WEBVTT 00:00:00.060 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.990 Your support will help MIT OpenCourseWare 00:00:05.990 --> 00:00:09.465 continue to offer high quality educational resources for free. 00:00:09.465 --> 00:00:12.600 To make a donation, or to view additional materials 00:00:12.600 --> 00:00:16.150 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.150 --> 00:00:21.270 at ocw.mit.edu. 00:00:21.270 --> 00:00:26.900 PROFESSOR: Now, today I need to get started by finishing up 00:00:26.900 --> 00:00:28.480 what I did last time. 00:00:28.480 --> 00:00:31.560 Namely, talking about numerical methods. 00:00:31.560 --> 00:00:34.680 And I want to just carry out one example. 00:00:34.680 --> 00:00:40.660 And then I want to fill in one loose end. 00:00:40.660 --> 00:00:48.870 And then we'll talk about the unit overall. 00:00:48.870 --> 00:01:01.720 We were talking, last time, about numerical integration. 00:01:01.720 --> 00:01:05.580 I'm going to illustrate this just with the simplest example 00:01:05.580 --> 00:01:07.960 that I can. 00:01:07.960 --> 00:01:14.070 We're going to look at the integral from 1 to 2 of dx / x. 00:01:14.070 --> 00:01:16.720 Which we know perfectly well already 00:01:16.720 --> 00:01:21.310 is the log of x evaluated between 1 and 2, which is ln 2 00:01:21.310 --> 00:01:23.250 - ln 1. 00:01:23.250 --> 00:01:26.080 Which is just ln 2. 00:01:26.080 --> 00:01:29.750 Now, if you punch that into your calculator, 00:01:29.750 --> 00:01:32.610 you're going to get something like this. 00:01:32.610 --> 00:01:41.030 I hope I saved it here. 00:01:41.030 --> 00:01:41.530 Yeah. 00:01:41.530 --> 00:01:53.250 It's about 0.693147. 00:01:53.250 --> 00:01:56.150 That's more digits than we're going to get in our discussion 00:01:56.150 --> 00:01:56.650 here. 00:01:56.650 --> 00:02:02.490 Anyway, that's about how big this number is. 00:02:02.490 --> 00:02:06.390 And the numerical integration methods 00:02:06.390 --> 00:02:10.760 will give you about as much accuracy as you 00:02:10.760 --> 00:02:13.270 can get on the function itself. 00:02:13.270 --> 00:02:15.100 And, of course, some functions we 00:02:15.100 --> 00:02:17.040 may have more trouble approximating. 00:02:17.040 --> 00:02:19.400 But the function 1 / x, we know pretty well how to do, 00:02:19.400 --> 00:02:21.440 because we know how to divide. 00:02:21.440 --> 00:02:26.410 So since the function that we're integrating here is 1 / x, 00:02:26.410 --> 00:02:28.874 it's going to be not too difficult to get 00:02:28.874 --> 00:02:29.540 some arithmetic. 00:02:29.540 --> 00:02:31.740 Nevertheless, I'm going to do this 00:02:31.740 --> 00:02:34.210 in the simplest possible case. 00:02:34.210 --> 00:02:44.040 Namely, just with two intervals. 00:02:44.040 --> 00:02:47.020 Now, you really can't expect things to work so well 00:02:47.020 --> 00:02:48.040 with two intervals. 00:02:48.040 --> 00:02:52.560 That's a pretty ridiculous approximation to your function. 00:02:52.560 --> 00:02:54.830 When you have two intervals, that 00:02:54.830 --> 00:02:58.490 means you're looking at the graph of this hyperbola. 00:02:58.490 --> 00:03:04.510 And you have 1 here, and you have 2 here and you have 3/2. 00:03:04.510 --> 00:03:07.910 And you're really only keeping track of the values 00:03:07.910 --> 00:03:10.880 at these three spots. 00:03:10.880 --> 00:03:14.630 So the idea that you can approximate the area just 00:03:14.630 --> 00:03:16.940 by knowing the values of three places 00:03:16.940 --> 00:03:21.120 is a little bit of a stretch of the imagination. 00:03:21.120 --> 00:03:24.850 But we're going to try it anyway. 00:03:24.850 --> 00:03:33.320 Now, the trapezoidal rule is the following formula. 00:03:33.320 --> 00:03:38.980 It's delta x (1/2 the first value + the second value 00:03:38.980 --> 00:03:42.290 + 1/2 the third value). 00:03:42.290 --> 00:03:47.930 In this case, the pattern is 1/2, 1, 1, 1, 1, 1, 1/2. 00:03:47.930 --> 00:03:53.540 And in this case, delta x = 1/2 because this interval's 00:03:53.540 --> 00:03:55.120 of length 1. 00:03:55.120 --> 00:03:57.680 The b - a, right. 00:03:57.680 --> 00:03:59.990 Let's just point that out here. 00:03:59.990 --> 00:04:05.410 Here, b = 2. a = 1. b - a = 1. 00:04:05.410 --> 00:04:08.500 And the number n is 2. 00:04:08.500 --> 00:04:16.160 And so, delta x, which is (b - a) / n, is 1/2. 00:04:16.160 --> 00:04:17.580 So here's what we get. 00:04:17.580 --> 00:04:19.850 And let's just see what this number is. 00:04:19.850 --> 00:04:23.540 It's 1/2 of the value at here. 00:04:23.540 --> 00:04:25.540 Well, so let's just check what these values are. 00:04:25.540 --> 00:04:29.560 This value is 1, this value over here is 2/3, 00:04:29.560 --> 00:04:33.320 and the last value is 1/2. 00:04:33.320 --> 00:04:36.766 Because the function, of course, was y = 1 / x. 00:04:36.766 --> 00:04:38.640 And those were the three values that we have. 00:04:38.640 --> 00:04:42.030 So y_0, this one is y_0, this one is y_1, 00:04:42.030 --> 00:04:47.480 and this one is y_2. 00:04:47.480 --> 00:05:00.780 Now, here we have 1/2* 1 + 2/3 + 1/2 * 1/2. 00:05:00.780 --> 00:05:03.900 Now, on an exam, I don't expect you 00:05:03.900 --> 00:05:07.930 to add up long messes of numbers like this. 00:05:07.930 --> 00:05:09.700 When you have two numbers, I expect 00:05:09.700 --> 00:05:12.670 you to add them up if they're reasonable, or subtract them. 00:05:12.670 --> 00:05:14.530 Just as we do when we take antiderivatives. 00:05:14.530 --> 00:05:16.660 Like, for example, I don't want you 00:05:16.660 --> 00:05:18.860 to leave the answer to an integration like this 00:05:18.860 --> 00:05:19.490 in this form. 00:05:19.490 --> 00:05:21.514 I want you to simplify it at least down to here. 00:05:21.514 --> 00:05:23.180 And I of course don't expect you to know 00:05:23.180 --> 00:05:24.880 the numerical approximation. 00:05:24.880 --> 00:05:27.417 But I certainly expect you to be able to do that. 00:05:27.417 --> 00:05:29.000 On the other hand, when the arithmetic 00:05:29.000 --> 00:05:32.360 gets a little bit long, you can relax a little bit. 00:05:32.360 --> 00:05:35.210 But I did carry this out on my calculator. 00:05:35.210 --> 00:05:41.240 Unless I'm mistaken, it's about 0.96. 00:05:41.240 --> 00:05:44.820 It's pretty far off. 00:05:44.820 --> 00:05:47.484 So remember what it was. 00:05:47.484 --> 00:05:49.650 It's what you get when you get these straight lines. 00:05:49.650 --> 00:05:52.170 And there are these little extra pieces of junk there. 00:05:52.170 --> 00:05:55.870 Now, don't trust that too much, but the point 00:05:55.870 --> 00:06:00.540 is that it's far off. 00:06:00.540 --> 00:06:09.760 So now, let's take a look at Simpson's Rule. 00:06:09.760 --> 00:06:13.500 And I claim that Simpson's Rule is surprisingly accurate. 00:06:13.500 --> 00:06:15.500 In this case, really, even a little more than it 00:06:15.500 --> 00:06:17.410 deserves to be. 00:06:17.410 --> 00:06:29.400 The formula is (delta x / 3) (y_0 + 4 y_1 + y_2). 