WEBVTT 00:00:00.060 --> 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 --> 00:00:10.230 continue to offer high quality educational resources for free. 00:00:10.230 --> 00:00:12.550 To make a donation or to view additional materials 00:00:12.550 --> 00:00:15.850 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.850 --> 00:00:21.950 at ocw.mit.edu. 00:00:21.950 --> 00:00:25.000 PROFESSOR: Today we're going to continue our discussion 00:00:25.000 --> 00:00:26.780 of parametric curves. 00:00:26.780 --> 00:00:29.960 I have to tell you about arc length. 00:00:29.960 --> 00:00:33.590 And let me remind me where we left off last time. 00:00:33.590 --> 00:00:45.820 This is parametric curves, continued. 00:00:45.820 --> 00:00:50.380 Last time, we talked about the parametric representation 00:00:50.380 --> 00:00:51.730 for the circle. 00:00:51.730 --> 00:00:55.960 Or one of the parametric representations for the circle. 00:00:55.960 --> 00:00:59.210 Which was this one here. 00:00:59.210 --> 00:01:05.590 And first we noted that this does parameterize, 00:01:05.590 --> 00:01:07.770 as we say, the circle. 00:01:07.770 --> 00:01:10.390 That satisfies the equation for the circle. 00:01:10.390 --> 00:01:17.990 And it's traced counterclockwise. 00:01:17.990 --> 00:01:20.590 The picture looks like this. 00:01:20.590 --> 00:01:22.350 Here's the circle. 00:01:22.350 --> 00:01:25.400 And it starts out here at t = 0 and it gets up 00:01:25.400 --> 00:01:31.820 to here at time t = pi / 2. 00:01:31.820 --> 00:01:41.330 So now I have to talk to you about arc length. 00:01:41.330 --> 00:01:43.840 In this parametric form. 00:01:43.840 --> 00:01:46.170 And the results should be the same as arc length 00:01:46.170 --> 00:01:48.830 around this circle ordinarily. 00:01:48.830 --> 00:01:55.840 And we start out with this basic differential relationship. 00:01:55.840 --> 00:02:00.080 ds^2 is dx^2 + dy^2. 00:02:00.080 --> 00:02:04.880 And then I'm going to take the square root, divide by dt, 00:02:04.880 --> 00:02:08.540 so the rate of change with respect to t of s 00:02:08.540 --> 00:02:10.780 is going to be the square root. 00:02:10.780 --> 00:02:13.220 Well, maybe I'll write it without dividing. 00:02:13.220 --> 00:02:15.130 Just write it as ds. 00:02:15.130 --> 00:02:24.640 So this would be (dx/dt)^2 + (dy/dt)^2, dt. 00:02:24.640 --> 00:02:27.050 So this is what you get formally from this equation. 00:02:27.050 --> 00:02:28.920 If you take its square roots and you 00:02:28.920 --> 00:02:32.470 divide by dt squared in the-- inside 00:02:32.470 --> 00:02:35.510 the square root, and you multiply by dt outside, 00:02:35.510 --> 00:02:36.630 so that those cancel. 00:02:36.630 --> 00:02:39.220 And this is the formal connection between the two. 00:02:39.220 --> 00:02:41.600 We'll be saying just a few more words 00:02:41.600 --> 00:02:48.430 in a few minutes about how to make sense of that rigorously. 00:02:48.430 --> 00:02:55.020 Alright so that's the set of formulas for the infinitesimal, 00:02:55.020 --> 00:02:57.110 the differential of arc length. 00:02:57.110 --> 00:03:00.820 And so to figure it out, I have to differentiate x with respect 00:03:00.820 --> 00:03:02.810 to t. 00:03:02.810 --> 00:03:04.225 And remember x is up here. 00:03:04.225 --> 00:03:11.330 It's defined by a cos t, so its derivative is -a sin t. 00:03:11.330 --> 00:03:19.850 And similarly, dy/dt = a cos t. 00:03:19.850 --> 00:03:22.000 And so I can plug this in. 00:03:22.000 --> 00:03:23.620 And I get the arc length element, 00:03:23.620 --> 00:03:36.920 which is the square root of (-a sin t)^2 + (a cos t)^2, dt. 00:03:36.920 --> 00:03:44.670 Which just becomes the square root of a^2, dt, or a dt. 00:03:44.670 --> 00:03:46.370 Now, I was about to divide by t. 00:03:46.370 --> 00:03:48.630 Let me do that now. 00:03:48.630 --> 00:03:52.050 We can also write the rate of change of arc length 00:03:52.050 --> 00:03:53.430 with respect to t. 00:03:53.430 --> 00:03:55.640 And that's a, in this case. 00:03:55.640 --> 00:04:01.420 And this gets interpreted as the speed 00:04:01.420 --> 00:04:03.180 of the particle going around. 00:04:03.180 --> 00:04:07.270 So not only, let me trade these two guys, 00:04:07.270 --> 00:04:14.020 not only do we have the direction is counterclockwise, 00:04:14.020 --> 00:04:20.010 but we also have that the speed is, if you like, it's uniform. 00:04:20.010 --> 00:04:21.740 It's constant speed. 00:04:21.740 --> 00:04:23.650 And the rate is a. 00:04:23.650 --> 00:04:26.810 So that's ds/dt. 00:04:26.810 --> 00:04:30.840 Travelling around. 00:04:30.840 --> 00:04:34.310 And that means that we can play around with the speed. 00:04:34.310 --> 00:04:37.650 And I just want to point out-- So the standard thing, what 00:04:37.650 --> 00:04:39.430 you'll have to get used to, and this 00:04:39.430 --> 00:04:42.520 is a standard presentation, you'll see this everywhere. 00:04:42.520 --> 00:04:45.710 In your physics classes and your other math classes, 00:04:45.710 --> 00:04:51.230 if you want to change the speed, so a new speed 00:04:51.230 --> 00:05:01.440 going around this would be, if I set up the equations this way. 00:05:01.440 --> 00:05:05.650 Now I'm tracing around the same circle. 00:05:05.650 --> 00:05:08.120 But the speed is going to turn out 00:05:08.120 --> 00:05:11.470 to be, if you figure it out, there'll 00:05:11.470 --> 00:05:13.200 be an extra factor of k. 00:05:13.200 --> 00:05:16.460 So it'll be ak. 00:05:16.460 --> 00:05:19.540 That's what we'll work out to be the speed. 00:05:19.540 --> 00:05:22.290 Provided k is positive and a is positive. 00:05:22.290 --> 00:05:30.490 So we're making these conventions. 00:05:30.490 --> 00:05:37.180 The constants that we're using are positive. 00:05:37.180 --> 00:05:40.980 Now, that's the first and most basic example. 00:05:40.980 --> 00:05:42.730 The one that comes up constantly. 00:05:42.730 --> 00:05:46.120 Now, let me just make those comments about notation 00:05:46.120 --> 00:05:47.860 that I wanted to make. 00:05:47.860 --> 00:05:51.884 And we've been treating these squared differentials here 00:05:51.884 --> 00:05:53.300 for a little while and I just want 00:05:53.300 --> 00:05:54.810 to pay attention a little bit more 00:05:54.810 --> 00:05:57.280 carefully to these manipulations. 00:05:57.280 --> 00:05:59.660 And what's allowed and what's not. 00:05:59.660 --> 00:06:01.890 And what's justified and what's not. 00:06:01.890 --> 00:06:06.680 So the basis for this was this approximate calculation that we 00:06:06.680 --> 00:06:11.440 had, that (delta s)^2 was (delta x)^2 + (delta y)^2. 00:06:11.440 --> 00:06:16.370 This is how we justified the arc length formula before. 00:06:16.370 --> 00:06:19.530 And let me just show you that the formula that I 00:06:19.530 --> 00:06:22.410 have up here, this basic formula for arc 00:06:22.410 --> 00:06:24.760 length in the parametric form, follows 00:06:24.760 --> 00:06:26.520 just as the other one did. 00:06:26.520 --> 00:06:31.370 And now I'm going to do it slightly more rigorously. 00:06:31.370 --> 00:06:34.400 I do the division really in disguise before I take 00:06:34.400 --> 00:06:36.310 the limit of the infinitesimal. 