1 = -a, and x_2 = a, you're getting 2 pi 00:37:55.900 --> 00:37:58.330 a times (a - (-a)). 00:38:02.640 --> 00:38:06.380 Which is 4 pi a^2. 00:38:06.380 --> 00:38:09.230 That's the whole thing. 00:38:09.230 --> 00:38:10.140 Yeah, question. 00:38:10.140 --> 00:38:21.780 STUDENT: [INAUDIBLE] 00:38:21.780 --> 00:38:23.340 PROFESSOR: The question is, would it 00:38:23.340 --> 00:38:27.730 be possible to rotate around the y-axis? 00:38:27.730 --> 00:38:30.330 And the answer is yes. 00:38:30.330 --> 00:38:34.550 It's legal to rotate around the y-axis. 00:38:34.550 --> 00:38:43.170 And there is-- If you use vertical slices as we did here, 00:38:43.170 --> 00:38:45.490 that is, well they're sort of tips of slices, 00:38:45.490 --> 00:38:46.790 it's a different idea. 00:38:46.790 --> 00:38:49.780 But anyway, it's using dx as the integral 00:38:49.780 --> 00:38:52.670 of the variable of integration. 00:38:52.670 --> 00:38:55.310 So we're checking each little piece, 00:38:55.310 --> 00:38:58.750 each little strip of that type. 00:38:58.750 --> 00:39:00.540 If we use dx here, we get this. 00:39:00.540 --> 00:39:02.710 If you did the same thing rotated the other way, 00:39:02.710 --> 00:39:06.140 and use dy as the variable, you get exactly the same answer. 00:39:06.140 --> 00:39:08.120 And it would be the same calculation. 00:39:08.120 --> 00:39:14.080 Because they're parallel. 00:39:14.080 --> 00:39:14.870 So you're, yep. 00:39:14.870 --> 00:39:17.280 STUDENT: [INAUDIBLE] 00:39:17.280 --> 00:39:19.770 PROFESSOR: Can you do surface area with shells? 00:39:19.770 --> 00:39:26.540 Well, the shell shape-- The short answer is not quite. 00:39:26.540 --> 00:39:30.851 The shell shape is a vertical shell 00:39:30.851 --> 00:39:32.600 which is itself already three-dimensional, 00:39:32.600 --> 00:39:34.850 and it has a thickness. 00:39:34.850 --> 00:39:37.270 So this is just a matter of terminology, though. 00:39:37.270 --> 00:39:41.210 This thickness is this dx, when we do this rotation here. 00:39:41.210 --> 00:39:43.820 And then there are two other dimensions. 00:39:43.820 --> 00:39:46.190 If we have a curved surface, there's 00:39:46.190 --> 00:39:56.460 no other dimension left to form a shell. 00:39:56.460 --> 00:39:59.421 But basically, you can chop things up into any bits 00:39:59.421 --> 00:40:00.670 that you can actually measure. 00:40:00.670 --> 00:40:04.020 That you can figure out what the area is. 00:40:04.020 --> 00:40:08.270 That's the main point. 00:40:08.270 --> 00:40:10.160 Now, I said we were going to, we've 00:40:10.160 --> 00:40:12.910 just launched into three-dimensional space. 00:40:12.910 --> 00:40:21.650 And I want to now move on to other space-like phenomena. 00:40:21.650 --> 00:40:26.780 But we're going to do this. 00:40:26.780 --> 00:40:31.510 So this is also a preparation for 18.02, 00:40:31.510 --> 00:40:34.840 where you'll be doing this a tremendous amount. 00:40:34.840 --> 00:40:49.530 We're going to talk now about parametric equations. 00:40:49.530 --> 00:40:57.102 Really just parametric curves. 00:40:57.102 --> 00:40:58.810 So you're going to see this now and we're 00:40:58.810 --> 00:41:00.539 going to interpret it a couple of times, 00:41:00.539 --> 00:41:02.580 and we're going to think about polar coordinates. 00:41:02.580 --> 00:41:05.920 These are all preparation for thinking in more variables, 00:41:05.920 --> 00:41:08.570 and thinking in a different way than you've 00:41:08.570 --> 00:41:09.720 been thinking before. 00:41:09.720 --> 00:41:11.970 So I want you to prepare your brain 00:41:11.970 --> 00:41:13.975 to make a transition here. 00:41:13.975 --> 00:41:15.600 This is the beginning of the transition 00:41:15.600 --> 00:41:21.580 to multivariable thinking. 00:41:21.580 --> 00:41:26.350 We're going to consider curves like this. 00:41:26.350 --> 00:41:29.850 Which are described with x being a function of t and y 00:41:29.850 --> 00:41:31.410 being a function of t. 00:41:31.410 --> 00:41:35.630 And this letter t is called the parameter. 