00:06:29.400 --> 00:06:32.460 So the pattern is 1, 4, 1, or 1, 4 00:06:32.460 --> 00:06:39.200 and then it alternates 2's and 4's until 4, 1 at the very end. 00:06:39.200 --> 00:06:43.060 And if I just plug in the numbers now, what I get 00:06:43.060 --> 00:06:47.340 is 1/6, because delta x = 1/2 again. 00:06:47.340 --> 00:06:50.600 And the value for y_0 was 1. 00:06:50.600 --> 00:06:55.120 And the value for y_1 was 2/3. 00:06:55.120 --> 00:06:59.350 And the value for y_2 was 1/2. 00:06:59.350 --> 00:07:03.740 So here's the estimate in this case. 00:07:03.740 --> 00:07:08.540 And this one I did carry out carefully. 00:07:08.540 --> 00:07:16.880 And it came out to 0.69444. 00:07:16.880 --> 00:07:18.810 Which is actually pretty impressive, 00:07:18.810 --> 00:07:20.490 if you think about it. 00:07:20.490 --> 00:07:26.050 Given what the logarithm is. 00:07:26.050 --> 00:07:31.740 Now, what's going on with Simpson's Rule in general 00:07:31.740 --> 00:07:33.240 is this. 00:07:33.240 --> 00:07:43.360 If you-- Simpson's minus the exact answer, 00:07:43.360 --> 00:07:46.970 in absolute value, is approximately of the size 00:07:46.970 --> 00:07:50.850 of (delta x)^4. 00:07:50.850 --> 00:07:52.730 That's really the way it behaves. 00:07:52.730 --> 00:08:03.300 Which means that if delta x is about 1/10, 00:08:03.300 --> 00:08:05.670 so if we had divided this up into 10 intervals, 00:08:05.670 --> 00:08:08.270 which we didn't, but if we'd divided it up into 10 00:08:08.270 --> 00:08:11.585 intervals, then you could expect that delta x-- 00:08:11.585 --> 00:08:15.320 the error would be about 10^(-4). 00:08:15.320 --> 00:08:21.090 In other words, four digits of accuracy here for this thing. 00:08:21.090 --> 00:08:25.940 But the exact analysis of this, a more careful analysis 00:08:25.940 --> 00:08:27.640 of this, is in your textbook. 00:08:27.640 --> 00:08:30.820 And I'm not going to do it. 00:08:30.820 --> 00:08:33.750 But I just want to point out that it is an effective method. 00:08:33.750 --> 00:08:35.830 It really does give you nice four-digit 00:08:35.830 --> 00:08:38.510 with manageable-- you could even really do it by hand. 00:08:38.510 --> 00:08:42.040 It's so convenient, the Simpson's Rule. 00:08:42.040 --> 00:08:46.140 Whereas the other rules aren't really that impressive 00:08:46.140 --> 00:08:51.790 as far as giving fairly accurate answers. 00:08:51.790 --> 00:08:54.330 The last little remark to make is 00:08:54.330 --> 00:08:57.520 that the reason is that Simpson's Rule is 00:08:57.520 --> 00:08:59.300 matching a parabola. 00:08:59.300 --> 00:09:03.650 And somehow the parabola follows this curve better. 00:09:03.650 --> 00:09:05.890 It's giving the exact answer. 00:09:05.890 --> 00:09:07.570 So I'll mention this. 00:09:07.570 --> 00:09:23.590 Simpson's Rule is derived using the exact answer 00:09:23.590 --> 00:09:32.770 for all degree 2 polynomials. 00:09:32.770 --> 00:09:36.680 In other words, parabolas. 00:09:36.680 --> 00:09:38.410 All parabolas. 00:09:38.410 --> 00:09:40.450 But even all the ones of lower degree. 00:09:40.450 --> 00:09:43.360 So straight lines would work, and constants 00:09:43.360 --> 00:09:44.390 would work as well. 00:09:44.390 --> 00:09:47.770 Whereas the other ones only work for, say, straight lines. 00:09:47.770 --> 00:09:51.200 The trapezoidal rule only works for straight lines. 00:09:51.200 --> 00:09:53.110 But there is a weird accident. 00:09:53.110 --> 00:09:55.650 It turns out that it also works for cubics. 00:09:55.650 --> 00:09:59.460 Once you get the formulas, it works for cubics. 00:09:59.460 --> 00:10:06.630 So it's also exact for cubics. 00:10:06.630 --> 00:10:12.400 And that's what explains the fourth order validity. 00:10:12.400 --> 00:10:14.900 The last thing that I want to point out 00:10:14.900 --> 00:10:17.820 is that this is extremely vague, what I said there. 00:10:17.820 --> 00:10:20.340 And you should be a little bit cautious about it. 00:10:20.340 --> 00:10:30.890 You need to watch out for 1/x for x near 0. 00:10:30.890 --> 00:10:34.120 All bets are off if the function is singular. 00:10:34.120 --> 00:10:36.250 And there's a lot of area under there. 00:10:36.250 --> 00:10:40.740 And it's also true that if the derivative messes up, 00:10:40.740 --> 00:10:41.770 you're in trouble too. 00:10:41.770 --> 00:10:44.480 You really need for the function to be nice and smooth in order 00:10:44.480 --> 00:10:46.700 for Simpson's Rule to work. 00:10:46.700 --> 00:10:48.740 This is wath out. 00:10:48.740 --> 00:10:55.920 That's a real wath out, but we'll try to-- Watch out. 00:10:55.920 --> 00:10:58.290 Watch out for whenever x near 0. 00:10:58.290 --> 00:11:00.770 Then this thing doesn't work. 00:11:00.770 --> 00:11:03.710 This thing really depends on bounds on derivatives. 00:11:03.710 --> 00:11:05.780 But I'm going to be relatively vague about that. 00:11:05.780 --> 00:11:12.780 I'm not attempting to give you an error analysis here. 00:11:12.780 --> 00:11:16.520 OK, so if you were doing this on an exam, 00:11:16.520 --> 00:11:21.600 how do you remember this strange pattern of numbers? 00:11:21.600 --> 00:11:27.250 The one thing that I want to recommend to you is, as a way 00:11:27.250 --> 00:11:35.620 of remembering it, so the one mnemonic device, 00:11:35.620 --> 00:11:39.340 we'll call it a mnemonic device here, 00:11:39.340 --> 00:11:42.850 for remembering what it is that you're doing, 00:11:42.850 --> 00:11:48.990 is to remind yourself of what happens for the simplest 00:11:48.990 --> 00:11:50.060 possible case. 00:11:50.060 --> 00:11:53.730 Which is f(x) = 1. 00:11:53.730 --> 00:11:57.260 It seems very modest, but if it doesn't give you 00:11:57.260 --> 00:11:59.410 the exact answer for f(x) = 1, you've 00:11:59.410 --> 00:12:02.220 got the wrong weightings. 00:12:02.220 --> 00:12:06.120 And here, if you check out what happens in the first formula 00:12:06.120 --> 00:12:14.110 here, y_0 / 2 + y_1 +..., well, we'll go all the way to y_(n-1) 00:12:14.110 --> 00:12:14.720 + y_n / 2. 00:12:17.770 --> 00:12:19.450 If you check that formula out here, 00:12:19.450 --> 00:12:21.440 this is the trapezoidal rule. 00:12:21.440 --> 00:12:25.220 If you check it out for this case, then what you get 00:12:25.220 --> 00:12:29.050 is that this is equal to delta x times what? 00:12:29.050 --> 00:12:30.629 Well, all of these are 1's. 00:12:30.629 --> 00:12:32.170 And how many are there in the middle? 00:12:32.170 --> 00:12:34.440 There are n - 1 of them in the middle. 00:12:34.440 --> 00:12:38.740 So it's 1/2 + n - 1 + 1/2. 00:12:38.740 --> 00:12:40.030 At the tail end. 00:12:40.030 --> 00:12:43.680 So all told it's delta x n. 00:12:43.680 --> 00:12:50.980 And I remind you that delta x = b - a / n. 00:12:50.980 --> 00:12:57.030 So, delta x, this thing, is equal to b - a. 00:12:57.030 --> 00:12:58.930 And that's just as it should be. 00:12:58.930 --> 00:13:03.130 What we just calculated is an approximation to this integral 00:13:03.130 --> 00:13:04.080 here. 00:13:04.080 --> 00:13:07.660 Which is just the area of the rectangle of base b - 00:13:07.660 --> 00:13:09.210 a and height 1. 