00:06:36.310 --> 00:06:40.349 So all I'm really doing is I'm doing this. 00:06:40.349 --> 00:06:42.390 Dividing through by this, and sorry this is still 00:06:42.390 --> 00:06:43.340 approximately equal. 00:06:43.340 --> 00:06:46.760 So I'm not dividing by something that's 0 or infinitesimal. 00:06:46.760 --> 00:06:49.320 I'm dividing by something nonzero. 00:06:49.320 --> 00:06:54.120 And here I have ((delta x)/(delta t))^2 + ((delta 00:06:54.120 --> 00:06:58.560 y)/(delta t))^2 And then in the limit, 00:06:58.560 --> 00:07:04.170 I have ds/dt is equal to the square root of this guy. 00:07:04.170 --> 00:07:13.820 Or, if you like, the square of it, so. 00:07:13.820 --> 00:07:17.200 So it's legal to divide by something that's almost 0 00:07:17.200 --> 00:07:20.340 and then take the limit as we go to 0. 00:07:20.340 --> 00:07:22.280 This is really what derivatives are all about. 00:07:22.280 --> 00:07:24.850 That we get a limit here. 00:07:24.850 --> 00:07:27.010 As the denominator goes to 0. 00:07:27.010 --> 00:07:31.510 Because the numerator's going to 0 too. 00:07:31.510 --> 00:07:32.770 So that's the notation. 00:07:32.770 --> 00:07:38.060 And now I want to warn you, maybe just a little bit, 00:07:38.060 --> 00:07:42.420 about misuses, if you like, of the notation. 00:07:42.420 --> 00:07:45.820 We don't do absolutely everything this way. 00:07:45.820 --> 00:07:49.280 This expression that came up with the squares, 00:07:49.280 --> 00:07:55.130 you should never write it as this. 00:07:55.130 --> 00:08:01.600 This, put it on the board but very quickly, never. 00:08:01.600 --> 00:08:02.420 OK. 00:08:02.420 --> 00:08:07.130 Don't do that. 00:08:07.130 --> 00:08:08.700 We use these square differentials, 00:08:08.700 --> 00:08:12.840 but we don't do it with these ratios here. 00:08:12.840 --> 00:08:15.900 But there was another place which is slightly confusing. 00:08:15.900 --> 00:08:17.680 It looks very similar, where we did 00:08:17.680 --> 00:08:20.247 use the square of the differential in a denominator. 00:08:20.247 --> 00:08:22.580 And I just want to point out to you that it's different. 00:08:22.580 --> 00:08:23.940 It's not the same. 00:08:23.940 --> 00:08:25.810 And it is OK. 00:08:25.810 --> 00:08:31.660 And that was this one. 00:08:31.660 --> 00:08:33.970 This thing here. 00:08:33.970 --> 00:08:36.900 This is a second derivative, it's something else. 00:08:36.900 --> 00:08:39.300 And it's got a dt^2 in the denominator. 00:08:39.300 --> 00:08:41.230 So it looks rather similar. 00:08:41.230 --> 00:08:49.230 But what this represents is the quantity d/dt squared. 00:08:49.230 --> 00:08:51.840 And you can see the squares came in. 00:08:51.840 --> 00:08:53.790 And squared the two expressions. 00:08:53.790 --> 00:08:58.970 And then there's also an x over here. 00:08:58.970 --> 00:09:00.600 So that's legal. 00:09:00.600 --> 00:09:02.550 Those are notations that we do use. 00:09:02.550 --> 00:09:04.070 And we can even calculate this. 00:09:04.070 --> 00:09:05.650 It has a perfectly good meaning. 00:09:05.650 --> 00:09:07.540 It's the same as the derivative with respect 00:09:07.540 --> 00:09:10.870 to t of the derivative of x, which we already know was minus 00:09:10.870 --> 00:09:17.530 sine-- sorry, a sin t, I guess. 00:09:17.530 --> 00:09:21.120 Not this example, but the previous one. 00:09:21.120 --> 00:09:21.870 Up here. 00:09:21.870 --> 00:09:24.340 So the derivative is this and so I can 00:09:24.340 --> 00:09:26.140 differentiate a second time. 00:09:26.140 --> 00:09:29.850 And I get -a cos t. 00:09:29.850 --> 00:09:31.760 So that's a perfectly legal operation. 00:09:31.760 --> 00:09:33.440 Everything in there makes sense. 00:09:33.440 --> 00:09:39.880 Just don't use that. 00:09:39.880 --> 00:09:41.770 There's another really unfortunate thing, 00:09:41.770 --> 00:09:45.820 right which is that the 2 creeps in funny places with sines. 00:09:45.820 --> 00:09:48.080 You have sine squared. 00:09:48.080 --> 00:09:50.150 It would be out here, it comes up here 00:09:50.150 --> 00:09:51.810 for some strange reason. 00:09:51.810 --> 00:09:54.730 This is just because typographers are lazy 00:09:54.730 --> 00:09:56.880 or somebody somewhere in the history 00:09:56.880 --> 00:10:00.620 of mathematical typography decided to let the 2 migrate. 00:10:00.620 --> 00:10:04.650 It would be like putting the 2 over here. 00:10:04.650 --> 00:10:07.940 There's inconsistency in mathematics, right. 00:10:07.940 --> 00:10:11.570 We're not perfect and people just develop these notations. 00:10:11.570 --> 00:10:14.450 So we have to live with them. 00:10:14.450 --> 00:10:20.210 The ones that people accept as conventions. 00:10:20.210 --> 00:10:23.230 The next example that I want to give you 00:10:23.230 --> 00:10:24.920 is just slightly different. 00:10:24.920 --> 00:10:29.140 It'll be a non-constant speed parameterization. 00:10:29.140 --> 00:10:32.470 Here x = 2 sin t. 00:10:32.470 --> 00:10:37.590 And y is, say, cos t. 00:10:37.590 --> 00:10:40.450 And let's keep track of what this one does. 00:10:40.450 --> 00:10:43.605 Now, this is a skill which I'm going 00:10:43.605 --> 00:10:45.170 to ask you about quite a bit. 00:10:45.170 --> 00:10:46.690 And it's one of several skills. 00:10:46.690 --> 00:10:48.970 You'll have to connect this with some kind 00:10:48.970 --> 00:10:50.330 of rectangular equation. 00:10:50.330 --> 00:10:51.725 An equation for x and y. 00:10:51.725 --> 00:10:54.230 And we'll be doing a certain amount of this today. 00:10:54.230 --> 00:10:56.240 In another context. 00:10:56.240 --> 00:11:00.510 Right here, to see the pattern, we know that the relationship 00:11:00.510 --> 00:11:04.160 we're going to want to use is that sin^2 + cos^2 = 1. 00:11:04.160 --> 00:11:11.910 So in fact the right thing to do here is to take 1/4 x^2 + y^2. 00:11:11.910 --> 00:11:17.020 And that's going to turn out to be sin^2 t + cos^2 t. 00:11:17.020 --> 00:11:18.310 Which is 1. 00:11:18.310 --> 00:11:19.500 So there's the equation. 00:11:19.500 --> 00:11:24.690 Here's the rectangular equation for this parametric curve. 00:11:24.690 --> 00:11:32.030 And this describes an ellipse. 00:11:32.030 --> 00:11:35.570 That's not the only information that we can get here. 00:11:35.570 --> 00:11:37.180 The other information that we can get 00:11:37.180 --> 00:11:39.570 is this qualitative information of where 00:11:39.570 --> 00:11:42.360 we start, where we're going, the direction. 00:11:42.360 --> 00:11:46.540 It starts out, I claim, at t = 0. 00:11:46.540 --> 00:11:54.630 That's when t = 0, this is (2 sin 0, cos 0), right? 00:11:54.630 --> 00:12:00.330 (2 sin 0, cos 0) is equal to the point (0, 1). 00:12:00.330 --> 00:12:02.360 So it starts up up here. 00:12:02.360 --> 00:12:05.140 At (0, 1). 00:12:05.140 --> 00:12:08.520 And then the next little place, so this is one thing 00:12:08.520 --> 00:12:11.400 that certainly you want to do. t = pi/2 00:12:11.400 --> 00:12:14.510 is maybe the next easy point to plot. 00:12:14.510 --> 00:12:22.830 And that's going to be (2 sin(pi/2), cos(pi/2)). 00:12:22.830 --> 00:12:27.880 And that's just (2, 0). 00:12:27.880 --> 00:12:31.490 And so that's over here somewhere. 00:12:31.490 --> 00:12:34.422 This is (2, 0). 00:12:34.422 --> 00:12:36.130 And we know it travels along the ellipse. 