WEBVTT 00:00:00.000 --> 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 --> 00:00:05.980 Your support will help MIT OpenCourseWare 00:00:05.980 --> 00:00:09.930 continue to offer high quality educational resources for free. 00:00:09.930 --> 00:00:12.550 To make a donation or to view additional materials 00:00:12.550 --> 00:00:16.180 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.180 --> 00:00:21.970 at ocw.mit.edu. 00:00:21.970 --> 00:00:25.330 PROFESSOR: Today I want to get started 00:00:25.330 --> 00:00:30.340 by correcting a mistake that I made last time. 00:00:30.340 --> 00:00:37.410 And this was mistaken terminology. 00:00:37.410 --> 00:00:41.290 I said that what we were computing, 00:00:41.290 --> 00:00:44.480 when we computed in this candy bar example, 00:00:44.480 --> 00:00:47.510 was energy and not heat. 00:00:47.510 --> 00:00:49.270 But it's both. 00:00:49.270 --> 00:00:51.010 They're the same thing. 00:00:51.010 --> 00:00:55.320 And in fact, energy, heat and work 00:00:55.320 --> 00:00:59.470 are all the same thing in physics. 00:00:59.470 --> 00:01:03.895 I was foolishly considering the much more 00:01:03.895 --> 00:01:08.350 - what am I trying to say - the intuitive feeling of heat 00:01:08.350 --> 00:01:10.720 as just being the same as temperature. 00:01:10.720 --> 00:01:17.890 But in physics, usually heat is measured in calories 00:01:17.890 --> 00:01:20.480 and energy can be in lots of things. 00:01:20.480 --> 00:01:26.610 Maybe kilowatt-hours or ergs, these are various of the units. 00:01:26.610 --> 00:01:31.840 And work would be in things like foot-pounds. 00:01:31.840 --> 00:01:35.500 That is, lifting some weight some distance. 00:01:35.500 --> 00:01:39.240 And the amount of force you have to apply. 00:01:39.240 --> 00:01:42.070 And these all have conversions between them. 00:01:42.070 --> 00:01:45.880 They're all the same quantity, in different units. 00:01:45.880 --> 00:01:54.030 OK, so these are the same quantity. 00:01:54.030 --> 00:02:00.850 Different units. 00:02:00.850 --> 00:02:08.850 So that's about as much physics as we'll do for today. 00:02:08.850 --> 00:02:11.290 And sorry about that. 00:02:11.290 --> 00:02:18.800 Now, the example that I was starting to discuss last time 00:02:18.800 --> 00:02:21.980 and that I'm going to carry out today 00:02:21.980 --> 00:02:32.390 was this dartboard example. 00:02:32.390 --> 00:02:40.160 We have a dartboard, which is some kind of target. 00:02:40.160 --> 00:02:46.380 And we have a person, your little brother, 00:02:46.380 --> 00:02:48.540 who's standing over there. 00:02:48.540 --> 00:02:50.680 And somebody is throwing darts. 00:02:50.680 --> 00:02:55.790 And the question is, how likely is he to be hit. 00:02:55.790 --> 00:02:59.950 So I want to describe to you how we're 00:02:59.950 --> 00:03:02.550 going to make this problem into a math problem. 00:03:02.550 --> 00:03:03.050 Yep. 00:03:03.050 --> 00:03:03.904 STUDENT: [INAUDIBLE] 00:03:03.904 --> 00:03:06.070 PROFESSOR: What topic is this that we're going over. 00:03:06.070 --> 00:03:08.250 We're going over an example. 00:03:08.250 --> 00:03:11.650 Which is a dartboard example. 00:03:11.650 --> 00:03:16.090 And it has to do with probability. 00:03:16.090 --> 00:03:17.370 OK. 00:03:17.370 --> 00:03:38.260 So what is the probability that this guy, your little brother, 00:03:38.260 --> 00:03:47.700 gets hit by a dart. 00:03:47.700 --> 00:03:52.020 Now, we have to put some assumptions into this problem 00:03:52.020 --> 00:03:53.630 in order to make it a math problem. 00:03:53.630 --> 00:03:55.940 And I'm really going to try to make them 00:03:55.940 --> 00:03:58.630 pretty realistic assumptions. 00:03:58.630 --> 00:04:09.580 So the first assumption is that the number of hits is 00:04:09.580 --> 00:04:21.520 proportional to some constant times e^(-r^2). 00:04:24.430 --> 00:04:27.910 So that's actually a kind of a normal distribution. 00:04:27.910 --> 00:04:30.560 That's the bell curve. 00:04:30.560 --> 00:04:34.250 But as a function of the radius. 00:04:34.250 --> 00:04:37.060 So this is the assumption that I'm 00:04:37.060 --> 00:04:39.990 going to make in this problem. 00:04:39.990 --> 00:04:44.970 And a problem in probability is a problem 00:04:44.970 --> 00:04:48.540 of the ratio of the part to the whole. 00:04:48.540 --> 00:04:53.580 So the part is where this little guy is standing. 00:04:53.580 --> 00:04:56.670 And the whole is all the possible places 00:04:56.670 --> 00:04:58.232 where the dart might hit. 00:04:58.232 --> 00:04:59.690 Which is maybe the whole blackboard 00:04:59.690 --> 00:05:02.670 or extending beyond, depending on how good an aim 00:05:02.670 --> 00:05:07.780 you think that this older child has. 00:05:07.780 --> 00:05:11.730 So, if you like, the part is-- We'll start simply. 00:05:11.730 --> 00:05:14.290 I mean, this doesn't sweep all the way around. 00:05:14.290 --> 00:05:17.170 But we're going to talk about some section. 00:05:17.170 --> 00:05:18.930 Like this. 00:05:18.930 --> 00:05:22.400 Where this is some radius r_1, and this other radius 00:05:22.400 --> 00:05:24.460 is some longer radius, r_2. 00:05:24.460 --> 00:05:27.830 And the part that we'll first keep track of 00:05:27.830 --> 00:05:29.470 is everything around there. 00:05:29.470 --> 00:05:31.470 That's not very well centered, but it's supposed 00:05:31.470 --> 00:05:35.060 to be two concentric circles. 00:05:35.060 --> 00:05:38.840 Maybe I should fix that a bit so that it's a little bit easier 00:05:38.840 --> 00:05:39.784 to read here. 00:05:39.784 --> 00:05:41.260 So here we go. 00:05:41.260 --> 00:05:46.700 So here I have radius r_1, and here I have radius r_2. 00:05:46.700 --> 00:05:49.610 And then the region in between is what we're 00:05:49.610 --> 00:05:56.050 going to try to keep track of. 00:05:56.050 --> 00:05:59.170 So I claim that we'll be able to get-- 00:05:59.170 --> 00:06:04.100 So this is what I'm calling the part, to start out with. 00:06:04.100 --> 00:06:05.570 And then we'll take a section of it 00:06:05.570 --> 00:06:12.220 to get the place where the person is standing. 00:06:12.220 --> 00:06:22.157 Now, I want to take a side view of e^(-r^2). 00:06:22.157 --> 00:06:23.740 The function that we're talking about. 