00:13:09.210 --> 00:13:12.080 Which of course is b - a. 00:13:12.080 --> 00:13:15.540 So this is the check that you got your weighted average 00:13:15.540 --> 00:13:18.610 correct here. 00:13:18.610 --> 00:13:20.880 You've put the correct weightings on everything. 00:13:20.880 --> 00:13:23.400 And you can do this same thing with Simpson's Rule. 00:13:23.400 --> 00:13:31.030 And match up those quantities. 00:13:31.030 --> 00:13:36.390 There was a question in the room at some point. 00:13:36.390 --> 00:13:38.440 No, OK. 00:13:38.440 --> 00:13:42.680 So now, the next thing I want to do for you 00:13:42.680 --> 00:13:49.730 is the loose end which I left hanging. 00:13:49.730 --> 00:13:55.280 Namely, I want to compute that mysterious constant square root 00:13:55.280 --> 00:13:56.680 of pi / 2. 00:13:56.680 --> 00:14:00.260 This is really one of the most famous computations 00:14:00.260 --> 00:14:02.420 in calculus. 00:14:02.420 --> 00:14:04.740 And it's a very, very clever trick. 00:14:04.740 --> 00:14:09.160 I certainly don't expect you to come up with this trick. 00:14:09.160 --> 00:14:11.360 I certainly wouldn't have myself. 00:14:11.360 --> 00:14:15.750 But it's an important thing to calculate. 00:14:15.750 --> 00:14:17.560 And it's just very useful. 00:14:17.560 --> 00:14:19.310 So I'm going to tell you about it. 00:14:19.310 --> 00:14:22.400 And it's just on the subject that we're dealing with 00:14:22.400 --> 00:14:28.260 in this unit; namely, slicing. 00:14:28.260 --> 00:14:30.400 Or adding up. 00:14:30.400 --> 00:14:38.850 So the first step, which is just something that we already did, 00:14:38.850 --> 00:14:50.800 was that we found the volume under this curve. 00:14:50.800 --> 00:14:54.180 This bell-shaped curve, e^(-r^2). 00:14:54.180 --> 00:15:00.300 But rotated around an axis. 00:15:00.300 --> 00:15:09.020 Rotated around this axis. 00:15:09.020 --> 00:15:10.680 Around this way. 00:15:10.680 --> 00:15:12.360 So we figured that out. 00:15:12.360 --> 00:15:15.280 And that was a relatively short computation. 00:15:15.280 --> 00:15:18.870 I'm just going to remind you, it goes by shells. 00:15:18.870 --> 00:15:23.500 We integrate the whole range from 0 to infinity. 00:15:23.500 --> 00:15:28.530 And we have 2 pi r 2 pi r e^(-r^2) dr. 00:15:28.530 --> 00:15:30.532 So this again is the circumference of the shell. 00:15:30.532 --> 00:15:32.240 This is the height of the shell, and this 00:15:32.240 --> 00:15:34.650 is the thickness of the shell. 00:15:34.650 --> 00:15:45.280 Circumference, height, thickness. 00:15:45.280 --> 00:15:47.790 So we're just taking a little piece here and sweeping it 00:15:47.790 --> 00:15:48.520 around. 00:15:48.520 --> 00:15:52.000 And then adding up. 00:15:52.000 --> 00:15:58.250 And then this antiderivative is pi-- -pi e^(-r^2), 00:15:58.250 --> 00:16:00.420 evaluated at 0 and infinity. 00:16:00.420 --> 00:16:03.140 And we worked this out last time. 00:16:03.140 --> 00:16:06.520 This is pi. 00:16:06.520 --> 00:16:12.700 It's pi (1 - 0). 00:16:12.700 --> 00:16:13.200 Which is pi. 00:16:13.200 --> 00:16:15.700 So the conclusion is that V = pi. 00:16:15.700 --> 00:16:20.220 We already know that. 00:16:20.220 --> 00:16:22.770 Now, the problem that we want to deal with now 00:16:22.770 --> 00:16:31.460 is the problem not of a volume, but an area. 00:16:31.460 --> 00:16:32.990 And this looks quite different. 00:16:32.990 --> 00:16:37.340 And of course the answer is going to be different. 00:16:37.340 --> 00:16:38.250 But let's do it. 00:16:38.250 --> 00:16:40.610 So this is this question mark here. 00:16:40.610 --> 00:16:47.970 And I'm going to do the one from minus infinity to infinity. 00:16:47.970 --> 00:16:50.000 And I'll relate it to what we talked 00:16:50.000 --> 00:16:54.210 about earlier in this unit, in just a couple of minutes 00:16:54.210 --> 00:17:00.520 when I show you the procedure that we're going to follow. 00:17:00.520 --> 00:17:02.710 So here's the quantity and now, what 00:17:02.710 --> 00:17:07.210 this is interpreted as is the area under this bell curve. 00:17:07.210 --> 00:17:22.950 This time, Q is really an area. 00:17:22.950 --> 00:17:26.370 Now, what's going to turn out to happen, is this. 00:17:26.370 --> 00:17:27.920 This is the trick. 00:17:27.920 --> 00:17:31.480 We're going to compute V in a different way. 00:17:31.480 --> 00:17:35.920 And you'll see it laid out in just a second. 00:17:35.920 --> 00:17:42.870 We will compute V by slices. 00:17:42.870 --> 00:17:45.370 We're going to slice it like a piece of bread here. 00:17:45.370 --> 00:17:48.640 We're going to solve for that same thing here. 00:17:48.640 --> 00:17:52.120 And then, amazingly, what's going to happen is that we will 00:17:52.120 --> 00:17:57.720 discover that V = Q^2. 00:17:57.720 --> 00:18:03.170 That's going to be what's going to come out. 00:18:03.170 --> 00:18:07.009 And that's the end of the computation that we want. 00:18:07.009 --> 00:18:08.800 Because actually we already know what V is. 00:18:08.800 --> 00:18:11.290 We don't want to read this equation forward. 00:18:11.290 --> 00:18:12.890 We want to read it the other way. 00:18:12.890 --> 00:18:18.310 We want to say Q^2 = V, which we already know is pi. 00:18:18.310 --> 00:18:29.770 And so Q is equal to the square root of pi. 00:18:29.770 --> 00:18:33.680 I haven't shown this yet, this is the weird part. 00:18:33.680 --> 00:18:35.560 And I'm going to put it in a little box 00:18:35.560 --> 00:18:38.220 so that we know that this is what we need to check. 00:18:38.220 --> 00:18:43.070 We need to check this fact here. 00:18:43.070 --> 00:18:45.000 We haven't done that yet. 00:18:45.000 --> 00:18:51.500 Now, let me connect this with what we did a few days ago. 00:18:51.500 --> 00:18:54.690 With what I called one of the important functions 00:18:54.690 --> 00:19:02.540 of mathematics besides the ones you already know. 00:19:02.540 --> 00:19:04.810 And so the function that we were faced with, 00:19:04.810 --> 00:19:15.080 and that we discussed, was this one. 00:19:15.080 --> 00:19:19.380 And then, we were interested in the value at infinity. 00:19:19.380 --> 00:19:23.780 We were interested in this. 00:19:23.780 --> 00:19:26.930 Which, if you draw a picture of it, 00:19:26.930 --> 00:19:29.650 and you draw the same bell curve, 00:19:29.650 --> 00:19:32.610 that's the area under half. of it. 00:19:32.610 --> 00:19:35.850 That's the area starting from 0 and going to infinity. 00:19:35.850 --> 00:19:37.470 That's the area under half. 00:19:37.470 --> 00:19:46.270 So this chunk is F of infinity. 00:19:46.270 --> 00:19:50.000 And now I hope that this part of the connection 00:19:50.000 --> 00:19:52.140 is not meant to be fancy. 