00:12:36.130 --> 00:12:40.120 And we know the minor axis is 1, and the major axis is 2, 00:12:40.120 --> 00:12:43.000 so it's doing this. 00:12:43.000 --> 00:12:45.090 So this is what happens at t = 0. 00:12:45.090 --> 00:12:48.390 This is where we are at t = pi/2. 00:12:48.390 --> 00:12:51.510 And it continues all the way around, etc. 00:12:51.510 --> 00:12:53.370 To the rest of the ellipse. 00:12:53.370 --> 00:12:57.750 This is the direction. 00:12:57.750 --> 00:13:09.620 So this one happens to be clockwise. 00:13:09.620 --> 00:13:12.640 Alright, now let's keep track of its speed. 00:13:12.640 --> 00:13:25.610 Let's keep track of the speed, and also the arc length. 00:13:25.610 --> 00:13:32.830 So the speed is the square root of the derivatives here. 00:13:32.830 --> 00:13:38.580 That would be (2 cos t)^2 + (sin t)^2. 00:13:42.160 --> 00:13:48.060 And the arc length is what? 00:13:48.060 --> 00:13:49.840 Well, if we want to go all the way around, 00:13:49.840 --> 00:13:53.540 we need to know that that takes a total of 2 pi. 00:13:53.540 --> 00:13:55.990 So 0 to 2 pi. 00:13:55.990 --> 00:13:59.310 And then we have to integrate ds, which is this expression, 00:13:59.310 --> 00:14:02.620 or ds/dt, dt. 00:14:02.620 --> 00:14:11.630 So that's the square root of 4 cos^2 t + sin^2 t, dt. 00:14:20.820 --> 00:14:26.580 The bad news, if you like, is that this is not 00:14:26.580 --> 00:14:38.524 an elementary integral. 00:14:38.524 --> 00:14:39.940 In other words, no matter how long 00:14:39.940 --> 00:14:44.240 you try to figure out how to antidifferentiate 00:14:44.240 --> 00:14:47.180 this expression, no matter how many substitutions you try, 00:14:47.180 --> 00:14:50.420 you will fail. 00:14:50.420 --> 00:14:52.030 That's the bad news. 00:14:52.030 --> 00:14:58.430 The good news is this is not an elementary integral. 00:14:58.430 --> 00:14:59.810 It's not an elementary integral. 00:14:59.810 --> 00:15:03.330 Which means that this is the answer to a question. 00:15:03.330 --> 00:15:06.230 Not something that you have to work on. 00:15:06.230 --> 00:15:11.680 So if somebody asks you for this arc length, you stop here. 00:15:11.680 --> 00:15:14.550 That's the answer, so it's actually better than it looks. 00:15:14.550 --> 00:15:17.700 And we'll try to-- I mean, I don't 00:15:17.700 --> 00:15:21.230 expect you to know already what all of the integrals 00:15:21.230 --> 00:15:22.680 are that are impossible. 00:15:22.680 --> 00:15:24.680 And which ones are hard and which ones are easy. 00:15:24.680 --> 00:15:27.200 So we'll try to coach you through when 00:15:27.200 --> 00:15:28.390 you face these things. 00:15:28.390 --> 00:15:31.670 It's not so easy to decide. 00:15:31.670 --> 00:15:34.345 I'll give you a few clues, but. 00:15:34.345 --> 00:15:34.845 OK. 00:15:34.845 --> 00:15:38.310 So this is the arc length. 00:15:38.310 --> 00:15:42.270 Now, I want to move on to the last thing that we did. 00:15:42.270 --> 00:15:44.410 Last type of thing that we did last time. 00:15:44.410 --> 00:15:54.140 Which is the surface area. 00:15:54.140 --> 00:15:55.230 And yeah, question. 00:15:55.230 --> 00:16:03.609 STUDENT: [INAUDIBLE] 00:16:03.609 --> 00:16:05.650 PROFESSOR: The question, this is a good question. 00:16:05.650 --> 00:16:08.290 The question is, when you draw the ellipse, 00:16:08.290 --> 00:16:11.650 do you not take into account what t is. 00:16:11.650 --> 00:16:16.460 The answer is that this is in disguise. 00:16:16.460 --> 00:16:20.360 What's going on here is we have a trouble 00:16:20.360 --> 00:16:24.720 with plotting in the plane what's really happening. 00:16:24.720 --> 00:16:29.210 So in other words, it's kind of in trouble. 00:16:29.210 --> 00:16:33.940 So the point is that we have two functions of t, not one. 00:16:33.940 --> 00:16:35.660 x(t) and y(t). 00:16:35.660 --> 00:16:38.970 So one thing that I can do if I plot things in the plane. 00:16:38.970 --> 00:16:41.710 In other words, the main point to make here 00:16:41.710 --> 00:16:45.150 is that we're not talking about the situation 00:16:45.150 --> 00:16:46.480 y is a function of x. 00:16:46.480 --> 00:16:47.870 We're out of that realm now. 00:16:47.870 --> 00:16:49.620 We're somewhere in a different part 00:16:49.620 --> 00:16:51.450 of the universe in our thought. 00:16:51.450 --> 00:16:54.760 And you should drop this point of view. 00:16:54.760 --> 00:16:56.990 So this depiction is not y as a function of x. 00:16:56.990 --> 00:17:00.940 Well, that's obvious because there are two values here, 00:17:00.940 --> 00:17:01.710 as opposed to one. 00:17:01.710 --> 00:17:02.960 So we're in trouble with that. 00:17:02.960 --> 00:17:04.750 And we have that background parameter, 00:17:04.750 --> 00:17:07.170 and that's exactly why we're using it. 00:17:07.170 --> 00:17:08.230 This parameter t. 00:17:08.230 --> 00:17:10.430 So that we can depict the entire curve. 00:17:10.430 --> 00:17:14.200 And deal with it as one thing. 00:17:14.200 --> 00:17:17.590 So since I can't really draw it, and since t is nowhere 00:17:17.590 --> 00:17:19.840 on the map, you should sort of imagine it as time, 00:17:19.840 --> 00:17:21.760 and there's some kind of trajectory which is travelling 00:17:21.760 --> 00:17:22.260 around. 00:17:22.260 --> 00:17:25.590 And then I just labelled a couple of the places. 00:17:25.590 --> 00:17:28.370 If somebody asked you to draw a picture of this, 00:17:28.370 --> 00:17:31.230 well, I'll tell you exactly where you need the picture 00:17:31.230 --> 00:17:33.250 in just one second, alright. 00:17:33.250 --> 00:17:36.690 It's going to come up right now in surface area. 00:17:36.690 --> 00:17:39.450 But otherwise, if nobody asks you to, 00:17:39.450 --> 00:17:44.020 you don't even have to put down t = 0 and t = pi / 2 here. 00:17:44.020 --> 00:17:46.402 Because nobody demanded it of you. 00:17:46.402 --> 00:17:47.110 Another question. 00:17:47.110 --> 00:17:51.842 STUDENT: [INAUDIBLE] 00:17:51.842 --> 00:17:53.550 PROFESSOR: So, another very good question 00:17:53.550 --> 00:17:55.890 which is exactly connected to this picture. 00:17:55.890 --> 00:17:58.060 So how is it that we're going to use the picture, 00:17:58.060 --> 00:18:02.190 and how is it we're going to use the notion of the t. 00:18:02.190 --> 00:18:07.040 The question was, why is this from t = 0 to t = 2 pi? 00:18:07.040 --> 00:18:11.000 That does use the t information on this diagram. 00:18:11.000 --> 00:18:13.150 the point is, we do know that t starts here. 00:18:13.150 --> 00:18:17.620 This is pi / 2, this is pi, this is 3 pi / 2, and this is 2 pi. 00:18:17.620 --> 00:18:19.380 When you go all the way around once, 00:18:19.380 --> 00:18:21.240 it's going to come back to itself. 00:18:21.240 --> 00:18:23.720 These are periodic functions of period 2 pi. 00:18:23.720 --> 00:18:26.780 And they come back to themselves exactly at 2 pi. 00:18:26.780 --> 00:18:29.030 And so that's why we know in order to get around once, 00:18:29.030 --> 00:18:32.250 we need to go from 0 to 2 pi. 00:18:32.250 --> 00:18:34.640 And the same thing is going to come up with surface area 00:18:34.640 --> 00:18:35.370 right now. 00:18:35.370 --> 00:18:38.720 That's going to be the issue, is what range of t 00:18:38.720 --> 00:18:45.620 we're going to need when we compute the surface area. 00:18:45.620 --> 00:18:52.144 STUDENT: [INAUDIBLE] 00:18:52.144 --> 00:18:54.185 PROFESSOR: In a question, what you might be asked 00:18:54.185 --> 00:18:56.180 is what's the rectangular equation 00:18:56.180 --> 00:18:57.650 for a parametric curve? 