00:06:23.740 --> 00:06:28.650 Again, that's the bell curve. 00:06:28.650 --> 00:06:31.900 And it sort of looks like this. 00:06:31.900 --> 00:06:36.090 This is the top value, this is now the r-axis. 00:06:36.090 --> 00:06:39.490 And this is up, or at least-- So you 00:06:39.490 --> 00:06:41.540 should think of this in terms of the fact 00:06:41.540 --> 00:06:46.900 that the horizontal here is the plane of the dartboard. 00:06:46.900 --> 00:06:49.390 And the vertical is measuring how likely 00:06:49.390 --> 00:06:52.460 it is that there will be darts piling up here. 00:06:52.460 --> 00:06:55.780 So if they were balls tumbling down 00:06:55.780 --> 00:06:57.690 or something else falling on, many of them 00:06:57.690 --> 00:06:58.530 would pile up here. 00:06:58.530 --> 00:07:02.890 Many fewer of them would be piling up farther away. 00:07:02.890 --> 00:07:06.430 And the chunk that we're keeping track of 00:07:06.430 --> 00:07:09.200 is the chunk between r_1 and r_2. 00:07:09.200 --> 00:07:11.590 That's the corresponding region here. 00:07:11.590 --> 00:07:14.180 And in order to calculate this part, 00:07:14.180 --> 00:07:20.500 we have to calculate this volume of revolution. 00:07:20.500 --> 00:07:21.350 Sweeping around. 00:07:21.350 --> 00:07:24.160 Because really, in disguise, this is a ring. 00:07:24.160 --> 00:07:26.480 This is a side view, it's really a ring. 00:07:26.480 --> 00:07:33.100 Because we're rotating it around this axis here. 00:07:33.100 --> 00:07:34.990 So we're trying to figure out what 00:07:34.990 --> 00:07:36.800 the total volume of that ring is. 00:07:36.800 --> 00:07:39.930 And that's going to be our weighting, our likelihood 00:07:39.930 --> 00:07:41.390 for whether the hits are occurring 00:07:41.390 --> 00:07:51.240 in this section, or in this ring, versus the rest. 00:07:51.240 --> 00:07:53.750 To set this up, I remind you we're going 00:07:53.750 --> 00:07:57.570 to use the method of shells. 00:07:57.570 --> 00:08:03.030 That's really the only one that's going to work here. 00:08:03.030 --> 00:08:07.990 And we want to integrate between r_1 and r_2. 00:08:07.990 --> 00:08:12.280 And the range here is that-- Because this is, if you like, 00:08:12.280 --> 00:08:13.470 a solid of revolution. 00:08:13.470 --> 00:08:17.250 So the variable r is the same as what we used to call x. 00:08:17.250 --> 00:08:18.924 And it's ranging between r_1 and r_2, 00:08:18.924 --> 00:08:20.340 and then we're sweeping it around. 00:08:20.340 --> 00:08:25.920 And the circumference of a little piece, 00:08:25.920 --> 00:08:35.040 so at a fixed distance r here, the circumference 00:08:35.040 --> 00:08:37.950 is going to be 2 pi r. 00:08:37.950 --> 00:08:40.240 So 2 pi r is the circumference. 00:08:40.240 --> 00:08:42.710 And then the height is the height of that green stem 00:08:42.710 --> 00:08:43.210 there. 00:08:43.210 --> 00:08:45.460 That's e^(-r^2). 00:08:45.460 --> 00:08:50.460 And then we're multiplying by the thickness, which is dr. 00:08:50.460 --> 00:08:52.530 So the thickness of the green is dr. 00:08:52.530 --> 00:08:57.860 The height of this little green guy is e^(-r^2), 00:08:57.860 --> 00:09:01.310 and the circumference is 2 pi times the radius, 00:09:01.310 --> 00:09:03.701 when we sweep the circle around. 00:09:03.701 --> 00:09:04.200 Question. 00:09:04.200 --> 00:09:21.970 STUDENT: [INAUDIBLE] 00:09:21.970 --> 00:09:23.850 PROFESSOR: So the question is, why 00:09:23.850 --> 00:09:28.020 is what we're interested in not this pink area. 00:09:28.020 --> 00:09:31.810 And the reason is an interpretation 00:09:31.810 --> 00:09:33.430 of what I meant by this. 00:09:33.430 --> 00:09:38.310 What I meant by this is that if you wanted to add up 00:09:38.310 --> 00:09:42.190 what the likelihood is that this thing will be here versus here, 00:09:42.190 --> 00:09:44.940 I want it to be, really, the proportions 00:09:44.940 --> 00:09:47.780 are the number of hits times the, if you like, 00:09:47.780 --> 00:09:49.590 I wanted it d area. 00:09:49.590 --> 00:09:51.850 That's really what I meant here. 00:09:51.850 --> 00:09:56.740 The number of hits in a little chunk. 00:09:56.740 --> 00:10:01.370 So little, maybe I'll call it delta A. A little chunk. 00:10:01.370 --> 00:10:04.480 Is proportional to the chunk times that. 00:10:04.480 --> 00:10:05.920 So there's already an area factor. 00:10:05.920 --> 00:10:06.920 And there's a height. 00:10:06.920 --> 00:10:09.580 So there are a total of three dimensions involved. 00:10:09.580 --> 00:10:13.120 There's the area and then this height. 00:10:13.120 --> 00:10:18.640 So it's a matter of the-- what I was given, what I 00:10:18.640 --> 00:10:21.720 intended to say the problem is. 00:10:21.720 --> 00:10:22.720 Yeah, another question. 00:10:22.720 --> 00:10:25.084 STUDENT: [INAUDIBLE] 00:10:25.084 --> 00:10:26.000 PROFESSOR: Yeah, yeah. 00:10:26.000 --> 00:10:27.060 Exactly. 00:10:27.060 --> 00:10:30.570 The height is c times that, whatever this c is. 00:10:30.570 --> 00:10:32.220 In fact, we don't know what the c is, 00:10:32.220 --> 00:10:34.460 but because we're going to have a part and a whole, 00:10:34.460 --> 00:10:37.490 we'll divide, the c will always cancel. 00:10:37.490 --> 00:10:39.240 So I'm throwing the c out. 00:10:39.240 --> 00:10:42.750 I don't know what it is, and in the end it won't matter. 00:10:42.750 --> 00:10:44.330 That's a very important question. 00:10:44.330 --> 00:10:44.830 Yes. 00:10:44.830 --> 00:10:49.180 STUDENT: [INAUDIBLE] 00:10:49.180 --> 00:10:50.180 PROFESSOR: Say it again? 00:10:50.180 --> 00:11:02.040 STUDENT: PROFESSOR: 00:11:02.040 --> 00:11:05.920 PROFESSOR: So what I mean is the number of hits in some chunk. 00:11:05.920 --> 00:11:09.520 That is, suppose you imagine, the question 00:11:09.520 --> 00:11:11.720 is, what does this left-hand side mean. 00:11:11.720 --> 00:11:12.250 That right? 00:11:12.250 --> 00:11:17.690 Is that the question that's you're asking? 00:11:17.690 --> 00:11:21.040 When I try to understand what the distribution 00:11:21.040 --> 00:11:25.420 of dartboard hits is, I should imagine my dartboard. 00:11:25.420 --> 00:11:28.400 And very often there'll be a whole bunch 00:11:28.400 --> 00:11:29.700 of holes in some places. 