00:19:52.140 --> 00:19:58.760 The idea here is that Q = 2 F(infinity). 00:19:58.760 --> 00:20:00.190 This number here. 00:20:00.190 --> 00:20:04.380 And so F F(infinity) is equal to the square root of pi 00:20:04.380 --> 00:20:10.900 over 2, if we believe what we said on the last panel. 00:20:10.900 --> 00:20:14.490 And that was the thing that I drew a picture of on the board. 00:20:14.490 --> 00:20:18.390 Namely, the graph of F looked like this. 00:20:18.390 --> 00:20:20.670 And there was this asymptote, which 00:20:20.670 --> 00:20:25.020 was the limit F(x) tends to square root of pi 00:20:25.020 --> 00:20:27.650 over 2, as x goes to infinity. 00:20:27.650 --> 00:20:30.560 That was that limiting value. 00:20:30.560 --> 00:20:33.910 Which is F of infinity. 00:20:33.910 --> 00:20:40.460 So this is the asymptote. 00:20:40.460 --> 00:20:43.350 And now I've explained the connection 00:20:43.350 --> 00:20:47.260 between what we claimed before, which was quite mysterious, 00:20:47.260 --> 00:20:50.840 and what we're actually going to be able to check now. 00:20:50.840 --> 00:21:07.170 Concretely, by making this computation. 00:21:07.170 --> 00:21:09.980 So how in the world can you get something like this. 00:21:09.980 --> 00:21:14.490 What's in that orange box there, that V = Q^2. 00:21:14.490 --> 00:21:19.300 Again, the technique is to use slices here. 00:21:19.300 --> 00:21:21.860 And I'm going to have to draw you a 3-D picture 00:21:21.860 --> 00:21:24.970 to visualize the slice. 00:21:24.970 --> 00:21:27.360 Let's do that. 00:21:27.360 --> 00:21:30.520 I'm going to draw three axes now, 00:21:30.520 --> 00:21:33.640 because we're now going to be in three-dimensional space, 00:21:33.640 --> 00:21:37.250 and I want you to imagine the x-axis as coming out 00:21:37.250 --> 00:21:40.320 of the blackboard, the y-axis is horizontal, 00:21:40.320 --> 00:21:41.980 and there's a new axis, which I'll call 00:21:41.980 --> 00:21:45.090 the z-axis, which is going up. 00:21:45.090 --> 00:21:46.860 So what's happening here is that I'm 00:21:46.860 --> 00:21:50.810 thinking of this-- This is, if you like, some kind of side 00:21:50.810 --> 00:21:51.320 view. 00:21:51.320 --> 00:21:53.653 And this is a view where I've tilted things a little bit 00:21:53.653 --> 00:21:57.080 up to the top. 00:21:57.080 --> 00:21:59.974 Now, the distribution, or you could 00:21:59.974 --> 00:22:01.640 think of this target in the plane, where 00:22:01.640 --> 00:22:04.110 the most likely places to hit were in the middle 00:22:04.110 --> 00:22:05.510 and it died off. 00:22:05.510 --> 00:22:07.390 As we went down. 00:22:07.390 --> 00:22:11.400 Now, I want to draw a picture of this graph. 00:22:11.400 --> 00:22:15.240 I'm going to draw a picture of e^(-r^2). 00:22:15.240 --> 00:22:18.090 And it's basically a hump. 00:22:18.090 --> 00:22:22.420 So I'm going to take the first-- the slice along y = 0. 00:22:22.420 --> 00:22:23.880 The y = 0 slice. 00:22:23.880 --> 00:22:26.300 And I claim that that goes up like this. 00:22:26.300 --> 00:22:28.310 And then comes back down. 00:22:28.310 --> 00:22:32.100 Let me shade this in, so that you can see 00:22:32.100 --> 00:22:34.140 what kind of a slice this is. 00:22:34.140 --> 00:22:38.679 This is supposed to be along this vertical plane here. 00:22:38.679 --> 00:22:40.220 Which is coming out of the blackboard 00:22:40.220 --> 00:22:42.409 and coming towards you. 00:22:42.409 --> 00:22:43.200 And that's a slice. 00:22:43.200 --> 00:22:46.630 Now, I'm going to draw one more slice so that you 00:22:46.630 --> 00:22:49.000 can see what's happening. 00:22:49.000 --> 00:22:52.310 I'm going to draw a slice at another place. 00:22:52.310 --> 00:22:52.860 Along here. 00:22:52.860 --> 00:22:55.030 This will be y = b. 00:22:55.030 --> 00:22:56.709 Some other level. 00:22:56.709 --> 00:22:58.500 And now I'm going to show you what happens. 00:22:58.500 --> 00:23:02.960 What happens is that the hump dies down a little bit. 00:23:02.960 --> 00:23:06.750 So the bump is just a little bit lower. 00:23:06.750 --> 00:23:10.284 And it's going to look a little bit the same way. 00:23:10.284 --> 00:23:11.950 But it's just going to be a bit smaller. 00:23:11.950 --> 00:23:15.380 So there's another slice here. 00:23:15.380 --> 00:23:16.940 Like that. 00:23:16.940 --> 00:23:20.890 And I want to give a name to these slices. 00:23:20.890 --> 00:23:25.090 I'm going to call this A(b). 00:23:25.090 --> 00:23:36.130 That is, the area of the b slice. 00:23:36.130 --> 00:23:39.790 Under the surface. 00:23:39.790 --> 00:23:41.010 OK? 00:23:41.010 --> 00:23:41.870 Yes, question. 00:23:41.870 --> 00:23:47.256 STUDENT: [INAUDIBLE] 00:23:47.256 --> 00:23:48.380 PROFESSOR: Yeah, the solid. 00:23:48.380 --> 00:23:48.890 Yeah. 00:23:48.890 --> 00:23:52.830 We're trying to figure out this volume here, 00:23:52.830 --> 00:23:56.780 which is the one we started out with, by slices. 00:23:56.780 --> 00:23:59.500 So first I have to think of-- I'm going to visualize-- 00:23:59.500 --> 00:24:01.970 So here I didn't even visualize. 00:24:01.970 --> 00:24:03.490 I took a cross section and I thought 00:24:03.490 --> 00:24:06.580 about how to spin it around without actually doing that 00:24:06.580 --> 00:24:07.954 in three-dimensional space. 00:24:07.954 --> 00:24:10.120 But now I'm going to take a different kind of slice. 00:24:10.120 --> 00:24:12.260 I'm going to take that same bump, which 00:24:12.260 --> 00:24:13.680 is a three-dimensional object. 00:24:13.680 --> 00:24:16.050 I'm going to lay it down on a plane. 00:24:16.050 --> 00:24:17.540 Which looks like this. 00:24:17.540 --> 00:24:20.850 And then it's a bump here. 00:24:20.850 --> 00:24:22.260 It's a hump. 00:24:22.260 --> 00:24:27.350 And now I'm going to try to slice it by various planes. 00:24:27.350 --> 00:24:29.370 STUDENT: [INAUDIBLE] 00:24:29.370 --> 00:24:34.270 PROFESSOR: So one way of defining 00:24:34.270 --> 00:24:37.609 the bump, as you just suggested, is you take this curve 00:24:37.609 --> 00:24:39.150 and you rotate it around this z-axis. 00:24:39.150 --> 00:24:41.400 So in other words, you make this the axis of rotation, 00:24:41.400 --> 00:24:43.505 you spin it around. 00:24:43.505 --> 00:24:44.130 That's correct. 00:24:44.130 --> 00:24:48.020 So that shows you that the peaks as you go down here 00:24:48.020 --> 00:24:50.740 are going to descend the same way. 00:24:50.740 --> 00:24:54.140 But I don't want to draw those lines. 00:24:54.140 --> 00:24:57.780 I want to imagine what the parallel slices are. 00:24:57.780 --> 00:24:59.610 Because I don't want to get cross slices. 00:24:59.610 --> 00:25:01.950 I want all slices parallel to the same thing. 00:25:01.950 --> 00:25:04.997 STUDENT: [INAUDIBLE] 00:25:04.997 --> 00:25:05.580 PROFESSOR: OK. 00:25:05.580 --> 00:25:13.070 This is not particularly easy to visualize. 00:25:13.070 --> 00:25:17.960 Now, here's the formula for volume by slices. 