00:18:57.650 --> 00:19:01.720 So that would be 1/4 x^2 + y^2 = 1. 00:19:01.720 --> 00:19:03.430 And then you might be asked, plot it. 00:19:03.430 --> 00:19:06.960 Well, that would be a picture of the ellipse. 00:19:06.960 --> 00:19:10.380 OK, those are types of questions that are legal questions. 00:19:10.380 --> 00:19:27.439 STUDENT: [INAUDIBLE] 00:19:27.439 --> 00:19:28.980 PROFESSOR: The question is, do I need 00:19:28.980 --> 00:19:30.600 to know any specific formulas? 00:19:30.600 --> 00:19:33.250 Any formulas that you know and remember will help you. 00:19:33.250 --> 00:19:35.400 They may be of limited use. 00:19:35.400 --> 00:19:37.640 I'm not going to ask you to memorize anything 00:19:37.640 --> 00:19:40.820 except, I guarantee you that the circle is going to come up. 00:19:40.820 --> 00:19:43.500 Not the ellipse, the circle will come up everywhere 00:19:43.500 --> 00:19:44.320 in your life. 00:19:44.320 --> 00:19:47.710 So at least at MIT, your life at MIT. 00:19:47.710 --> 00:19:52.002 We're very round here. 00:19:52.002 --> 00:19:52.960 Yeah, another question. 00:19:52.960 --> 00:19:56.811 STUDENT: I'm just a tiny bit confused back to the basics. 00:19:56.811 --> 00:19:58.810 This is more a question from yesterday, I guess. 00:19:58.810 --> 00:20:04.390 But when you have your original ds^2 = dx^2 + dy^2, 00:20:04.390 --> 00:20:10.060 and then you integrate that to get arc length, how are you, 00:20:10.060 --> 00:20:14.360 the integral has dx's and dy's. 00:20:14.360 --> 00:20:18.590 So how are you just integrating with respect to dx? 00:20:18.590 --> 00:20:22.560 PROFESSOR: OK, the question is how are we just integrating 00:20:22.560 --> 00:20:24.200 with respect to x? 00:20:24.200 --> 00:20:26.880 So this is a question which goes back to last time. 00:20:26.880 --> 00:20:29.120 And what is it with arc length. 00:20:29.120 --> 00:20:30.370 So. 00:20:30.370 --> 00:20:35.442 I'm going to have to answer that question in connection 00:20:35.442 --> 00:20:36.400 with what we did today. 00:20:36.400 --> 00:20:38.420 So this is a subtle question. 00:20:38.420 --> 00:20:42.450 But I want you to realize that this is actually 00:20:42.450 --> 00:20:44.290 an important conceptual step here. 00:20:44.290 --> 00:20:49.810 So shhh, everybody, listen. 00:20:49.810 --> 00:20:53.340 If you're representing one-dimensional objects, 00:20:53.340 --> 00:20:56.050 which are curves, maybe, in space. 00:20:56.050 --> 00:20:58.270 Or in two dimensions. 00:20:58.270 --> 00:21:00.759 When you're keeping track of arc length, 00:21:00.759 --> 00:21:02.800 you're going to have to have an integral which is 00:21:02.800 --> 00:21:05.180 with respect to some variable. 00:21:05.180 --> 00:21:08.510 But that variable, you get to pick. 00:21:08.510 --> 00:21:12.310 And we're launching now into this variety of choices 00:21:12.310 --> 00:21:13.950 of variables with respect to which you 00:21:13.950 --> 00:21:15.980 can represent something. 00:21:15.980 --> 00:21:17.580 Now, there are some disadvantages 00:21:17.580 --> 00:21:19.370 on the circle to representing things 00:21:19.370 --> 00:21:21.480 with respect to the variable x. 00:21:21.480 --> 00:21:24.732 Because there are two points on the circle here. 00:21:24.732 --> 00:21:26.190 On the other hand, you actually can 00:21:26.190 --> 00:21:27.480 succeed with half the circle. 00:21:27.480 --> 00:21:29.620 So you can figure out the arc length that way. 00:21:29.620 --> 00:21:32.521 And then you can set it up as an integral dx. 00:21:32.521 --> 00:21:34.770 But you can also set it up as an integral with respect 00:21:34.770 --> 00:21:37.240 to any parameter you want. 00:21:37.240 --> 00:21:40.170 And the uniform parameter is perhaps the easiest one. 00:21:40.170 --> 00:21:43.210 This one is perhaps the easiest one. 00:21:43.210 --> 00:21:47.970 And so now the thing that's strange about this perspective 00:21:47.970 --> 00:21:51.470 - and I'm going to make this point later in the lecture 00:21:51.470 --> 00:21:55.810 as well - is that the letters x and y-- As I say, 00:21:55.810 --> 00:22:00.630 you should drop this notion that y is a function of x. 00:22:00.630 --> 00:22:03.950 This is what we're throwing away at this point. 00:22:03.950 --> 00:22:05.770 What we're thinking of is, you can 00:22:05.770 --> 00:22:08.110 describe things in terms of any coordinate you want. 00:22:08.110 --> 00:22:11.340 You just have to say what each one is in terms of the others. 00:22:11.340 --> 00:22:15.300 And these x and y over here are where 00:22:15.300 --> 00:22:18.380 we are in the Cartesian coordinate system. 00:22:18.380 --> 00:22:20.490 They're not-- And in this case they're 00:22:20.490 --> 00:22:24.610 functions of some other variable. 00:22:24.610 --> 00:22:25.720 Some other variable. 00:22:25.720 --> 00:22:27.150 So they're each functions. 00:22:27.150 --> 00:22:29.480 So the letters x and y just changed on you. 00:22:29.480 --> 00:22:33.710 They mean something different. x is no longer the variable. 00:22:33.710 --> 00:22:36.810 It's the function. 00:22:36.810 --> 00:22:38.542 Right? 00:22:38.542 --> 00:22:40.250 You're going to have to get used to that. 00:22:40.250 --> 00:22:42.380 That's because we run out of letters. 00:22:42.380 --> 00:22:44.870 And we kind of want to use all of them the way we want. 00:22:44.870 --> 00:22:48.290 I'll say some more about that later. 00:22:48.290 --> 00:22:51.220 So now I want to do this surface area example. 00:22:51.220 --> 00:22:59.150 I'm going to just take the surface area of the ellipsoid. 00:22:59.150 --> 00:23:11.340 The surface of the ellipsoid formed 00:23:11.340 --> 00:23:19.910 by revolving this previous example, which was Example 2. 00:23:19.910 --> 00:23:28.020 Around the y-axis. 00:23:28.020 --> 00:23:30.410 So we want to set up that surface area integral here 00:23:30.410 --> 00:23:32.490 for you. 00:23:32.490 --> 00:23:38.160 Now, I remind you that the area element looks like this. 00:23:38.160 --> 00:23:41.190 If you're revolving around the y-axis, 00:23:41.190 --> 00:23:42.815 that means you're going around this way 00:23:42.815 --> 00:23:43.720 and you have some curve. 00:23:43.720 --> 00:23:44.990 In this case it's this piece of an ellipse. 00:23:44.990 --> 00:23:46.475 If you sweep it around you're going 00:23:46.475 --> 00:23:48.770 to get what's called an ellipsoid. 00:23:48.770 --> 00:23:53.890 And there's a little chunk here, that you're wrapping around. 00:23:53.890 --> 00:23:58.430 And the important thing you need besides this ds, this arc 00:23:58.430 --> 00:24:04.120 length piece over here, is the distance to the axis. 00:24:04.120 --> 00:24:06.320 So that's this horizontal distance here. 00:24:06.320 --> 00:24:09.850 I'll draw it in another color. 00:24:09.850 --> 00:24:15.520 And that horizontal distance now has a name. 00:24:15.520 --> 00:24:18.670 And this is, again, the virtue of this coordinate system. 