00:11:29.700 --> 00:11:30.810 And fewer holes else. 00:11:30.810 --> 00:11:33.010 I'm trying to figure out what the whole distribution 00:11:33.010 --> 00:11:35.590 of those marks is. 00:11:35.590 --> 00:11:38.410 And so some places will have more hits on them 00:11:38.410 --> 00:11:41.060 and some places will have fewer hits on them. 00:11:41.060 --> 00:11:43.980 And so what I want to measure is, on average, 00:11:43.980 --> 00:11:44.910 the number of hits. 00:11:44.910 --> 00:11:47.387 So this would really be some-- This constant 00:11:47.387 --> 00:11:49.470 of proportionality is ambiguous because it depends 00:11:49.470 --> 00:11:50.553 on how many times you try. 00:11:50.553 --> 00:11:52.470 If you throw a thousand times, it'll 00:11:52.470 --> 00:11:54.320 be much more densely packed. 00:11:54.320 --> 00:11:58.780 And if you have only a hundred times it'll be fewer. 00:11:58.780 --> 00:12:01.850 So that's where this constant comes in. 00:12:01.850 --> 00:12:04.610 But given that you have a certain number of times 00:12:04.610 --> 00:12:06.790 that you tried, say, a thousand times, 00:12:06.790 --> 00:12:09.500 there will be a whole bunch more piled in the middle. 00:12:09.500 --> 00:12:12.020 And fewer and fewer as you get farther away. 00:12:12.020 --> 00:12:15.800 Assuming that the person's aim is reasonable. 00:12:15.800 --> 00:12:17.024 So that's what we're saying. 00:12:17.024 --> 00:12:19.440 So we're thinking in terms of-- The person's always aiming 00:12:19.440 --> 00:12:20.065 for the center. 00:12:20.065 --> 00:12:22.660 So it's most likely that the person will hit the center. 00:12:22.660 --> 00:12:24.820 But on the other hand, it's a fallible person, 00:12:24.820 --> 00:12:26.440 so the person may miss. 00:12:26.440 --> 00:12:29.272 And so it's less and less likely as you go farther out. 00:12:29.272 --> 00:12:31.480 And we're just counting how many times this gets hit, 00:12:31.480 --> 00:12:33.220 how many time this and so on. 00:12:33.220 --> 00:12:34.830 In proportion to the area. 00:12:34.830 --> 00:12:40.460 STUDENT: [INAUDIBLE] 00:12:40.460 --> 00:12:42.252 PROFESSOR: Yeah. r_1 and r_2 are arbitrary. 00:12:42.252 --> 00:12:44.251 We're going to make this calculation in general. 00:12:44.251 --> 00:12:46.240 We're going to calculate what the likelihood is 00:12:46.240 --> 00:12:50.290 that we hit any possible band. 00:12:50.290 --> 00:12:53.010 And I want to leave those as just letters for now. 00:12:53.010 --> 00:12:54.990 The r_1 and the r_2. 00:12:54.990 --> 00:12:57.110 Because I want to be able to try various different 00:12:57.110 --> 00:12:57.990 possibilities. 00:12:57.990 --> 00:13:03.730 STUDENT: [INAUDIBLE] 00:13:03.730 --> 00:13:06.310 PROFESSOR: Say it again, why do we have to take volume? 00:13:06.310 --> 00:13:08.830 So this is what we were addressing before. 00:13:08.830 --> 00:13:13.900 It's a volume because it's number of hit per unit area. 00:13:13.900 --> 00:13:17.590 So there's a height, that is, number of hits. 00:13:17.590 --> 00:13:19.820 And then there's an area and the product of those 00:13:19.820 --> 00:13:22.610 is-- So this is, if you like, a histogram 00:13:22.610 --> 00:13:25.100 of the number of hits. 00:13:25.100 --> 00:13:26.910 But this should be measured per area. 00:13:26.910 --> 00:13:29.400 Not per length of r. 00:13:29.400 --> 00:13:32.020 Because on the real diagram, it's going all the way around. 00:13:32.020 --> 00:13:35.760 There's a lot more area to this red band 00:13:35.760 --> 00:13:41.150 than just the distance implies. 00:13:41.150 --> 00:13:42.940 OK. 00:13:42.940 --> 00:13:47.460 So, having discussed the setup, this 00:13:47.460 --> 00:13:49.770 is a pretty standard setup-- Oh, one more question. 00:13:49.770 --> 00:13:50.270 Yes. 00:13:50.270 --> 00:14:13.564 STUDENT: [INAUDIBLE] 00:14:13.564 --> 00:14:14.230 PROFESSOR: Yeah. 00:14:14.230 --> 00:14:17.700 So the question is, why is this a realistic. 00:14:17.700 --> 00:14:21.260 Why is this choice of function here, e^(-r^2), 00:14:21.260 --> 00:14:25.670 a realistic choice of function for the darts. 00:14:25.670 --> 00:14:30.480 So I can answer this with an analogy. 00:14:30.480 --> 00:14:32.800 When people were asking themselves 00:14:32.800 --> 00:14:38.120 where the V-2 rockets from Germany hit London, 00:14:38.120 --> 00:14:39.170 they used this model. 00:14:39.170 --> 00:14:43.100 It turned out to be the one which was the most accurate. 00:14:43.100 --> 00:14:48.630 So that gives you an idea that this is actually real. 00:14:48.630 --> 00:14:53.670 The question, this makes it look like people are masters. 00:14:53.670 --> 00:14:56.940 That is, that they'll all hit in the center 00:14:56.940 --> 00:14:58.870 more often than elsewhere. 00:14:58.870 --> 00:15:03.960 But that's actually somewhat deceptive. 00:15:03.960 --> 00:15:08.040 There's a difference between the mode, 00:15:08.040 --> 00:15:13.510 that is, the most likely spot, and what happens on average. 00:15:13.510 --> 00:15:17.960 So in other words, the single most likely spot is the center. 00:15:17.960 --> 00:15:20.720 But there's rather little area in here, and in fact 00:15:20.720 --> 00:15:24.860 the likelihood of hitting that is some little tiny bit. 00:15:24.860 --> 00:15:25.620 In here. 00:15:25.620 --> 00:15:29.850 In fact, you're much more likely to be out here. 00:15:29.850 --> 00:15:31.539 So if you take the total of the volume, 00:15:31.539 --> 00:15:32.955 you'll see that much of the volume 00:15:32.955 --> 00:15:34.260 is contributed from out here. 00:15:34.260 --> 00:15:37.780 And in fact, the person hits rather rarely near the center. 00:15:37.780 --> 00:15:42.737 So this is not a ridiculous thing to do. 00:15:42.737 --> 00:15:45.320 If you think of it in terms of somebody's aiming at the center 00:15:45.320 --> 00:15:46.950 but there's some random thing which 00:15:46.950 --> 00:15:49.770 is throwing the person off, then there 00:15:49.770 --> 00:15:51.710 is likely to be to left or to the right, 00:15:51.710 --> 00:15:54.080 or they might even get lucky and all those errors 00:15:54.080 --> 00:15:55.470 cancel themselves and they happen 00:15:55.470 --> 00:15:59.090 to hit pretty much near the center. 00:15:59.090 --> 00:16:00.240 Yeah, another question. 