00:25:17.960 --> 00:25:23.900 The formula for volume by slices is that you add up 00:25:23.900 --> 00:25:28.770 the areas of the slices. 00:25:28.770 --> 00:25:31.050 That's how you do it. 00:25:31.050 --> 00:25:32.465 You take each slice. 00:25:32.465 --> 00:25:33.840 You add the cross-sectional area, 00:25:33.840 --> 00:25:36.170 and then you take a little thickness, dy, 00:25:36.170 --> 00:25:38.330 and then you add all of them up. 00:25:38.330 --> 00:25:40.950 Because this is extending over the whole plane, 00:25:40.950 --> 00:25:43.630 we're going to have to go all the way from minus infinity 00:25:43.630 --> 00:25:45.100 to plus infinity. 00:25:45.100 --> 00:25:56.940 And this is the formula for volumes by slicing. 00:25:56.940 --> 00:26:00.300 And now our goal, in order to do this calculation, 00:26:00.300 --> 00:26:04.620 we're going to just fix y is equal to some b. 00:26:04.620 --> 00:26:06.840 We're just going to fix one of these slices. 00:26:06.840 --> 00:26:13.454 And we're going to calculate A(b). 00:26:13.454 --> 00:26:14.870 That's what we need to do in order 00:26:14.870 --> 00:26:23.030 to make this procedure succeed. 00:26:23.030 --> 00:26:27.740 This is the only place where this method works. 00:26:27.740 --> 00:26:32.120 But it's an important one. 00:26:32.120 --> 00:26:34.195 In order to make it work, I'm going 00:26:34.195 --> 00:26:37.760 to have to again draw the plot from a different point of view. 00:26:37.760 --> 00:26:43.550 I'm going to do the top view. 00:26:43.550 --> 00:26:48.750 So I want to look down on this x-y plane here. 00:26:48.750 --> 00:26:52.020 This is the x-direction, and here's the y-direction. 00:26:52.020 --> 00:26:56.930 And then again I want to draw my slice. 00:26:56.930 --> 00:27:00.760 My slice is here. 00:27:00.760 --> 00:27:04.860 At y = b. 00:27:04.860 --> 00:27:07.430 So we're just right on top of it. 00:27:07.430 --> 00:27:09.404 And it's coming up at some kind of bump. 00:27:09.404 --> 00:27:11.070 Here, with a little higher in the middle 00:27:11.070 --> 00:27:15.970 and going down on the sides. 00:27:15.970 --> 00:27:20.210 Now, the formula for the height is this. 00:27:20.210 --> 00:27:25.290 If I take a distance r here, the formula for the height 00:27:25.290 --> 00:27:27.290 of the bump is e^(-r^2). 00:27:31.200 --> 00:27:33.390 I'll store that over here. 00:27:33.390 --> 00:27:37.910 e^(-r^2) is the height at this place. 00:27:37.910 --> 00:27:41.020 If this distance to the origin is r. 00:27:41.020 --> 00:27:43.360 That's true all the way around. 00:27:43.360 --> 00:27:48.622 And in terms of b and x, we can figure out that 00:27:48.622 --> 00:27:49.580 by this right triangle. 00:27:49.580 --> 00:27:52.400 This height is b, and this distance is x. 00:27:52.400 --> 00:27:55.940 So r^2 = b^2 + x^2. 00:28:02.060 --> 00:28:02.560 Question. 00:28:02.560 --> 00:28:15.350 STUDENT: [INAUDIBLE] 00:28:15.350 --> 00:28:19.105 PROFESSOR: The question is, is that the x-y plane. 00:28:19.105 --> 00:28:23.280 So the answer is that over here I cleverly used the letter r. 00:28:23.280 --> 00:28:27.010 I avoided using y's and z's or anything. 00:28:27.010 --> 00:28:28.970 And over here, this is the distance r. 00:28:28.970 --> 00:28:31.650 And you like, this is z, going up. 00:28:31.650 --> 00:28:33.430 That's the way to think of it. 00:28:33.430 --> 00:28:36.180 So that all of the letters are consistent. 00:28:36.180 --> 00:28:39.170 So I just avoided giving it a name. 00:28:39.170 --> 00:28:41.960 That's good, that's exactly the point. 00:28:41.960 --> 00:28:43.160 Alright. 00:28:43.160 --> 00:28:46.870 So now, I claim I have a formula for r^2. 00:28:46.870 --> 00:28:48.397 And so I can write this down. 00:28:48.397 --> 00:28:49.230 This e^(-b^2 + x^2). 00:28:58.620 --> 00:29:01.934 But now I'm going to use the rule of exponents. 00:29:01.934 --> 00:29:04.350 Which is that this is the same as e^(-b^2) times e^(-x^2). 00:29:07.120 --> 00:29:11.330 And that's going to be the main way in which we 00:29:11.330 --> 00:29:15.590 use the particular function that we're dealing with here. 00:29:15.590 --> 00:29:20.940 That's really the main step, amazingly. 00:29:20.940 --> 00:29:33.470 So now I get to compute what A(b) is. 00:29:33.470 --> 00:29:37.960 A(b) is the area under a curve. 00:29:37.960 --> 00:29:40.300 So it's going to be, let me write it over here, 00:29:40.300 --> 00:29:47.160 A(b) is the area under this curve here. 00:29:47.160 --> 00:29:51.580 Which is some constant times-- so if you imagine, 00:29:51.580 --> 00:29:53.270 call this thing the name c. 00:29:53.270 --> 00:29:57.070 Under some curve, ce^(-x^2). 00:29:57.070 --> 00:29:59.040 Where the c is equal to e^(-b^2). 00:30:06.380 --> 00:30:07.560 That's what our slice is. 00:30:07.560 --> 00:30:09.060 In fact, it looks like one of those. 00:30:09.060 --> 00:30:12.970 It looks like one of those bumps. 00:30:12.970 --> 00:30:14.630 Here's its formula again. 00:30:14.630 --> 00:30:17.850 It's the integral from minus infinity to infinity 00:30:17.850 --> 00:30:19.500 of e^(-b^2) e^(-x^2) dx. 00:30:28.960 --> 00:30:31.560 We just recopied what I had up there. 00:30:31.560 --> 00:30:37.430 And this is the height at each value of x, with b fixed. 00:30:37.430 --> 00:30:42.630 And now, so we have a lot of steps here. 00:30:42.630 --> 00:30:44.680 But each of them is very elementary. 00:30:44.680 --> 00:30:47.380 The first one was just that law of exponents. 00:30:47.380 --> 00:30:49.520 That we could split the two into products. 00:30:49.520 --> 00:30:51.970 Now I'm going to make that splitting even further. 00:30:51.970 --> 00:30:53.420 This is a constant. 00:30:53.420 --> 00:30:55.150 It's not varying with x. 00:30:55.150 --> 00:30:59.060 So I'm going to factor it out of the integral. 00:30:59.060 --> 00:31:02.230 This is e^(-b^2) times the integral from minus infinity 00:31:02.230 --> 00:31:03.850 to infinity of e^(-x^2) dx. 00:31:06.390 --> 00:31:09.890 So this might look frightening, but actually it's just 00:31:09.890 --> 00:31:11.110 the property of an integral. 00:31:11.110 --> 00:31:13.180 All integrals have this kind of property. 00:31:13.180 --> 00:31:17.090 You can always factor out a constant. 00:31:17.090 --> 00:31:20.900 And now here comes the remarkable thing. 00:31:20.900 --> 00:31:26.920 This is e^(-b^2) times a number which is now familiar to us. 00:31:26.920 --> 00:31:27.980 What is this number? 00:31:27.980 --> 00:31:29.280 This is what we're looking for. 00:31:29.280 --> 00:31:40.550 This is our unknown, Q. So I've computed A(b), 00:31:40.550 --> 00:31:44.260 and now I'm ready to finish the problem off. 00:31:44.260 --> 00:31:48.720 A(b) = e^(-b^2) Q. Q is that strange number which we 00:31:48.720 --> 00:31:49.380 don't know yet. 00:31:49.380 --> 00:31:51.850 What it is. 00:31:51.850 --> 00:31:55.570 So now I'm going to compute the whole volume. 