00:24:18.670 --> 00:24:20.170 The t is something else. 00:24:20.170 --> 00:24:21.020 This has a name. 00:24:21.020 --> 00:24:22.760 This distance has a name. 00:24:22.760 --> 00:24:27.080 This distance is called x. 00:24:27.080 --> 00:24:29.570 And it even has a formula. 00:24:29.570 --> 00:24:36.090 Its formula is 2 sin t. 00:24:36.090 --> 00:24:38.550 In terms of t. 00:24:38.550 --> 00:24:41.530 So the full formula up for the integral 00:24:41.530 --> 00:24:46.039 here is, I have to take the circumference when 00:24:46.039 --> 00:24:47.080 I spin this thing around. 00:24:47.080 --> 00:24:48.950 And this little arc length element. 00:24:48.950 --> 00:24:53.660 So I have here 2 pi times 2 sin t. 00:24:53.660 --> 00:24:55.640 That's the x variable here. 00:24:55.640 --> 00:25:00.560 And then I have here ds, which is kind of a mess. 00:25:00.560 --> 00:25:04.170 So unfortunately I don't quite have room for it. 00:25:04.170 --> 00:25:05.650 Plan ahead. 00:25:05.650 --> 00:25:15.200 Square root of 4 cos^2 t + sin^2 t, is that what it was, dt. 00:25:15.200 --> 00:25:17.740 Alright, I guess I squeezed it in there. 00:25:17.740 --> 00:25:20.090 So that was the arc length, which I re-copied 00:25:20.090 --> 00:25:21.620 from this board above. 00:25:21.620 --> 00:25:24.310 That was the ds piece. 00:25:24.310 --> 00:25:29.760 It's this whole thing including the dt. 00:25:29.760 --> 00:25:32.360 That's the answer except for one thing. 00:25:32.360 --> 00:25:33.590 What else do we need? 00:25:33.590 --> 00:25:35.350 We don't just need the integrand, 00:25:35.350 --> 00:25:37.720 this is half of setting up an integral. 00:25:37.720 --> 00:25:40.990 The other half of setting up an integral is the limits. 00:25:40.990 --> 00:25:42.840 We need specific limits here. 00:25:42.840 --> 00:25:46.760 Otherwise we don't have a number that we can get out. 00:25:46.760 --> 00:25:50.370 So we now have to think about what the limits are. 00:25:50.370 --> 00:25:52.550 And maybe somebody can see. 00:25:52.550 --> 00:25:54.429 It has something to do with this diagram 00:25:54.429 --> 00:25:55.470 of the ellipse over here. 00:25:55.470 --> 00:25:58.520 Can somebody guess what it is? 00:25:58.520 --> 00:25:59.480 0 to pi. 00:25:59.480 --> 00:26:02.070 Well, that was quick. 00:26:02.070 --> 00:26:02.620 That's it. 00:26:02.620 --> 00:26:04.709 Because we go from the top to the bottom, 00:26:04.709 --> 00:26:06.250 but we don't want to continue around. 00:26:06.250 --> 00:26:07.640 We don't want to go from 0 to 2 pi, 00:26:07.640 --> 00:26:09.723 because that would be duplicating what we're going 00:26:09.723 --> 00:26:12.020 to get when we spin around. 00:26:12.020 --> 00:26:13.730 And we know that we start at 0. 00:26:13.730 --> 00:26:15.540 It's interesting because it descends 00:26:15.540 --> 00:26:17.090 when you change variables to think 00:26:17.090 --> 00:26:20.360 of it in terms of the y variable it's going the opposite way. 00:26:20.360 --> 00:26:24.550 But anyway, just one piece of this is what we want. 00:26:24.550 --> 00:26:27.660 So that's this setup. 00:26:27.660 --> 00:26:36.230 And now I claim that this is actually a doable integral. 00:26:36.230 --> 00:26:37.850 However, it's long. 00:26:37.850 --> 00:26:39.830 I'm going to spare you, I'll just tell you 00:26:39.830 --> 00:26:41.330 how you would get started. 00:26:41.330 --> 00:26:45.970 You would use the substitution u = cos t. 00:26:45.970 --> 00:26:53.620 And then the du is going to be -sin t dt. 00:26:53.620 --> 00:26:56.290 But then, unfortunately, there's a lot more. 00:26:56.290 --> 00:26:57.810 There's another trig substitution 00:26:57.810 --> 00:27:01.430 with some other multiple of the cosine and so forth. 00:27:01.430 --> 00:27:02.420 So it goes on and on. 00:27:02.420 --> 00:27:06.260 If you want to check it yourself, you can. 00:27:06.260 --> 00:27:08.710 There's an inverse trig substitution which 00:27:08.710 --> 00:27:11.590 isn't compatible with this one. 00:27:11.590 --> 00:27:17.090 But it can be done. 00:27:17.090 --> 00:27:22.690 Calculated. 00:27:22.690 --> 00:27:26.980 In elementary terms. 00:27:26.980 --> 00:27:30.547 Yeah, another question. 00:27:30.547 --> 00:27:31.380 STUDENT: [INAUDIBLE] 00:27:31.380 --> 00:27:33.490 PROFESSOR: So, if you get this on an exam, 00:27:33.490 --> 00:27:35.240 I'm going to have to coach you through it. 00:27:35.240 --> 00:27:37.640 Either I'm going to have to tell you don't evaluate it 00:27:37.640 --> 00:27:40.130 or, you're going to have to work really hard. 00:27:40.130 --> 00:27:42.500 Or here's the first step, and then the next step 00:27:42.500 --> 00:27:44.299 is, keep on going. 00:27:44.299 --> 00:27:44.840 Or something. 00:27:44.840 --> 00:27:47.890 I'll have to give you some cues. 00:27:47.890 --> 00:27:49.260 Because it's quite long. 00:27:49.260 --> 00:27:52.860 This is way too long for an exam, this particular one. 00:27:52.860 --> 00:27:53.650 OK. 00:27:53.650 --> 00:27:55.359 It's not too long for a problem set. 00:27:55.359 --> 00:27:57.650 This is where I would leave you off if I were giving it 00:27:57.650 --> 00:27:58.320 to you on a problem set. 00:27:58.320 --> 00:28:00.220 Just to give you an idea of the order of magnitude. 00:28:00.220 --> 00:28:02.761 Whereas one of the ones that I did yesterday, I wouldn't even 00:28:02.761 --> 00:28:11.120 give you on a problem set, it was so long. 00:28:11.120 --> 00:28:17.630 So now, our next job is to move on to polar coordinates. 00:28:17.630 --> 00:28:20.960 Now, polar coordinates involve the geometry of circles. 00:28:20.960 --> 00:28:23.392 As I said, we really love circles here. 00:28:23.392 --> 00:28:24.100 We're very round. 00:28:24.100 --> 00:28:28.210 Just as I love 0, the rest of the Institute loves circles. 00:28:28.210 --> 00:28:47.380 So we're going to do that right now. 00:28:47.380 --> 00:28:58.900 What we're going to talk about now is polar coordinates. 00:28:58.900 --> 00:29:01.010 Which are set up in the following way. 00:29:01.010 --> 00:29:04.640 It's a way of describing the points in the plane. 00:29:04.640 --> 00:29:07.460 Here is a point in a plane, and here's 00:29:07.460 --> 00:29:10.530 what we think of as the usual x-y axes. 00:29:10.530 --> 00:29:12.860 And now this point is going to be described 00:29:12.860 --> 00:29:15.260 by a different pair of coordinates, different pair 00:29:15.260 --> 00:29:16.190 of numbers. 00:29:16.190 --> 00:29:26.420 Namely, the distance to the origin. 00:29:26.420 --> 00:29:30.490 And the second parameter here, second number here, 00:29:30.490 --> 00:29:32.550 is this angle theta. 00:29:32.550 --> 00:29:41.500 Which is the angle of ray from origin 00:29:41.500 --> 00:29:48.670 with the horizontal axis. 00:29:48.670 --> 00:29:50.620 So that's what it is in language. 00:29:50.620 --> 00:29:53.690 And you should put this in quotation marks, 00:29:53.690 --> 00:29:57.320 because it's not a perfect match. 00:29:57.320 --> 00:30:00.800 This is geometrically what you should always think of, 00:30:00.800 --> 00:30:03.720 but the technical details involve 00:30:03.720 --> 00:30:06.530 dealing directly with formulas. 00:30:06.530 --> 00:30:09.880 The first formula is the formula for x. 00:30:09.880 --> 00:30:11.590 And this is the fundamental, these two 00:30:11.590 --> 00:30:12.750 are the fundamental ones. 00:30:12.750 --> 00:30:16.120 Namely, x = r cos theta. 