00:16:00.240 --> 00:16:06.266 STUDENT: [INAUDIBLE] PROFESSOR: How does the little brother 00:16:06.266 --> 00:16:06.890 come into play? 00:16:06.890 --> 00:16:09.390 The little brother is going to come into play as follows. 00:16:09.390 --> 00:16:11.300 I'll tell you in advance. 00:16:11.300 --> 00:16:15.500 So the thing is, the little brother was not so stupid 00:16:15.500 --> 00:16:19.760 as to stand in front of the target. 00:16:19.760 --> 00:16:22.220 I know. 00:16:22.220 --> 00:16:26.800 He stood about twice the radius of the target away. 00:16:26.800 --> 00:16:29.960 And so, we're going to approximate the location 00:16:29.960 --> 00:16:32.460 by some sector here. 00:16:32.460 --> 00:16:35.920 Which is just going to be some chunk of one of these things. 00:16:35.920 --> 00:16:38.010 We'll just break off a piece of it. 00:16:38.010 --> 00:16:39.850 And that's how we're going to capture. 00:16:39.850 --> 00:16:42.030 So the point is, the target is here. 00:16:42.030 --> 00:16:46.000 But there is the possibility that the seven-year-old 00:16:46.000 --> 00:16:48.550 who's throwing the darts actually missed the target. 00:16:48.550 --> 00:16:50.780 That actually happens a lot. 00:16:50.780 --> 00:16:54.660 So, does that answer your question? 00:16:54.660 --> 00:16:55.530 Alright. 00:16:55.530 --> 00:16:57.170 Are we ready now to to do this? 00:16:57.170 --> 00:16:57.950 One more question. 00:16:57.950 --> 00:17:05.250 STUDENT: [INAUDIBLE] 00:17:05.250 --> 00:17:08.320 PROFESSOR: I'm giving you this property here. 00:17:08.320 --> 00:17:10.110 I'm telling you that-- This is what's 00:17:10.110 --> 00:17:12.540 called a mathematical model, when you give 00:17:12.540 --> 00:17:13.900 somebody something like this. 00:17:13.900 --> 00:17:16.450 In fact, that requires further justification. 00:17:16.450 --> 00:17:19.120 It's an interesting issue. 00:17:19.120 --> 00:17:19.620 Yeah. 00:17:19.620 --> 00:17:26.084 STUDENT: [INAUDIBLE] 00:17:26.084 --> 00:17:27.750 PROFESSOR: I'm giving it to you for now. 00:17:27.750 --> 00:17:30.120 And it's something which really has to be justified. 00:17:30.120 --> 00:17:32.080 In certain circumstances it is justified. 00:17:32.080 --> 00:17:37.620 But, OK. 00:17:37.620 --> 00:17:43.800 So anyway, here's our part. 00:17:43.800 --> 00:17:46.200 This is going to be our chunk, for now, 00:17:46.200 --> 00:17:50.760 that we're going to estimate the importance, 00:17:50.760 --> 00:17:56.210 the relative importance, of. 00:17:56.210 --> 00:18:01.460 And now, this is something whose antiderivative we can just do 00:18:01.460 --> 00:18:03.600 by substitution or by guessing. 00:18:03.600 --> 00:18:07.870 It's just -pi e^(-r^2). 00:18:07.870 --> 00:18:10.650 If you differentiate that, you get a -2r, which 00:18:10.650 --> 00:18:12.730 cancels the minus sign here. 00:18:12.730 --> 00:18:15.370 So you get 2pi r e^(-r^2). 00:18:15.370 --> 00:18:18.400 So that's the antiderivative. 00:18:18.400 --> 00:18:24.710 And we're evaluating it at r_1 and r_2. 00:18:24.710 --> 00:18:28.490 So with the minus sign that's going to get reversed. 00:18:28.490 --> 00:18:33.350 The answer is going to be pi times e^(-r_1^2), 00:18:33.350 --> 00:18:39.090 that's the bottom one, minus e^(r_2^2), that's the top. 00:18:39.090 --> 00:18:45.777 So this is what our part gives us. 00:18:45.777 --> 00:18:48.110 And, more technically, if you wanted to multiply through 00:18:48.110 --> 00:18:50.350 by c, it would be c times this. 00:18:50.350 --> 00:18:55.140 I'll say that in just a second. 00:18:55.140 --> 00:18:59.280 OK, now I want to work-- So, if you like, 00:18:59.280 --> 00:19:06.490 the part is equal to, maybe even I'll call it c pi, 00:19:06.490 --> 00:19:07.740 times e^(-r_1^2) - e^(-r_2^2). 00:19:12.429 --> 00:19:14.470 That's what it really is if I put in this factor. 00:19:14.470 --> 00:19:17.690 So now there's no prejudice as to how many attempts we make. 00:19:17.690 --> 00:19:20.490 Whether it was a thousand attempts or a million 00:19:20.490 --> 00:19:22.740 attempts at the target. 00:19:22.740 --> 00:19:25.480 Now, the most important second feature 00:19:25.480 --> 00:19:27.250 here of these kinds of modeling problems 00:19:27.250 --> 00:19:30.560 is, there is always some kind of idealization. 00:19:30.560 --> 00:19:33.070 And the next thing that I want to discuss with you 00:19:33.070 --> 00:19:36.940 is the interpretation of the whole. 00:19:36.940 --> 00:19:41.870 That is, what's the family of all possibilities. 00:19:41.870 --> 00:19:45.790 And in this case, what I'm going to claim 00:19:45.790 --> 00:19:50.160 is that the reasonable way to think of the whole 00:19:50.160 --> 00:19:55.940 is it's that r can range all the way from 0 to infinity. 00:19:55.940 --> 00:19:57.210 Now, you may not like this. 00:19:57.210 --> 00:20:01.730 But these are maybe my first and third favorite number, 00:20:01.730 --> 00:20:03.810 my second favorite number being 1. 00:20:03.810 --> 00:20:07.930 So infinity is a really useful concept. 00:20:07.930 --> 00:20:12.010 Of course, it's nonsense in the context of the darts. 00:20:12.010 --> 00:20:14.770 Because if you think of the basement wall 00:20:14.770 --> 00:20:17.140 where the kid might miss a target, 00:20:17.140 --> 00:20:19.619 he'd probably hit the wall. 00:20:19.619 --> 00:20:22.160 He's probably not going to hit one of the walls to the right, 00:20:22.160 --> 00:20:25.080 and anyway he's certainly not going to hit over there. 00:20:25.080 --> 00:20:28.870 So there's something artificial about sending 00:20:28.870 --> 00:20:32.390 the possibility of hits all the way out to infinity. 00:20:32.390 --> 00:20:34.850 On the other hand, the shape of this curve 00:20:34.850 --> 00:20:38.310 is such that the real tail ends here, 00:20:38.310 --> 00:20:40.990 because of this exponential decrease, are tiny. 00:20:40.990 --> 00:20:42.630 And that's negligible. 00:20:42.630 --> 00:20:45.700 And the point is that actually the value, 00:20:45.700 --> 00:20:47.550 if you go all the way out to infinity, 00:20:47.550 --> 00:20:50.020 is the easiest value to calculate. 00:20:50.020 --> 00:20:52.697 So by doing this, I'm idealizing the problem 00:20:52.697 --> 00:20:54.530 but I'm actually making the numbers come out 00:20:54.530 --> 00:20:55.890 much more cleanly. 