00:31:55.570 --> 00:31:57.850 The whole volume, remember, it's over there, 00:31:57.850 --> 00:32:02.540 it's minus infinity to infinity, A(y) dy. 00:32:02.540 --> 00:32:05.300 And now I'm just going to plug in the formula that we've found 00:32:05.300 --> 00:32:07.860 for A. Now I'm doing this for each b, 00:32:07.860 --> 00:32:09.740 so I'm doing it varying over all b's. 00:32:09.740 --> 00:32:12.710 So I have the integral from minus infinity to infinity. 00:32:12.710 --> 00:32:15.490 And here I have e^(-y^2). 00:32:15.490 --> 00:32:17.620 I've replaced b by y. 00:32:17.620 --> 00:32:21.640 And now I have Q. And I have dy. 00:32:21.640 --> 00:32:23.970 I just recopied what I had over there 00:32:23.970 --> 00:32:27.100 into the formula for slicing. 00:32:27.100 --> 00:32:29.920 And now, I'm going to do this trick of factoring out 00:32:29.920 --> 00:32:32.260 the constant a second time. 00:32:32.260 --> 00:32:33.860 This is a constant. 00:32:33.860 --> 00:32:35.180 It doesn't depend on y. 00:32:35.180 --> 00:32:38.030 It's the same for all y, it just will factor out. 00:32:38.030 --> 00:32:40.880 So this is the same as Q times the integral from minus 00:32:40.880 --> 00:32:47.350 infinity to infinity, e^(-y^2) dy. 00:32:47.350 --> 00:32:52.200 And now, lo and behold, this expression here. 00:32:52.200 --> 00:32:54.510 Of course, notice how I defined Q. Let's 00:32:54.510 --> 00:32:56.950 go back carefully to where Q is defined. 00:32:56.950 --> 00:33:02.900 Here's Q. This t is a dummy variable. 00:33:02.900 --> 00:33:04.370 It doesn't matter what I call it. 00:33:04.370 --> 00:33:05.790 I can call it x, I can call it u, 00:33:05.790 --> 00:33:07.620 I can call it v. In this case, I've 00:33:07.620 --> 00:33:10.020 given it two different names. 00:33:10.020 --> 00:33:13.030 At this stage, I called it x. 00:33:13.030 --> 00:33:14.860 And at this stage I'm calling it y. 00:33:14.860 --> 00:33:17.970 But it's the same variable. 00:33:17.970 --> 00:33:20.650 And so this little chunk is Q and altogether I have two 00:33:20.650 --> 00:33:23.320 of them, for Q^2 being the total. 00:33:23.320 --> 00:33:32.050 And that's the end of the argument. 00:33:32.050 --> 00:33:33.010 It's a real miracle. 00:33:33.010 --> 00:33:45.920 STUDENT: [INAUDIBLE] 00:33:45.920 --> 00:33:47.580 PROFESSOR: Great question. 00:33:47.580 --> 00:33:49.490 The question is, wait a minute. 00:33:49.490 --> 00:33:53.620 As y changes, doesn't x change. 00:33:53.620 --> 00:33:58.120 And so then this wouldn't be a constant. 00:33:58.120 --> 00:34:01.740 So that's the way in which we've used the letters x and y 00:34:01.740 --> 00:34:04.110 in this whole course. 00:34:04.110 --> 00:34:07.710 When you get to 18.02, you'll almost never do that. 00:34:07.710 --> 00:34:10.807 Always y and x will be different variables. 00:34:10.807 --> 00:34:12.640 And they won't have to depend on each other. 00:34:12.640 --> 00:34:15.700 Now, let me show you where on this picture the x and the y 00:34:15.700 --> 00:34:17.610 are. 00:34:17.610 --> 00:34:21.000 We've got a whole x-y plane, and here I'm fixing y = b, 00:34:21.000 --> 00:34:23.220 y isn't varying. 00:34:23.220 --> 00:34:25.340 Whereas x is changing. 00:34:25.340 --> 00:34:29.000 So, in other words, I don't have a relationship between x and y, 00:34:29.000 --> 00:34:29.900 unless I fix it. 00:34:29.900 --> 00:34:32.690 In this case I've decided that y is going to be constant. 00:34:32.690 --> 00:34:35.400 For all x. 00:34:35.400 --> 00:34:39.110 Over here, I made a computation. 00:34:39.110 --> 00:34:41.380 And I have a Q, which is just a single number. 00:34:41.380 --> 00:34:44.830 No matter which b I took, it didn't matter which. 00:34:44.830 --> 00:34:46.130 No matter which y equals b. 00:34:46.130 --> 00:34:47.930 Of course, I changed the name to b so it 00:34:47.930 --> 00:34:50.630 wouldn't be so jarring to you. 00:34:50.630 --> 00:34:53.920 But in fact this b was y all along. 00:34:53.920 --> 00:34:57.460 It's just that the x varied completely independently 00:34:57.460 --> 00:34:57.960 of the y. 00:34:57.960 --> 00:35:00.300 I could fix the y and vary the x, I could fix the x 00:35:00.300 --> 00:35:03.390 and vary the y. 00:35:03.390 --> 00:35:06.590 So it's a different use of the letters. 00:35:06.590 --> 00:35:07.720 From what you're used to. 00:35:07.720 --> 00:35:10.360 It happens that y is not a function of x. 00:35:10.360 --> 00:35:11.171 In this case. 00:35:11.171 --> 00:35:11.670 Yes. 00:35:11.670 --> 00:35:16.455 STUDENT: [INAUDIBLE] 00:35:16.455 --> 00:35:17.080 PROFESSOR: Yes. 00:35:17.080 --> 00:35:23.030 STUDENT: [INAUDIBLE] 00:35:23.030 --> 00:35:25.610 PROFESSOR: The question is, because I'm 00:35:25.610 --> 00:35:29.000 rotating around the z-axis, doesn't x change exactly 00:35:29.000 --> 00:35:30.210 as much as y does. 00:35:30.210 --> 00:35:33.790 What happens is that x and y are symmetric variables. 00:35:33.790 --> 00:35:36.870 They can be treated equally. 00:35:36.870 --> 00:35:41.660 But if I decide to take slices with respect to y being fixed 00:35:41.660 --> 00:35:45.070 and x varying, then of course they're now separated, 00:35:45.070 --> 00:35:47.410 and I have a separate role for the x and a separate role 00:35:47.410 --> 00:35:48.870 for the y. 00:35:48.870 --> 00:35:50.420 Or if I'd sliced it the other way, 00:35:50.420 --> 00:35:51.920 I would have gotten the same answer. 00:35:51.920 --> 00:35:54.560 I just would have reversed the roles of x and y. 00:35:54.560 --> 00:35:56.100 So what's happening is that x and y 00:35:56.100 --> 00:36:01.150 are on equal footing with each other in this picture, 00:36:01.150 --> 00:36:02.650 and I could've sliced the other way. 00:36:02.650 --> 00:36:04.610 I would have gotten the same answer. 00:36:04.610 --> 00:36:07.660 That's more or less the answer to your question. 00:36:07.660 --> 00:36:13.180 OK. 00:36:13.180 --> 00:36:16.300 Now I have given you a review sheet, 00:36:16.300 --> 00:36:19.150 and I want to run through, briefly, 00:36:19.150 --> 00:36:23.390 what's going to be on the exam. 00:36:23.390 --> 00:36:28.140 And this list of exam questions is 00:36:28.140 --> 00:36:30.120 what's going to be on the exam. 00:36:30.120 --> 00:36:33.390 There are, sorry this is not displayed correctly. 00:36:33.390 --> 00:36:35.110 So, exam questions, but now I'm just 00:36:35.110 --> 00:36:36.490 going to show you what they are. 00:36:36.490 --> 00:36:38.090 There are five questions on the exam. 00:36:38.090 --> 00:36:46.580 They are completely parallel to what you got last year. 00:36:46.580 --> 00:36:48.100 So you should look at that test. 00:36:48.100 --> 00:36:50.570 It's worth looking at. 00:36:50.570 --> 00:36:54.750 And you'll see in the descriptions on this sheet 00:36:54.750 --> 00:37:00.410 that what I'm describing is what's on that test. 00:37:00.410 --> 00:37:04.160 So what's going to happen is - and this is also 00:37:04.160 --> 00:37:06.790 posted on the Web - is that you'll 00:37:06.790 --> 00:37:11.020 be expected to calculate some definite integrals using 00:37:11.020 --> 00:37:13.750 the fundamental theorem of calculus. 