00:30:16.120 --> 00:30:17.860 The second formula is the formula 00:30:17.860 --> 00:30:21.380 for y, which is r sin theta. 00:30:21.380 --> 00:30:25.420 So these are the unambiguous definitions 00:30:25.420 --> 00:30:27.100 of polar coordinates. 00:30:27.100 --> 00:30:28.790 This is it. 00:30:28.790 --> 00:30:32.590 And this is the thing from which all other almost correct 00:30:32.590 --> 00:30:37.180 statements almost follow. 00:30:37.180 --> 00:30:39.320 But this is the one you should trust always. 00:30:39.320 --> 00:30:44.980 This is the unambiguous statement. 00:30:44.980 --> 00:30:47.360 So let me give you an example something that's 00:30:47.360 --> 00:30:52.040 close to being a good formula and is certainly 00:30:52.040 --> 00:30:57.530 useful in its way. 00:30:57.530 --> 00:31:04.180 Namely, you can think of r as being the square root of x^2 + 00:31:04.180 --> 00:31:05.810 y^2. 00:31:05.810 --> 00:31:07.336 That's easy enough to derive, it's 00:31:07.336 --> 00:31:08.460 the distance to the origin. 00:31:08.460 --> 00:31:11.320 That's pretty obvious. 00:31:11.320 --> 00:31:14.690 And the formula for theta, which you can also derive, 00:31:14.690 --> 00:31:17.480 which is that it's the inverse tangent of y y/x. 00:31:21.050 --> 00:31:24.310 However, let me just warn you that these formulas are 00:31:24.310 --> 00:31:26.870 slightly ambiguous. 00:31:26.870 --> 00:31:33.357 So somewhat ambiguous. 00:31:33.357 --> 00:31:35.440 In other words, you can't just apply them blindly. 00:31:35.440 --> 00:31:37.023 You actually have to look at a picture 00:31:37.023 --> 00:31:38.180 in order to get them right. 00:31:38.180 --> 00:31:43.690 In particular, r could be plus or minus here. 00:31:43.690 --> 00:31:47.950 And when you take the inverse tangent, 00:31:47.950 --> 00:31:52.510 there's an ambiguity between, it's the same as the inverse 00:31:52.510 --> 00:31:56.330 tangent of (-y)/(-x). 00:31:56.330 --> 00:32:00.550 So these minus signs are a plague on your existence. 00:32:00.550 --> 00:32:05.050 And you're not going to get a completely unambiguous answer 00:32:05.050 --> 00:32:07.760 out of these formulas without paying attention 00:32:07.760 --> 00:32:08.430 to the diagram. 00:32:08.430 --> 00:32:10.550 On the other hand, the formula up in the box 00:32:10.550 --> 00:32:14.337 there always works. 00:32:14.337 --> 00:32:15.920 So when people mean polar coordinates, 00:32:15.920 --> 00:32:17.370 they always mean that. 00:32:17.370 --> 00:32:22.370 And then they have conventions, which sometimes match things up 00:32:22.370 --> 00:32:27.550 with the formulas over on this next board. 00:32:27.550 --> 00:32:32.670 Let me give you various examples here first. 00:32:32.670 --> 00:32:36.260 But maybe first I should I should draw 00:32:36.260 --> 00:32:38.100 the two coordinate systems. 00:32:38.100 --> 00:32:40.560 So the coordinate system that we're used to 00:32:40.560 --> 00:32:43.360 is the rectangular coordinate system. 00:32:43.360 --> 00:32:49.190 And maybe I'll draw it in orange and green here. 00:32:49.190 --> 00:32:59.430 So these are the coordinate lines y = 0, y = 1, y = 2. 00:32:59.430 --> 00:33:01.950 That's how the coordinate system works. 00:33:01.950 --> 00:33:08.427 And over here we have the rest of the coordinate system. 00:33:08.427 --> 00:33:10.510 And this is the way we're thinking of x and y now. 00:33:10.510 --> 00:33:12.570 We're no longer thinking of y as a function of x and x 00:33:12.570 --> 00:33:13.986 as a function of y, we're thinking 00:33:13.986 --> 00:33:16.960 of x as a label of a place in a plane. 00:33:16.960 --> 00:33:20.900 And y as a label of a place in a plane. 00:33:20.900 --> 00:33:27.770 So here we have x = 0, x = 1, x = 2, etc. 00:33:27.770 --> 00:33:30.740 Here's x = -1. 00:33:30.740 --> 00:33:31.900 So forth. 00:33:31.900 --> 00:33:37.100 So that's what the rectangular coordinate system looks like. 00:33:37.100 --> 00:33:41.380 And now I should draw the other coordinate system that we have. 00:33:41.380 --> 00:33:47.900 Which is this guy here. 00:33:47.900 --> 00:33:49.610 Well, close enough. 00:33:49.610 --> 00:33:54.720 And these guys here. 00:33:54.720 --> 00:33:57.730 Kind of this bulls-eye or target operation. 00:33:57.730 --> 00:34:01.480 And this one is, say, theta = pi/2. 00:34:01.480 --> 00:34:03.870 This is theta = 0. 00:34:03.870 --> 00:34:07.710 This is theta = -pi/4. 00:34:07.710 --> 00:34:11.380 For instance, so I've just labeled for you three 00:34:11.380 --> 00:34:17.870 of the rays on this diagram. 00:34:17.870 --> 00:34:23.130 It's kind of like a radar screen. 00:34:23.130 --> 00:34:28.840 And then in pink, this is maybe r = 2, the radius 2. 00:34:28.840 --> 00:34:33.980 And inside is r = 1. 00:34:33.980 --> 00:34:38.090 So it's a different coordinate system for the plane. 00:34:38.090 --> 00:34:42.120 And again, the letter r represents measuring 00:34:42.120 --> 00:34:44.930 how far we are from the origin. 00:34:44.930 --> 00:34:47.060 The theta represents something about the angle, 00:34:47.060 --> 00:34:50.250 which ray we're on. 00:34:50.250 --> 00:34:52.260 And they're just two different variables. 00:34:52.260 --> 00:35:10.880 And this is a very different kind of coordinate system. 00:35:10.880 --> 00:35:15.391 OK so, our main job is just to get used to this. 00:35:15.391 --> 00:35:15.890 For now. 00:35:15.890 --> 00:35:18.350 You will be using this a lot in 18.02. 00:35:18.350 --> 00:35:20.570 It's very useful in physics. 00:35:20.570 --> 00:35:25.680 And our job is just to get started with it. 00:35:25.680 --> 00:35:29.990 And so, let's try a few examples here. 00:35:29.990 --> 00:35:31.220 Tons of examples. 00:35:31.220 --> 00:35:34.590 We'll start out very slow. 00:35:34.590 --> 00:35:41.860 If you have (x, y) = (1, -1), that's a point in the plane. 00:35:41.860 --> 00:35:44.380 I can draw that point. 00:35:44.380 --> 00:35:46.460 It's down here, right? 00:35:46.460 --> 00:35:50.630 This is -1 and this is 1, and here's my point, (1, -1). 00:35:50.630 --> 00:35:53.550 I can figure out what the representative is 00:35:53.550 --> 00:35:56.670 of this in polar coordinates. 00:35:56.670 --> 00:36:03.040 So in polar coordinates, there are actually 00:36:03.040 --> 00:36:05.130 a bunch of choices here. 00:36:05.130 --> 00:36:09.250 First of all, I'll tell you one choice. 00:36:09.250 --> 00:36:10.970 If I start with the angle horizontally, 00:36:10.970 --> 00:36:14.200 I wrap all the way around, that would 00:36:14.200 --> 00:36:19.350 be to this ray here-- Let's do it in green again. 00:36:19.350 --> 00:36:21.820 Alright, I labeled it actually as -pi/4, 00:36:21.820 --> 00:36:27.310 but another way of looking at it is that it's this angle here. 00:36:27.310 --> 00:36:31.440 So that would be r = square root of 2. 00:36:31.440 --> 00:36:34.210 Theta = 7pi/4. 00:36:38.150 --> 00:36:41.750 So that's one possibility of the angle and the distance. 00:36:41.750 --> 00:36:45.380 I know the distance is a square root of 2, that's not hard. 00:36:45.380 --> 00:36:47.930 Another way of looking at it is the way 00:36:47.930 --> 00:36:49.640 which was suggested when I labeled this 00:36:49.640 --> 00:36:51.230 with a negative angle. 00:36:51.230 --> 00:36:56.850 And that would be r = square root of 2, theta = -pi/4. 