00:20:55.890 --> 00:20:58.180 And this is just always done in mathematics. 00:20:58.180 --> 00:21:00.290 That's what we did when we went from differences 00:21:00.290 --> 00:21:02.460 to differentials, to differentiation 00:21:02.460 --> 00:21:04.640 and infinitesimals. 00:21:04.640 --> 00:21:06.282 So we like that. 00:21:06.282 --> 00:21:08.740 Because it makes things easier, not because it makes things 00:21:08.740 --> 00:21:10.320 harder. 00:21:10.320 --> 00:21:12.240 So we're just going to pretend the whole is 00:21:12.240 --> 00:21:13.450 from 0 to infinity. 00:21:13.450 --> 00:21:15.550 And now let's just see what it is. 00:21:15.550 --> 00:21:21.330 It's c pi times, the starting place is e^(-0^2). 00:21:21.330 --> 00:21:24.630 That's r_1, right, this is the role that r_1 plays is this, 00:21:24.630 --> 00:21:26.374 and the r_2 is this value. 00:21:26.374 --> 00:21:27.290 Minus e^(-infinity^2). 00:21:30.180 --> 00:21:33.940 Which is that negligibly small number, 0. 00:21:33.940 --> 00:21:36.750 So this is just c pi. 00:21:36.750 --> 00:21:39.700 Because this number is 1, and this other number is 0. 00:21:39.700 --> 00:21:48.420 This is just (1 - 0), in the parentheses there. 00:21:48.420 --> 00:21:52.300 And now I can tell you from these two numbers 00:21:52.300 --> 00:21:54.990 what the probability is. 00:21:54.990 --> 00:21:58.450 The probability that we landed on the target 00:21:58.450 --> 00:22:04.250 in a radius between r_1 and r_2, so that's this annulus here, 00:22:04.250 --> 00:22:12.350 is the ratio of the part to the whole. 00:22:12.350 --> 00:22:16.060 Which in this case just cancels the c pi. 00:22:16.060 --> 00:22:21.280 So it's e^(-r_1^2) - e^(-r_2^2). 00:22:21.280 --> 00:22:25.520 There's the formula for the probability. 00:22:25.520 --> 00:22:28.100 So the c canceled and the pi canceled. 00:22:28.100 --> 00:22:32.170 It's all gone. 00:22:32.170 --> 00:22:34.040 Now, again, let me just emphasize 00:22:34.040 --> 00:22:35.750 the way this formula is supposed to work. 00:22:35.750 --> 00:22:42.210 The total probability of every possibility 00:22:42.210 --> 00:22:45.900 here is supposed to be set up to be 1. 00:22:45.900 --> 00:22:48.530 This is some fraction of 1. 00:22:48.530 --> 00:22:51.191 If you like, it's a percent. 00:22:51.191 --> 00:22:51.690 Yes. 00:22:51.690 --> 00:22:57.149 STUDENT: [INAUDIBLE] 00:22:57.149 --> 00:22:59.690 PROFESSOR: This one is giving-- The question is, doesn't this 00:22:59.690 --> 00:23:00.780 just give the probability of the ring? 00:23:00.780 --> 00:23:02.620 This gives you the probability of the ring, 00:23:02.620 --> 00:23:05.510 but this is a very, very wide ring. 00:23:05.510 --> 00:23:08.479 This is a ring starting with 0, nothing, on the inside. 00:23:08.479 --> 00:23:09.770 And then going all the way out. 00:23:09.770 --> 00:23:11.880 So that's everything. 00:23:11.880 --> 00:23:13.380 So this corresponds to everything. 00:23:13.380 --> 00:23:22.650 This corresponds to a ring. 00:23:22.650 --> 00:23:24.990 So now, let's see. 00:23:24.990 --> 00:23:26.390 Where do I want to go from here. 00:23:26.390 --> 00:23:28.550 So in order to make progress here, 00:23:28.550 --> 00:23:31.230 I still have to give you one more piece of information. 00:23:31.230 --> 00:23:35.650 And this is, again, supposed to be realistic. 00:23:35.650 --> 00:23:40.350 When I was three years old and my brother's friend Ralph 00:23:40.350 --> 00:23:43.920 was seven, I watched him throwing darts a lot. 00:23:43.920 --> 00:23:54.200 And I would say that for Ralph, so for Ralph, at age seven, 00:23:54.200 --> 00:23:57.080 anyway, later on he got a little better at it. 00:23:57.080 --> 00:23:59.950 But Ralph at age seven, the probability that he 00:23:59.950 --> 00:24:08.440 hit the target was about 1/2. 00:24:08.440 --> 00:24:09.660 Right? 00:24:09.660 --> 00:24:12.040 So he hit the target about half the time. 00:24:12.040 --> 00:24:14.560 And the other times, there was cement 00:24:14.560 --> 00:24:17.870 on the walls of the basement, it wasn't that bad. 00:24:17.870 --> 00:24:19.250 Just bounced off. 00:24:19.250 --> 00:24:22.000 That also meant that the points got a little blunter as time 00:24:22.000 --> 00:24:22.560 went on. 00:24:22.560 --> 00:24:26.150 So it was a little less dangerous when they hit. 00:24:26.150 --> 00:24:26.650 Alright. 00:24:26.650 --> 00:24:33.210 So now, so here's the extra assumption that I want to make. 00:24:33.210 --> 00:24:41.260 So a is going to be the radius of the target. 00:24:41.260 --> 00:24:45.660 Now, the other realistic assumption that I want to make 00:24:45.660 --> 00:24:49.170 is where this little kid would be standing. 00:24:49.170 --> 00:24:51.650 And now, here, I want to get very specific 00:24:51.650 --> 00:24:54.280 and just do the computation in one case. 00:24:54.280 --> 00:25:00.430 We're going to imagine the target is here. 00:25:00.430 --> 00:25:03.200 And the kid is standing, say, between-- So we'll just 00:25:03.200 --> 00:25:04.430 do a section of this. 00:25:04.430 --> 00:25:13.130 This is between 3 o'clock and 5 o'clock. 00:25:13.130 --> 00:25:16.740 There's more of him, but it's lower down and maybe negligible 00:25:16.740 --> 00:25:17.350 here. 00:25:17.350 --> 00:25:20.400 So this section is the part, the chunk 00:25:20.400 --> 00:25:23.330 that we want to see about. 00:25:23.330 --> 00:25:31.210 And this is a, and then this distance here is 2a. 00:25:31.210 --> 00:25:35.330 And then the longest distance here is 3a. 00:25:35.330 --> 00:25:39.000 So the longest distance is 3a. 00:25:39.000 --> 00:25:40.870 So in other words, what I'm saying 00:25:40.870 --> 00:25:50.180 is that the probability, if you're standing too close, 00:25:50.180 --> 00:26:06.660 the chance Ralph hits younger brother is about 1/6-- Right? 00:26:06.660 --> 00:26:12.820 Because 2/12 = 1/6. 00:26:12.820 --> 00:26:22.020 1/6 of the probability that we're between a and 3a. 00:26:22.020 --> 00:26:29.052 That's the number that we're looking for. 00:26:29.052 --> 00:26:29.760 Another question. 00:26:29.760 --> 00:26:33.900 STUDENT: [INAUDIBLE] 00:26:33.900 --> 00:26:36.680 PROFESSOR: The 2/12 came from the fact that we assumed. 