00:37:13.750 --> 00:37:16.000 Do a numerical approximation. 00:37:16.000 --> 00:37:17.930 There'll be a Riemann, a trapezoidal rule 00:37:17.930 --> 00:37:19.550 and a Simpson's Rule. 00:37:19.550 --> 00:37:22.770 Calculate areas and volumes. 00:37:22.770 --> 00:37:25.510 And then some other cumulative sum. 00:37:25.510 --> 00:37:29.990 Either an average value or probability or perhaps work. 00:37:29.990 --> 00:37:32.410 And sketch a function which is given 00:37:32.410 --> 00:37:37.380 in this form as an integral. 00:37:37.380 --> 00:37:42.020 So those are the questions, and you'll 00:37:42.020 --> 00:37:46.080 see by the example of last year's exam exactly the style. 00:37:46.080 --> 00:37:48.540 They're really going to be very similar. 00:37:48.540 --> 00:37:49.450 Yes, question. 00:37:49.450 --> 00:38:01.582 STUDENT: [INAUDIBLE] 00:38:01.582 --> 00:38:02.790 PROFESSOR: OK, good question. 00:38:02.790 --> 00:38:05.560 So the question is, for Riemann sums, 00:38:05.560 --> 00:38:08.850 what's the difference between upper and lower, 00:38:08.850 --> 00:38:19.750 and right and left? 00:38:19.750 --> 00:38:25.300 So here we have a Riemann sum. 00:38:25.300 --> 00:38:29.720 And I'm going to give you a picture which is, 00:38:29.720 --> 00:38:33.045 maybe this function y = 1 / x, which was the one that we 00:38:33.045 --> 00:38:37.900 were discussing earlier. 00:38:37.900 --> 00:38:40.420 If you take the function y = 1 / x 00:38:40.420 --> 00:38:43.540 and you break it up into pieces here, 00:38:43.540 --> 00:38:45.370 however it doesn't matter how many pieces, 00:38:45.370 --> 00:38:48.390 let's just say there are four of them. 00:38:48.390 --> 00:38:53.020 Then the lower Riemann sum is the staircase 00:38:53.020 --> 00:38:55.490 which fits underneath. 00:38:55.490 --> 00:39:01.270 So this one is a picture of the lower sum. 00:39:01.270 --> 00:39:03.220 It's always less. 00:39:03.220 --> 00:39:08.960 And in the case of a decreasing function, it's going to be, 00:39:08.960 --> 00:39:19.530 so since if you like, since 1 / x decreases, 00:39:19.530 --> 00:39:27.710 the lower sum equals the right sum. 00:39:27.710 --> 00:39:29.980 You can see that visually on this picture. 00:39:29.980 --> 00:39:32.730 The values you're going to select 00:39:32.730 --> 00:39:38.360 are going to be the right ends of the rectangles. 00:39:38.360 --> 00:39:40.980 The upper sum is the left one. 00:39:40.980 --> 00:39:43.340 Now, if the function wiggles up and down, 00:39:43.340 --> 00:39:45.700 then you have to pick whichever side is appropriate. 00:39:45.700 --> 00:39:47.366 Or maybe it'll be a point in the middle, 00:39:47.366 --> 00:39:54.360 if the maximum is achieved in the middle. 00:39:54.360 --> 00:39:55.370 Yeah, another question. 00:39:55.370 --> 00:40:02.549 STUDENT: [INAUDIBLE] 00:40:02.549 --> 00:40:03.340 PROFESSOR: Correct. 00:40:03.340 --> 00:40:09.630 If the function is increasing, then the lower sum 00:40:09.630 --> 00:40:11.170 is the left sum. 00:40:11.170 --> 00:40:13.630 So it just exactly reverses what's here. 00:40:13.630 --> 00:40:17.920 So this is decreasing, lower sum is right-hand sum. 00:40:17.920 --> 00:40:24.400 Increasing, lower sum is left-hand sum. 00:40:24.400 --> 00:40:26.135 STUDENT: [INAUDIBLE] 00:40:26.135 --> 00:40:26.760 PROFESSOR: Yes. 00:40:26.760 --> 00:40:44.549 STUDENT: [INAUDIBLE] 00:40:44.549 --> 00:40:45.590 PROFESSOR: Good question. 00:40:45.590 --> 00:40:47.570 Suppose you're faced with a function like this 00:40:47.570 --> 00:40:48.660 in this last problem. 00:40:48.660 --> 00:40:52.270 Which, generally, these are the trickiest problems. 00:40:52.270 --> 00:40:54.150 And the question is, how are you ever going 00:40:54.150 --> 00:40:55.790 to be able to decide on an asymptote, 00:40:55.790 --> 00:40:58.110 even whether there is an asymptote. 00:40:58.110 --> 00:41:01.930 And the answer is, you're not. 00:41:01.930 --> 00:41:05.211 It's going to be pretty tricky to get 00:41:05.211 --> 00:41:07.460 keep track of what's happening as it goes to infinity. 00:41:07.460 --> 00:41:08.960 We had an example on the homework 00:41:08.960 --> 00:41:10.470 where is was oscillating and it's 00:41:10.470 --> 00:41:12.100 very unclear what's going on. 00:41:12.100 --> 00:41:15.100 You have to do a very long analysis for that. 00:41:15.100 --> 00:41:20.892 So in fact, just don't worry about that now. 00:41:20.892 --> 00:41:22.350 At the very end of the class, we'll 00:41:22.350 --> 00:41:24.180 talk a little bit about these asymptotes. 00:41:24.180 --> 00:41:27.620 And really, the first issue is whether they exist or not. 00:41:27.620 --> 00:41:30.275 And that's even something. 00:41:30.275 --> 00:41:32.150 That's a serious question which we'll address 00:41:32.150 --> 00:41:33.800 at the very end of this course. 00:41:33.800 --> 00:41:35.680 STUDENT: [INAUDIBLE] 00:41:35.680 --> 00:41:36.680 PROFESSOR: That's right. 00:41:36.680 --> 00:41:40.840 It's not going to be anything that complicated. 00:41:40.840 --> 00:41:42.350 Other questions? 00:41:42.350 --> 00:41:44.100 We we still have a five whole minutes, 00:41:44.100 --> 00:41:50.440 and I have an example to give, if nobody has a question. 00:41:50.440 --> 00:41:50.940 Yeah. 00:41:50.940 --> 00:41:51.773 STUDENT: [INAUDIBLE] 00:41:51.773 --> 00:41:55.710 PROFESSOR: The question, uh, will I tell you 00:41:55.710 --> 00:41:59.410 which one of what to use? 00:41:59.410 --> 00:42:00.910 STUDENT: [INAUDIBLE] PROFESSOR: When 00:42:00.910 --> 00:42:02.576 I tell you the numeric approximation is, 00:42:02.576 --> 00:42:04.450 you'll see on the exam. 00:42:04.450 --> 00:42:06.450 The practice exam that you have. 00:42:06.450 --> 00:42:09.220 I will ask you for all three. 00:42:09.220 --> 00:42:11.620 I will ask you for the Riemann sum, the trapezoidal rule, 00:42:11.620 --> 00:42:12.660 and the Simpson's rule. 00:42:12.660 --> 00:42:16.104 I'm guaranteeing you they'll all three be on the exam. 00:42:16.104 --> 00:42:18.020 I'm guaranteeing that every single thing which 00:42:18.020 --> 00:42:20.930 is on that piece of paper is on the exam. 00:42:20.930 --> 00:42:23.865 And you'll see it on the exam that you've got. 00:42:23.865 --> 00:42:27.420 It's exactly parallel to what's there. 00:42:27.420 --> 00:42:31.341 STUDENT: [INAUDIBLE] 00:42:31.341 --> 00:42:32.840 PROFESSOR: So with areas and volume, 00:42:32.840 --> 00:42:36.720 the question is will I tell you which method to use. 00:42:36.720 --> 00:42:48.340 So let's discuss that. 00:42:48.340 --> 00:42:58.180 So with areas and volumes, there's basically-- 00:42:58.180 --> 00:43:00.170 So this is always true with areas. 00:43:00.170 --> 00:43:07.040 And it's true with volumes of revolution. 00:43:07.040 --> 00:43:08.970 By the way you should read this sheet. 00:43:08.970 --> 00:43:12.740 Not everything that's on here have I said. 00:43:12.740 --> 00:43:15.020 But you should read it. 00:43:15.020 --> 00:43:16.470 Because it's all relevant. 