00:41:35.630 --> 00:41:38.080 In this case you should think of it-- the easiest way 00:41:38.080 --> 00:41:39.700 to think of it is as time. 00:41:39.700 --> 00:41:43.840 And what you have is what's called a trajectory. 00:41:43.840 --> 00:41:48.090 So this is also called a trajectory. 00:41:48.090 --> 00:41:54.590 And its location, let's say, at time 0, is this location here. 00:41:54.590 --> 00:41:58.310 (x(0), y(0)), that's a point in the plane. 00:41:58.310 --> 00:42:01.380 And then over here, for instance, maybe it's 00:42:01.380 --> 00:42:04.410 (x(1), y(1)). 00:42:04.410 --> 00:42:06.540 And I drew arrows along here to indicate 00:42:06.540 --> 00:42:10.110 that we're going from this place over to that place. 00:42:10.110 --> 00:42:16.190 These are later times. t = 1 is a later time than t = 0. 00:42:16.190 --> 00:42:20.170 So that's just a very casual, it's 00:42:20.170 --> 00:42:22.680 just the way we use these notations. 00:42:22.680 --> 00:42:31.670 Now let me give you the first example, which is x = a cos 00:42:31.670 --> 00:42:40.340 t, y = a sin t. 00:42:40.340 --> 00:42:43.630 And the first thing to figure out is what kind of curve 00:42:43.630 --> 00:42:45.490 this is. 00:42:45.490 --> 00:42:47.880 And to do that, we want to figure out 00:42:47.880 --> 00:42:52.000 what equation it satisfies in rectangular coordinates. 00:42:52.000 --> 00:42:53.650 So to figure out what curve this is, 00:42:53.650 --> 00:42:59.730 we recognize that if we square and add-- So we add x^2 to y^2, 00:42:59.730 --> 00:43:02.670 we're going to get something very nice and clean. 00:43:02.670 --> 00:43:08.700 We're going to get a^2 cos^2 t + a^2 sin^2 t. 00:43:11.990 --> 00:43:13.500 Yeah that's right, OK. 00:43:13.500 --> 00:43:19.230 Which is just a ^2 a^2 (cos^2 + sin^2), or in other words a^2. 00:43:19.230 --> 00:43:23.450 So lo and behold, what we have is a circle. 00:43:23.450 --> 00:43:27.840 And then we know what shape this is now. 00:43:27.840 --> 00:43:33.240 And the other thing I'd like to keep track of 00:43:33.240 --> 00:43:35.990 is which direction we're going on the circle. 00:43:35.990 --> 00:43:40.020 Because there's more to this parameter then just the shape. 00:43:40.020 --> 00:43:42.570 There's also where we are at what time. 00:43:42.570 --> 00:43:46.610 This would be, think of it like the trajectory of a planet. 00:43:46.610 --> 00:43:51.600 So here, I have to do this by plotting the picture 00:43:51.600 --> 00:43:53.250 and figuring out what happens. 00:43:53.250 --> 00:43:59.980 So at t = 0, we have (x, y) is equal to, 00:43:59.980 --> 00:44:06.190 plug in here (a cos 0, a sin 0). 00:44:06.190 --> 00:44:10.570 Which is just a * 1 + a * 0, so (a, 0). 00:44:10.570 --> 00:44:11.830 And that's here. 00:44:11.830 --> 00:44:14.660 That's the point (a, 0). 00:44:14.660 --> 00:44:18.474 We know that it's the circle of radius a. 00:44:18.474 --> 00:44:19.890 So we know that the curve is going 00:44:19.890 --> 00:44:22.140 to go around like this somehow. 00:44:22.140 --> 00:44:26.390 So let's see what happens at t = pi / 2. 00:44:26.390 --> 00:44:30.950 So at that point, we have (x, y) = (a cos(pi/2), a sin(pi/2)). 00:44:37.310 --> 00:44:41.580 Which is (0, a), because sine of pi / 2 is 1. 00:44:41.580 --> 00:44:43.190 So that's up here. 00:44:43.190 --> 00:44:46.100 So this is what happens at t = 0. 00:44:46.100 --> 00:44:49.220 This is what happens at t = pi / 2. 00:44:49.220 --> 00:44:51.390 And the trajectory clearly goes this way. 00:44:51.390 --> 00:44:54.800 In fact, this turns out to be t = pi, etc. 00:44:54.800 --> 00:44:58.330 And it repeats at t = 2pi. 00:44:58.330 --> 00:45:00.610 So the other feature that we have, 00:45:00.610 --> 00:45:11.560 which is qualitative feature, is that it's counterclockwise. 00:45:11.560 --> 00:45:17.020 Now the last little bit is going to be the arc length. 00:45:17.020 --> 00:45:19.030 Keeping track of the arc length. 00:45:19.030 --> 00:45:23.390 And we'll do that next time.