00:36:56.850 --> 00:36:58.370 And these are both legal. 00:36:58.370 --> 00:37:00.736 These are perfectly legal representatives. 00:37:00.736 --> 00:37:02.110 And that's what I meant by saying 00:37:02.110 --> 00:37:06.180 that these representations over here are somewhat ambiguous. 00:37:06.180 --> 00:37:08.900 There's more than one answer to this question, of what 00:37:08.900 --> 00:37:11.860 the polar representation is. 00:37:11.860 --> 00:37:17.190 A third possibility, which is even more dicey but also legal, 00:37:17.190 --> 00:37:21.890 is r equals minus square root of 2. 00:37:21.890 --> 00:37:25.360 Theta = 3pi/4. 00:37:25.360 --> 00:37:30.080 Now, what that corresponds to doing is going around to here. 00:37:30.080 --> 00:37:33.490 We're pointing out 3/4 pi direction. 00:37:33.490 --> 00:37:37.130 But then going negative square root of 2 distance. 00:37:37.130 --> 00:37:39.710 We're going backwards. 00:37:39.710 --> 00:37:42.250 So we're landing in the same place. 00:37:42.250 --> 00:37:44.380 So this is also legal. 00:37:44.380 --> 00:37:44.880 Yeah. 00:37:44.880 --> 00:37:51.324 STUDENT: [INAUDIBLE] 00:37:51.324 --> 00:37:53.240 PROFESSOR: The question is, don't the radiuses 00:37:53.240 --> 00:37:54.989 have to be positive because they represent 00:37:54.989 --> 00:37:56.620 a distance to the origin? 00:37:56.620 --> 00:38:00.620 The answer is I lied to you here. 00:38:00.620 --> 00:38:04.770 All of these things that I said are wrong, except for this. 00:38:04.770 --> 00:38:09.020 Which is the rule for what polar coordinates mean. 00:38:09.020 --> 00:38:21.170 So it's maybe plus or minus the distance, is what it is always. 00:38:21.170 --> 00:38:29.090 I try not to lie to you too much, but I do succeed. 00:38:29.090 --> 00:38:36.270 Now, let's do a little bit more practice here. 00:38:36.270 --> 00:38:38.330 There are some easy examples, which 00:38:38.330 --> 00:38:40.580 I will run through very quickly. r = a, 00:38:40.580 --> 00:38:44.100 we already know this is a circle. 00:38:44.100 --> 00:38:51.280 And the 3 theta equals a constant is a ray. 00:38:51.280 --> 00:38:54.820 However, this involves an implicit assumption, which 00:38:54.820 --> 00:38:57.360 I want to point out to you. 00:38:57.360 --> 00:38:59.040 So this is Example 3. 00:38:59.040 --> 00:39:01.060 Theta's equal to a constant is a ray. 00:39:01.060 --> 00:39:14.070 But this implicitly assumes 0 <= r < infinity. 00:39:14.070 --> 00:39:19.400 If you really wanted to allow minus infinity < r < infinity 00:39:19.400 --> 00:39:22.890 in this example, you would get a line. 00:39:22.890 --> 00:39:28.540 Gives the whole line. 00:39:28.540 --> 00:39:30.050 It gives everything behind. 00:39:30.050 --> 00:39:33.085 So you go out on some ray, you go backwards on that ray 00:39:33.085 --> 00:39:36.460 and you get the whole line through the origin, both ways. 00:39:36.460 --> 00:39:39.740 If you allow r going to minus infinity as well. 00:39:39.740 --> 00:39:42.310 So the typical conventions, so here 00:39:42.310 --> 00:39:49.680 are the typical conventions. 00:39:49.680 --> 00:39:53.140 And you will see people assume this without even telling you. 00:39:53.140 --> 00:39:55.340 So you need to watch out for it. 00:39:55.340 --> 00:39:57.450 The typical conventions are certainly this one, 00:39:57.450 --> 00:40:00.270 which is a nice thing to do. 00:40:00.270 --> 00:40:04.240 Pretty much all the time, although not all the time. 00:40:04.240 --> 00:40:05.360 Most of the time. 00:40:05.360 --> 00:40:11.950 And then you might have theta ranging from minus pi 00:40:11.950 --> 00:40:15.730 to pi, so in other words symmetric around 0. 00:40:15.730 --> 00:40:21.630 Or, another very popular choice is this one. 00:40:21.630 --> 00:40:25.890 Theta's >= 0 and strictly less than 2pi. 00:40:25.890 --> 00:40:29.660 So these are the two typical ranges 00:40:29.660 --> 00:40:33.930 in which all of these variables are chosen. 00:40:33.930 --> 00:40:34.900 But not always. 00:40:34.900 --> 00:40:43.210 You'll find that it's not consistent. 00:40:43.210 --> 00:40:46.010 As I said, our job is to get used to this. 00:40:46.010 --> 00:40:49.600 And I need to work up to some slightly more 00:40:49.600 --> 00:40:51.420 complicated examples. 00:40:51.420 --> 00:40:57.840 Some of which I'll give you on next Tuesday. 00:40:57.840 --> 00:41:05.780 But let's do a few more. 00:41:05.780 --> 00:41:10.820 So, I guess this is Example 4. 00:41:10.820 --> 00:41:14.980 Example 4, I'm going to take y = 1. 00:41:14.980 --> 00:41:20.650 That's awfully simple in rectangular coordinates. 00:41:20.650 --> 00:41:23.960 But interestingly, you might conceivably 00:41:23.960 --> 00:41:26.050 want to deal with it in polar coordinates. 00:41:26.050 --> 00:41:29.580 If you do, so here's how you make the translation. 00:41:29.580 --> 00:41:32.850 But this translation is not so terrible. 00:41:32.850 --> 00:41:39.080 What you do is, you plug in y = r sin(theta). 00:41:39.080 --> 00:41:40.710 That's all you have to do. 00:41:40.710 --> 00:41:42.760 And so that's going to be equal to 1. 00:41:42.760 --> 00:41:46.240 And that's going to give us our polar equation. 00:41:46.240 --> 00:41:50.330 The polar equation is r = 1 / sin(theta). 00:41:50.330 --> 00:41:54.360 There it is. 00:41:54.360 --> 00:41:58.120 And let's draw a picture of it. 00:41:58.120 --> 00:42:03.480 So here's a picture of the line y = 1. 00:42:03.480 --> 00:42:11.950 And now we see that if we take our rays going out from here, 00:42:11.950 --> 00:42:17.240 they collide with the line at various lengths. 00:42:17.240 --> 00:42:19.760 So if you take an angle, theta, here there'll 00:42:19.760 --> 00:42:21.364 be a distance r corresponding to that 00:42:21.364 --> 00:42:23.030 and you'll hit this in exactly one spot. 00:42:23.030 --> 00:42:26.600 For each theta you'll have a different radius. 00:42:26.600 --> 00:42:27.810 And it's a variable radius. 00:42:27.810 --> 00:42:30.740 It's given by this formula here. 00:42:30.740 --> 00:42:33.210 And so to trace this line out, you actually 00:42:33.210 --> 00:42:36.120 have to realize that there's one more thing involved. 00:42:36.120 --> 00:42:40.160 Which is the possible range of theta. 00:42:40.160 --> 00:42:41.730 Again, when you're doing integrations 00:42:41.730 --> 00:42:44.104 you're going to need to know those limits of integration. 00:42:44.104 --> 00:42:46.360 So you're going to need to know this. 00:42:46.360 --> 00:42:48.990 The range here goes from theta = 0, 00:42:48.990 --> 00:42:51.230 that's sort of when it's out at infinity. 00:42:51.230 --> 00:42:53.140 That's when the denominator is 0 here. 00:42:53.140 --> 00:42:55.800 And it goes all the way to pi. 00:42:55.800 --> 00:42:57.940 Swing around just one half-turn. 00:42:57.940 --> 00:43:03.610 So the range here is 0 < theta < pi. 00:43:03.610 --> 00:43:04.620 Yeah, question. 00:43:04.620 --> 00:43:09.676 STUDENT: [INAUDIBLE] 00:43:09.676 --> 00:43:11.050 PROFESSOR: The question is, is it 00:43:11.050 --> 00:43:13.940 typical to express r as a function of theta, 00:43:13.940 --> 00:43:16.550 or vice versa, or does it matter? 00:43:16.550 --> 00:43:19.790 The answer is that for the purposes of this course, 00:43:19.790 --> 00:43:24.420 we're almost always going to be writing things in this form. 00:43:24.420 --> 00:43:27.070 r as a function of theta. 