00:26:36.680 --> 00:26:39.250 So we made a very, very bold assumption here. 00:26:39.250 --> 00:26:41.530 We assumed that this human being, 00:26:41.530 --> 00:26:46.760 who is actually standing-- The floor is about down here. 00:26:46.760 --> 00:26:48.460 Maybe he wasn't that tall. 00:26:48.460 --> 00:26:51.560 But anyway, so he's really a little bigger than this. 00:26:51.560 --> 00:26:53.850 That the part of him that was close to the target 00:26:53.850 --> 00:27:00.500 covered about this section here, between radius 2a and 3a. 00:27:00.500 --> 00:27:02.740 As you'll see, actually from the computation, 00:27:02.740 --> 00:27:07.290 because the likelihood drops off pretty quickly, whatever 00:27:07.290 --> 00:27:09.140 of him was standing outside there 00:27:09.140 --> 00:27:11.269 wouldn't have mattered anyway. 00:27:11.269 --> 00:27:13.060 So we're just worried about the part that's 00:27:13.060 --> 00:27:18.677 closest to the target here. 00:27:18.677 --> 00:27:19.510 STUDENT: [INAUDIBLE] 00:27:19.510 --> 00:27:20.801 PROFESSOR: Why is it out of 12? 00:27:20.801 --> 00:27:22.650 Because I made it a clock. 00:27:22.650 --> 00:27:25.110 And I made it from 3 o'clock to 5 o'clock, 00:27:25.110 --> 00:27:28.790 so it's 2 of the 12 hours of a clock. 00:27:28.790 --> 00:27:32.640 It's just a way of me making a section that you can visibly 00:27:32.640 --> 00:27:36.400 see. 00:27:36.400 --> 00:27:40.060 So now, so here's what we're trying to calculate. 00:27:40.060 --> 00:27:47.790 And in order to figure this out, I need one more item. here. 00:27:47.790 --> 00:27:49.870 So maybe I'll leave myself a little bit of room. 00:27:49.870 --> 00:27:52.010 I have to figure out something about what 00:27:52.010 --> 00:27:53.830 our information gave us. 00:27:53.830 --> 00:28:00.380 Which is that the probability, sorry, this probability 00:28:00.380 --> 00:28:01.690 was 1/2. 00:28:01.690 --> 00:28:03.580 So let's remember what this is. 00:28:03.580 --> 00:28:09.470 This is going to be e^(-0^2) - e^(-a^2). 00:28:09.470 --> 00:28:11.180 That's what's this probability is. 00:28:11.180 --> 00:28:16.210 And that's equal to 1/2. 00:28:16.210 --> 00:28:21.330 So that means that - 1 - e^(-a^2). = 1/2. 00:28:21.330 --> 00:28:26.620 Which means that e^(-a^2) = 1/2. 00:28:26.620 --> 00:28:28.220 I'm not going to calculate a, this 00:28:28.220 --> 00:28:30.050 is the part about the information about a 00:28:30.050 --> 00:28:32.480 that I want to use. 00:28:32.480 --> 00:28:33.520 That's what I'll use. 00:28:33.520 --> 00:28:36.250 And now I'm going to calculate this other probability here. 00:28:36.250 --> 00:28:41.590 So the probability right up there is this. 00:28:41.590 --> 00:28:47.390 And that's going to be the same as e^(-(2a)^2) - e^(-(3a)^2). 00:28:51.610 --> 00:28:54.650 That's what we calculated. 00:28:54.650 --> 00:28:59.200 And now I want to use some of the arithmetic of exponents. 00:28:59.200 --> 00:29:01.380 This is (e^(-a^2))^2. 00:29:05.180 --> 00:29:11.890 Because it's really (2a)^2 is 4-- the quantity is 4a^2, 00:29:11.890 --> 00:29:14.580 and then I bring that exponent out. 00:29:14.580 --> 00:29:19.740 Minus e^(-a^2) to the 9th power, that's 3^2. 00:29:25.040 --> 00:29:28.560 And so this comes out to be (1/2)^4 - (1/2)^9. 00:29:36.190 --> 00:29:39.340 Which is approximately 1/16. 00:29:39.340 --> 00:29:40.380 This is negligible. 00:29:40.380 --> 00:29:41.180 This part here. 00:29:41.180 --> 00:29:43.110 And this is actually why these tails, 00:29:43.110 --> 00:29:48.130 as you go out to infinity, don't really matter that much. 00:29:48.130 --> 00:29:49.880 So this is a much smaller number. 00:29:49.880 --> 00:29:53.390 So the probability of the whole band is 1/16. 00:29:53.390 --> 00:29:55.720 And now I can answer the question up here. 00:29:55.720 --> 00:30:01.110 This is approximately 1/6 * 1/16. 00:30:01.110 --> 00:30:07.340 Which is about 1/100, or about 1%. 00:30:07.340 --> 00:30:13.340 So if I stood there for 100 attempts, 00:30:13.340 --> 00:30:18.720 then my chances of getting hit were pretty high. 00:30:18.720 --> 00:30:27.970 So that's the computation. 00:30:27.970 --> 00:30:34.940 That's a typical example of a problem in probability. 00:30:34.940 --> 00:30:36.860 And let me just make one more connection 00:30:36.860 --> 00:30:42.660 with what we did before. 00:30:42.660 --> 00:30:46.130 This is connected to weighted averages or integrals 00:30:46.130 --> 00:30:47.470 over weights. 00:30:47.470 --> 00:30:53.110 But the weight that's involved in this problem 00:30:53.110 --> 00:30:59.810 was w(r) is equal to-- So let's just look at what happened 00:30:59.810 --> 00:31:01.520 in all of those integrals. 00:31:01.520 --> 00:31:03.790 What happened in all the integrals was, we 00:31:03.790 --> 00:31:09.060 had this factor here, 2 pi r. 00:31:09.060 --> 00:31:23.330 And if I include this c, it was really 2 pi c r e^(-r^2). 00:31:23.330 --> 00:31:27.960 This was the weight that we were using. 00:31:27.960 --> 00:31:30.770 The relative importance of things. 00:31:30.770 --> 00:31:34.260 Now, this is not the same as the e^(-r^2) that we started out 00:31:34.260 --> 00:31:34.760 with. 00:31:34.760 --> 00:31:38.000 Because this is the one-dimensional histogram. 00:31:38.000 --> 00:31:40.850 And that has to do with the method of shells that gave us 00:31:40.850 --> 00:31:43.210 that extra factor of r here. 00:31:43.210 --> 00:31:45.090 So that also connects with the question 00:31:45.090 --> 00:31:47.610 at the beginning, which had to do with this paradox, 00:31:47.610 --> 00:31:50.420 that it looks like these places in the middle 00:31:50.420 --> 00:31:51.440 are the most likely. 00:31:51.440 --> 00:31:54.500 But that's the plot e^(-r^2). 00:31:54.500 --> 00:31:57.240 If you actually look at this plot here, 00:31:57.240 --> 00:32:00.330 you see that as r goes to 0, it's going to 0. 00:32:00.330 --> 00:32:03.010 This is a different graph here. 00:32:03.010 --> 00:32:07.720 And actually, so this is what's happening really. 00:32:07.720 --> 00:32:09.730 In terms of how likely it is that you'll 00:32:09.730 --> 00:32:18.392 get within a certain distance of the center of the target. 00:32:18.392 --> 00:32:20.850 Again, it's not the area under that curve that we're doing. 