00:43:16.470 --> 00:43:19.820 So with volumes of revolution, you always work your way 00:43:19.820 --> 00:43:25.600 back to some 2-D diagram. 00:43:25.600 --> 00:43:30.910 So there's some 2-D diagram which is always-- 00:43:30.910 --> 00:43:33.090 two-dimensional diagram, which is always 00:43:33.090 --> 00:43:34.960 connected with these problems. 00:43:34.960 --> 00:43:37.960 I mean, something this hard is really just too hard 00:43:37.960 --> 00:43:39.080 to do on an exam, right? 00:43:39.080 --> 00:43:41.538 I mean, I'm not going to ask you something this complicated 00:43:41.538 --> 00:43:42.472 on the exam. 00:43:42.472 --> 00:43:44.180 Because this involves a three-dimensional 00:43:44.180 --> 00:43:47.200 visualization. 00:43:47.200 --> 00:43:50.270 But once you're down to 2-D, you're 00:43:50.270 --> 00:43:52.180 supposed to be able to handle it. 00:43:52.180 --> 00:43:56.710 Now, what's the main issue after you've got your 2-D diagram? 00:43:56.710 --> 00:43:59.990 The main issue is, do you want to integrate with respect 00:43:59.990 --> 00:44:06.950 to dx or dy? 00:44:06.950 --> 00:44:13.850 And the answer is that it will depend. 00:44:13.850 --> 00:44:16.810 And if there's one that's going to cause 00:44:16.810 --> 00:44:21.310 you incredible difficulty, and I feel that you're not 00:44:21.310 --> 00:44:24.940 able to dodge it, then I might give you a hint 00:44:24.940 --> 00:44:27.750 and say you'd better use shells, or you'd better 00:44:27.750 --> 00:44:31.370 use disks or washers or something like that. 00:44:31.370 --> 00:44:34.760 But if I feel that you're grown up enough to figure out 00:44:34.760 --> 00:44:37.470 which one it is, because one of them is so ridiculous 00:44:37.470 --> 00:44:40.840 you say forget it, immediately, after thinking about it. 00:44:40.840 --> 00:44:43.460 Then I won't tell you which one. 00:44:43.460 --> 00:44:44.870 Because I figure, in other words, 00:44:44.870 --> 00:44:47.160 I don't want you to waste your time. 00:44:47.160 --> 00:44:50.580 But I'm willing to waste a minute or two of your time 00:44:50.580 --> 00:44:57.420 on a wild goose chase. 00:44:57.420 --> 00:45:01.380 So let me give you an example of this. 00:45:01.380 --> 00:45:14.670 Suppose you're looking at the curve y between 0 and x - x^3. 00:45:14.670 --> 00:45:17.380 So this is some kind of lump. 00:45:17.380 --> 00:45:18.460 Like this. 00:45:18.460 --> 00:45:25.080 It goes from 0 to 1, because the right-hand side is 0 at 0 and 1 00:45:25.080 --> 00:45:25.921 here. 00:45:25.921 --> 00:45:26.920 It's some kind of thing. 00:45:26.920 --> 00:45:29.850 And there are these two possibilities. 00:45:29.850 --> 00:45:33.220 One of them is to do shells. 00:45:33.220 --> 00:45:41.040 And then, so this is supposed to be rotated around the y-axis. 00:45:41.040 --> 00:45:45.940 In this case. 00:45:45.940 --> 00:45:48.519 And the same would apply, actually, to the area problem. 00:45:48.519 --> 00:45:50.560 So I'm doing a slightly more complicated problem. 00:45:50.560 --> 00:45:53.660 But you could ask for the area underneath this, and so forth. 00:45:53.660 --> 00:45:54.480 OK. 00:45:54.480 --> 00:45:57.400 So we can integrate this dx, or we can integrate this dy. 00:45:57.400 --> 00:46:00.130 This indicates that I'm deciding that this 00:46:00.130 --> 00:46:02.600 is going to be of thickness dx, and I'm integrating dx. 00:46:02.600 --> 00:46:04.940 So that's a choice that I'm making. 00:46:04.940 --> 00:46:07.160 Now, the minute I made that choice 00:46:07.160 --> 00:46:10.000 I know that these are shells. 00:46:10.000 --> 00:46:12.980 Because they sweep around this way and that makes them shells. 00:46:12.980 --> 00:46:15.460 Cylindrical shells. 00:46:15.460 --> 00:46:19.210 And if I do that, the setup is this. 00:46:19.210 --> 00:46:25.540 It's 2 2 pi x (x - x^3) dx. 00:46:25.540 --> 00:46:28.340 Now, I claim that when you get to this point, 00:46:28.340 --> 00:46:30.580 you already know you've won. 00:46:30.580 --> 00:46:32.920 Because this is an easy integral to calculate. 00:46:32.920 --> 00:46:34.640 So you're done here. 00:46:34.640 --> 00:46:37.840 You're happy. 00:46:37.840 --> 00:46:42.930 Now, if you happened to say, oh gee, I hate to do this. 00:46:42.930 --> 00:46:47.040 I want to do something clever, you 00:46:47.040 --> 00:46:54.220 could try to do it with cutting this way. 00:46:54.220 --> 00:46:56.500 Let's do this. 00:46:56.500 --> 00:47:01.610 And this would be the dy thickness. 00:47:01.610 --> 00:47:04.760 And then when you sweep this around, 00:47:04.760 --> 00:47:10.560 you get what we call a washer. 00:47:10.560 --> 00:47:14.710 Which is really just the difference of two disks. 00:47:14.710 --> 00:47:24.560 So the shape here is this thing swung around this axis. 00:47:24.560 --> 00:47:26.110 And it looks like this. 00:47:26.110 --> 00:47:29.270 So it's going to be the difference of radii. 00:47:29.270 --> 00:47:32.580 So what's the formula for this? 00:47:32.580 --> 00:47:37.070 It's some integral of pi times the right end, 00:47:37.070 --> 00:47:39.550 which I'll call x_2, and here the left end, 00:47:39.550 --> 00:47:40.720 which I'll call x_1. 00:47:40.720 --> 00:47:43.860 So this is pi pi (x_2^2 - x_1^2) dy. 00:47:47.800 --> 00:47:50.510 Now, already at this stage, you think to yourself 00:47:50.510 --> 00:47:54.740 this is more complicated than the other method. 00:47:54.740 --> 00:47:56.500 So you've already abandoned it. 00:47:56.500 --> 00:47:59.790 But I'm just going to go one step further into this one 00:47:59.790 --> 00:48:01.860 to see what it is that's happening. 00:48:01.860 --> 00:48:08.950 If you try to figure out what these values x_1 and x_2 are, 00:48:08.950 --> 00:48:14.740 that corresponds to solving for x_1 and x_2 in terms of y. 00:48:14.740 --> 00:48:17.200 So that's the following equation. 00:48:17.200 --> 00:48:25.590 x_1 and x_2 solve the equation that-- the curve, x - 00:48:25.590 --> 00:48:32.080 x^3 is equal y. 00:48:32.080 --> 00:48:33.790 Now, look at this equation. 00:48:33.790 --> 00:48:42.290 That's the equation x^3-- sorry, x^3 - x + y, I guess. 00:48:42.290 --> 00:48:43.320 Let's see. 00:48:43.320 --> 00:48:46.880 Yeah, that's right, is equal to 0. 00:48:46.880 --> 00:48:53.370 This is a cubic equation. 00:48:53.370 --> 00:48:55.060 Although there is a formula for this. 00:48:55.060 --> 00:48:57.351 You've never been taught the formula for this equation. 00:48:57.351 --> 00:48:58.920 So therefore, you will never, ever 00:48:58.920 --> 00:49:02.560 be able to get a formula for x_2 and x_1 as a function of y. 00:49:02.560 --> 00:49:05.030 And you'll never be able to compute this one. 00:49:05.030 --> 00:49:08.030 This is more than just a dead end, it's like crash, burn, 00:49:08.030 --> 00:49:11.300 and, you know, self-destruct. 00:49:11.300 --> 00:49:14.670 So there may be such a thing, so do the other way. 00:49:14.670 --> 00:49:17.490 Good luck, folks.