00:43:27.070 --> 00:43:30.050 And you can do whatever you want. 00:43:30.050 --> 00:43:33.920 This turns out to be what we'll be doing in this course, 00:43:33.920 --> 00:43:37.040 exclusively. 00:43:37.040 --> 00:43:40.570 As you'll see when we get to other examples, 00:43:40.570 --> 00:43:42.160 it's the traditional sort of thing 00:43:42.160 --> 00:43:45.060 to do when you're thinking about observing a planet 00:43:45.060 --> 00:43:48.650 or something like that. 00:43:48.650 --> 00:43:52.930 You see the angle, and then you guess far away it is. 00:43:52.930 --> 00:43:55.600 But it's not necessary. 00:43:55.600 --> 00:43:58.940 The formulas are often easier this way. 00:43:58.940 --> 00:44:00.370 For the examples that we have. 00:44:00.370 --> 00:44:02.610 Because it's usually a trig function of theta. 00:44:02.610 --> 00:44:05.110 Whereas the other way, it would be an inverse trig function. 00:44:05.110 --> 00:44:08.930 So it's an uglier expression. 00:44:08.930 --> 00:44:10.540 As you can see. 00:44:10.540 --> 00:44:12.860 The real reason is that we choose this thing that's 00:44:12.860 --> 00:44:19.410 easier to deal with. 00:44:19.410 --> 00:44:22.200 So now let me give you a slightly more complicated 00:44:22.200 --> 00:44:24.410 example of the same type. 00:44:24.410 --> 00:44:28.930 Where we use a shortcut. 00:44:28.930 --> 00:44:31.680 This is a standard example. 00:44:31.680 --> 00:44:33.960 And it comes up a lot. 00:44:33.960 --> 00:44:40.730 And so this is an off-center circle. 00:44:40.730 --> 00:44:44.000 A circle is really easy to describe, but not 00:44:44.000 --> 00:44:54.170 necessarily if the center is on the rim of the circle. 00:44:54.170 --> 00:44:56.550 So that's a different problem. 00:44:56.550 --> 00:44:59.990 And let's do this with a circle of radius a. 00:44:59.990 --> 00:45:06.120 So this is the point (a, 0) and this is (2a, 0). 00:45:06.120 --> 00:45:08.550 And actually, if you know these two numbers, 00:45:08.550 --> 00:45:11.080 you'll be able to remember the result of this calculation. 00:45:11.080 --> 00:45:13.780 Which you'll do about five or six times and then finally 00:45:13.780 --> 00:45:17.310 you'll memorize it during 18.02 when you will need it a lot. 00:45:17.310 --> 00:45:21.220 So this is a standard calculation here. 00:45:21.220 --> 00:45:24.350 So the starting place is the rectangular equation. 00:45:24.350 --> 00:45:27.170 And we're going to pass to the polar representation. 00:45:27.170 --> 00:45:33.550 The rectangular representation is (x-a)^2 + y^2 = a^2. 00:45:33.550 --> 00:45:40.290 So this is a circle centered at (a, 0) of radius a. 00:45:40.290 --> 00:45:44.110 And now, if you like, the slow way of doing this 00:45:44.110 --> 00:45:50.145 would be to plug in x = r cos(theta), y = r sin(theta). 00:45:50.145 --> 00:45:51.520 The way I did in this first step. 00:45:51.520 --> 00:45:53.500 And that works perfectly well. 00:45:53.500 --> 00:45:56.980 But I'm going to do it more quickly than that. 00:45:56.980 --> 00:46:00.070 Because I can sort of see in advance how it's going to work. 00:46:00.070 --> 00:46:09.810 I'm just going to expand this out. 00:46:09.810 --> 00:46:13.160 And now I see the a^2's cancel. 00:46:13.160 --> 00:46:17.120 And not only that, but x^2 + y^2 = r^2. 00:46:17.120 --> 00:46:19.670 So this becomes r^2. 00:46:19.670 --> 00:46:28.590 That's x^2 + y^2 - 2ax = 0. 00:46:28.590 --> 00:46:32.360 The r came from the fact that r^2 = x^2 + y^2. 00:46:36.100 --> 00:46:37.890 So I'm doing this the rapid way. 00:46:37.890 --> 00:46:40.260 You can do it by plugging in, as I said. 00:46:40.260 --> 00:46:43.900 r equals-- So now that I've simplified it, 00:46:43.900 --> 00:46:45.720 I am going to use that procedure. 00:46:45.720 --> 00:46:47.570 I'm going to plug in. 00:46:47.570 --> 00:46:57.120 So here I have r^2 - 2ar cos(theta) = 0. 00:46:57.120 --> 00:47:00.146 I just plugged in for x. 00:47:00.146 --> 00:47:02.270 As I said, I could have done that at the beginning. 00:47:02.270 --> 00:47:06.430 I just simplified first. 00:47:06.430 --> 00:47:11.780 And now, this is the same thing as r^2 = 2ar cos(theta). 00:47:11.780 --> 00:47:13.530 And we're almost done. 00:47:13.530 --> 00:47:19.230 There's a boring part of this equation, which is r = 0. 00:47:19.230 --> 00:47:21.530 And then there's, if I divide by r, 00:47:21.530 --> 00:47:23.430 there's the interesting part of the equation. 00:47:23.430 --> 00:47:25.830 Which is this. 00:47:25.830 --> 00:47:28.810 So this is or r = 0. 00:47:28.810 --> 00:47:33.690 Which is already included in that equation anyway. 00:47:33.690 --> 00:47:36.890 So I'm allowed to divide by r because in the case of r = 0, 00:47:36.890 --> 00:47:39.781 this is represented anyway. 00:47:39.781 --> 00:47:40.280 Question. 00:47:40.280 --> 00:47:44.390 STUDENT: [INAUDIBLE] 00:47:44.390 --> 00:47:46.270 PROFESSOR: r = 0 is just one case. 00:47:46.270 --> 00:47:48.380 That is, it's the union of these two. 00:47:48.380 --> 00:47:49.550 It's both. 00:47:49.550 --> 00:47:50.670 Both are possible. 00:47:50.670 --> 00:47:53.270 So r = 0 is one point on it. 00:47:53.270 --> 00:47:56.150 And this is all of it. 00:47:56.150 --> 00:48:01.230 So we can just ignore this. 00:48:01.230 --> 00:48:04.500 So now I want to say one more important thing. 00:48:04.500 --> 00:48:06.600 You need to understand the range of this. 00:48:06.600 --> 00:48:10.840 So wait a second and we're going to figure out the range here. 00:48:10.840 --> 00:48:13.710 The range is very important, because otherwise you'll 00:48:13.710 --> 00:48:18.280 never be able to integrate using this representation here. 00:48:18.280 --> 00:48:19.840 So this is the representation. 00:48:19.840 --> 00:48:25.190 But notice when theta = 0, we're out here at 2a. 00:48:25.190 --> 00:48:26.780 That's consistent, and that's actually 00:48:26.780 --> 00:48:29.020 how you remember this factor 2a here. 00:48:29.020 --> 00:48:31.570 Because if you remember this picture and where you land when 00:48:31.570 --> 00:48:34.830 theta = 0. 00:48:34.830 --> 00:48:36.370 So that's the theta = 0 part. 00:48:36.370 --> 00:48:39.440 But now as I tip up like this, you 00:48:39.440 --> 00:48:43.780 see that when we get to vertical, we're done. 00:48:43.780 --> 00:48:44.630 With the circle. 00:48:44.630 --> 00:48:46.463 It's gotten shorter and shorter and shorter, 00:48:46.463 --> 00:48:49.020 and at theta = pi/2, we're down at 0. 00:48:49.020 --> 00:48:51.720 Because that's cos(pi/2) = 0. 00:48:51.720 --> 00:48:53.770 So it swings up like this. 00:48:53.770 --> 00:48:55.400 And it gets up to pi/2. 00:48:55.400 --> 00:48:57.110 Similarly, we swing down like this. 00:48:57.110 --> 00:48:59.000 And then we're done. 00:48:59.000 --> 00:49:04.510 So the range is -pi/2 < theta < pi/2. 00:49:04.510 --> 00:49:06.650 Or, if you want to throw in the r = 0 case, 00:49:06.650 --> 00:49:08.700 you can throw in this, this is repeating, 00:49:08.700 --> 00:49:11.200 if you like, at the ends. 00:49:11.200 --> 00:49:14.100 So this is the range of this circle. 00:49:14.100 --> 00:49:17.150 And let's see. 00:49:17.150 --> 00:49:21.300 Next time we'll figure out area in polar coordinates.