00:32:20.850 --> 00:32:27.940 It's that volume of revolution. 00:32:27.940 --> 00:32:29.494 We're going to change subjects now. 00:32:29.494 --> 00:32:30.410 OK, one more question. 00:32:30.410 --> 00:32:31.937 Yes. 00:32:31.937 --> 00:32:32.770 STUDENT: [INAUDIBLE] 00:32:32.770 --> 00:32:36.120 PROFESSOR: Yeah, that's supposed to be the graph of w(r). 00:32:36.120 --> 00:32:47.359 STUDENT: [INAUDIBLE] 00:32:47.359 --> 00:32:48.900 PROFESSOR: Well, so, the question is, 00:32:48.900 --> 00:32:50.983 wouldn't the importance of the center be greatest? 00:32:50.983 --> 00:32:53.790 It's a question of which variable you're using. 00:32:53.790 --> 00:32:57.057 According to pure radius, it's not. 00:32:57.057 --> 00:32:58.640 It turns out that there are some bands 00:32:58.640 --> 00:33:00.930 in radius which are more important, more likely 00:33:00.930 --> 00:33:06.206 for hits than others. 00:33:06.206 --> 00:33:07.580 It really has to do with the fact 00:33:07.580 --> 00:33:10.980 that the center, or the core, of the target is really tiny. 00:33:10.980 --> 00:33:11.950 So it's harder to hit. 00:33:11.950 --> 00:33:16.170 Whereas a whole band around the outside has a lot more area. 00:33:16.170 --> 00:33:18.500 Many, many ways to hit that band. 00:33:18.500 --> 00:33:19.810 So it's a much larger area. 00:33:19.810 --> 00:33:24.220 So there's a competition there between those two things. 00:33:24.220 --> 00:33:25.910 So we're going to move on. 00:33:25.910 --> 00:33:28.170 And I want to talk now about a different kind 00:33:28.170 --> 00:33:30.550 of weighted average. 00:33:30.550 --> 00:33:36.410 These weighted averages are going to be much simpler. 00:33:36.410 --> 00:34:01.260 And they come up in what's called numerical integration. 00:34:01.260 --> 00:34:04.597 There are many, many methods of integrating numerically. 00:34:04.597 --> 00:34:06.680 And they're important because many, many integrals 00:34:06.680 --> 00:34:08.510 don't have formulas. 00:34:08.510 --> 00:34:13.980 And so you have to compute them with a calculator or a machine. 00:34:13.980 --> 00:34:24.400 So the first type that we've already done are Riemann sums. 00:34:24.400 --> 00:34:28.170 They turned out to be incredibly inefficient. 00:34:28.170 --> 00:34:30.270 They're lousy. 00:34:30.270 --> 00:34:32.120 The next rule that I'm going to describe 00:34:32.120 --> 00:34:33.370 is a little improvement. 00:34:33.370 --> 00:34:38.130 It's called the trapezoidal rule. 00:34:38.130 --> 00:34:41.340 And this one is much more reasonable 00:34:41.340 --> 00:34:43.370 than the Riemann sum. 00:34:43.370 --> 00:34:49.140 Unfortunately, it's actually also pretty lousy. 00:34:49.140 --> 00:34:53.150 There's another rule which is just a slightly trickier rule. 00:34:53.150 --> 00:34:56.030 And it's actually amazing that it exists. 00:34:56.030 --> 00:35:01.750 And it's called Simpson's Rule. 00:35:01.750 --> 00:35:06.690 And this one is actually pretty good. 00:35:06.690 --> 00:35:07.770 It's clever. 00:35:07.770 --> 00:35:12.320 So let's get started with these. 00:35:12.320 --> 00:35:15.030 And the way I'll get started is by reminding you 00:35:15.030 --> 00:35:16.030 what the Riemann sum is. 00:35:16.030 --> 00:35:17.779 So this is a good review, because you need 00:35:17.779 --> 00:35:19.790 to know all three of these. 00:35:19.790 --> 00:35:26.540 And you're going to want to see them all laid out in parallel. 00:35:26.540 --> 00:35:31.800 So here's the setup. 00:35:31.800 --> 00:35:33.580 OK, so here we go. 00:35:33.580 --> 00:35:36.660 We have our graph, we have our function, it starts out at a, 00:35:36.660 --> 00:35:38.240 it ends up at b. 00:35:38.240 --> 00:35:40.160 Maybe goes on, but we're only paying attention 00:35:40.160 --> 00:35:41.380 to this interval. 00:35:41.380 --> 00:35:44.760 This is a function y = f(x). 00:35:44.760 --> 00:35:48.160 And we split it up when we do Riemann sums. 00:35:48.160 --> 00:35:50.450 So this is 1, this is Riemann sums. 00:35:50.450 --> 00:35:53.930 We start out with a, which is a point we call x_0, 00:35:53.930 --> 00:35:57.370 and then we go a certain distance. 00:35:57.370 --> 00:35:58.710 We go all the way over to b. 00:35:58.710 --> 00:36:04.150 And we subdivide this thing with these delta x's. 00:36:04.150 --> 00:36:08.130 Which are the step sizes. 00:36:08.130 --> 00:36:12.840 So we have these little steps of size delta x. 00:36:12.840 --> 00:36:18.860 Corresponding to these x values, we have y values. y_0 = f(x_0), 00:36:18.860 --> 00:36:21.090 that's the point above a. 00:36:21.090 --> 00:36:26.840 Then y_1 = f(x_1), that's the point above x_1. 00:36:26.840 --> 00:36:33.120 And so forth, all the way up to y_n, which is f(x_n). 00:36:33.120 --> 00:36:35.650 In order to figure out the area, you really 00:36:35.650 --> 00:36:37.400 need to know something about the function. 00:36:37.400 --> 00:36:39.217 You need to be able to evaluate it. 00:36:39.217 --> 00:36:40.300 So that's what we've done. 00:36:40.300 --> 00:36:43.330 We've evaluated here at n + 1 points. 00:36:43.330 --> 00:36:46.150 Enumerated 0 through n. 00:36:46.150 --> 00:36:48.910 And those are the numbers out of which 00:36:48.910 --> 00:36:53.390 we're going to get all of our approximations to the integral. 00:36:53.390 --> 00:36:56.370 So somehow we want average these numbers. 00:36:56.370 --> 00:37:01.780 So here's our goal. 00:37:01.780 --> 00:37:14.450 Our goal is to average, or add, I'm using average very loosely 00:37:14.450 --> 00:37:14.950 here. 00:37:14.950 --> 00:37:18.830 But I was going to say add up these numbers 00:37:18.830 --> 00:37:25.750 to get an approximation. 00:37:25.750 --> 00:37:30.730 To average or add the y's. 00:37:30.730 --> 00:37:37.670 To get an approximation to the integral. 00:37:37.670 --> 00:37:43.410 Which we know is the area under the curve. 00:37:43.410 --> 00:37:49.730 So here's what the Riemann sum is. 00:37:49.730 --> 00:37:51.020 It's the following thing. 00:37:51.020 --> 00:37:56.030 You take y_0 plus y_1 plus... up to y_(n-1). 00:37:56.030 --> 00:37:57.520 And you multiply by delta x. 00:37:57.520 --> 00:38:00.040 That's it. 00:38:00.040 --> 00:38:07.010 Now, this one is the one with left endpoints. 00:38:07.010 --> 00:38:09.350 So the left-hand sum. 00:38:09.350 --> 00:38:11.120 There's also a right one. 00:38:11.120 --> 00:38:13.210 Which is, if you start at the right-hand ends. 00:38:13.210 --> 00:38:17.830 And that will go from y_1 to y_n. 00:38:17.830 --> 00:38:22.170 OK, so this one is the right-hand. 00:38:22.170 --> 00:38:25.170 Right Riemann sum. 00:38:25.170 --> 00:38:33.110 Those are the two that we did before. 00:38:33.110 --> 00:38:35.940 Now I'm going to describe to you the next two. 00:38:35.940 --> 00:38:42.410 They have a similar pattern to them. 00:38:42.410 --> 00:38:51.200 And the one with trapezoids requires a picture. 00:38:51.200 --> 00:38:53.110 Here's a shape. 00:38:53.110 --> 00:39:00.690 And here's a bunch of values. 00:39:00.690 --> 00:39:08.870 And we're trying to estimate the size of these chunks. 00:39:08.870 --> 00:39:10.950 And now, instead of doing something stupid, which 00:39:10.950 --> 00:39:14.050 is to draw horizontal lines in rectangles, 00:39:14.050 --> 00:39:16.320 we're going to do something slightly more clever. 00:39:16.320 --> 00:39:21.550 Which is to draw straight lines that are diagonal. 00:39:21.550 --> 00:39:23.177 You see that many of them actually 00:39:23.177 --> 00:39:25.510 coincide probably pretty closely with what I drew there. 00:39:25.510 --> 00:39:29.650 Although if they're curved, they miss by a little bit. 00:39:29.650 --> 00:39:32.270 So this is called the trapezoidal rule. 00:39:32.270 --> 00:39:35.660 Because if you pick one of these shapes, say this is y_2 00:39:35.660 --> 00:39:40.120 and this is y_3, if you pick one of these shapes, 00:39:40.120 --> 00:39:44.440 this height here is y_2, and this height is y_3, 00:39:44.440 --> 00:39:47.720 and this base is delta x. 00:39:47.720 --> 00:39:52.590 This is a trapezoid. 00:39:52.590 --> 00:40:01.500 So this being a trapezoid, I can figure out its area. 00:40:01.500 --> 00:40:02.760 And what do I get? 00:40:02.760 --> 00:40:12.020 I get the base times the average height. 00:40:12.020 --> 00:40:14.470 If you think about if, you work out 00:40:14.470 --> 00:40:16.060 what happens when you do something 00:40:16.060 --> 00:40:18.220 with a straight line on top, like that, 00:40:18.220 --> 00:40:21.080 you'll get this average. 00:40:21.080 --> 00:40:31.450 So this is the average height of the trapezoid. 00:40:31.450 --> 00:40:34.580 And now I want to add up. 00:40:34.580 --> 00:40:41.600 I want to add them all up to get my formula 00:40:41.600 --> 00:40:45.190 for the trapezoidal rule. 00:40:45.190 --> 00:40:46.090 So what do I do? 00:40:46.090 --> 00:40:50.210 I have delta x times the first one. 00:40:50.210 --> 00:40:53.040 Which is (y_0 + y_1) / 2. 00:40:53.040 --> 00:40:55.010 That's the first trapezoid. 00:40:55.010 --> 00:40:59.520 The next one is (y_1 + y_2) / 2. 00:40:59.520 --> 00:41:01.710 And this keeps on going. 00:41:01.710 --> 00:41:07.790 And at the end, I have (y_(n-2) + y_(n-1)) / 2. 00:41:07.790 --> 00:41:10.110 And then last of all, I have (y_(n-1) + y_n) / 2. 00:41:14.610 --> 00:41:16.560 That's a very long formula here. 00:41:16.560 --> 00:41:21.390 We're going to simplify it quite a bit in just a second. 00:41:21.390 --> 00:41:22.600 What's this equal to? 00:41:22.600 --> 00:41:28.820 Well, notice that I get y_0 / 2 to start out with. 00:41:28.820 --> 00:41:32.930 And now, y_1 got mentioned twice. 00:41:32.930 --> 00:41:35.400 Each time with a factor of 1/2. 00:41:35.400 --> 00:41:39.405 So we get a whole y_1 in here. 00:41:39.405 --> 00:41:40.780 And the same thing is going to be 00:41:40.780 --> 00:41:43.940 true of all the middle terms. 00:41:43.940 --> 00:41:48.580 You're going to get y_2 and all the way up to y_(n-1). 00:41:48.580 --> 00:41:53.750 But then, the last one is unmatched. y_n is only 1/2, 00:41:53.750 --> 00:41:59.790 only counts 1/2. 00:41:59.790 --> 00:42:17.590 So here is what's known as the trapezoidal rule. 00:42:17.590 --> 00:42:19.430 Now, I'd like to compare it for you 00:42:19.430 --> 00:42:25.480 to the Riemann sums, which are sitting just to the left here. 00:42:25.480 --> 00:42:28.900 Here's the left one, and here's the right one. 00:42:28.900 --> 00:42:32.570 If you take the average of the left and the right, 00:42:32.570 --> 00:42:34.960 that is, a half of this plus a half of that, 00:42:34.960 --> 00:42:36.370 there's an overlap. 00:42:36.370 --> 00:42:39.250 The y_1 through y_n things are listed in both. 00:42:39.250 --> 00:42:41.860 But the y_0 only gets counted 1/2 00:42:41.860 --> 00:42:44.170 and the y_n only gets counted 1/2. 00:42:44.170 --> 00:42:46.740 So what this is, is this is the symmetric compromise 00:42:46.740 --> 00:42:49.710 between the two Riemann sums. 00:42:49.710 --> 00:42:54.410 This is actually equal to the left Riemann 00:42:54.410 --> 00:43:02.020 sum plus the right Riemann sum divided by 2. 00:43:02.020 --> 00:43:10.600 It's the average of them. 00:43:10.600 --> 00:43:13.340 Now, this would be great and it does look like it's closer. 00:43:13.340 --> 00:43:16.500 But actually it's not as impressive as it looks. 00:43:16.500 --> 00:43:18.910 If you actually do it in practice, 00:43:18.910 --> 00:43:23.190 it's not very efficient. 00:43:23.190 --> 00:43:25.120 Although it's way better than a Riemann sum, 00:43:25.120 --> 00:43:27.560 it's still not good enough. 00:43:27.560 --> 00:43:37.250 So now I need to describe to you the fancier rule. 00:43:37.250 --> 00:43:52.010 Which is known as Simpson's Rule. 00:43:52.010 --> 00:44:00.160 And so, this is, if you like, 3, Method 3. 00:44:00.160 --> 00:44:09.770 The idea is again to divide things into chunks. 00:44:09.770 --> 00:44:16.690 But now it always needs n to be even. 00:44:16.690 --> 00:44:20.750 In other words, we're going to deal not with just one box, 00:44:20.750 --> 00:44:24.800 we're going to deal with pairs of boxes. 00:44:24.800 --> 00:44:28.370 Here's delta x, and here's delta x again. 00:44:28.370 --> 00:44:34.720 And we're going to study the area of this piece here. 00:44:34.720 --> 00:44:39.270 So let me focus just on that part. 00:44:39.270 --> 00:44:50.700 Let's reproduce it over here. 00:44:50.700 --> 00:44:52.684 And here's the delta x, here's delta x. 00:44:52.684 --> 00:44:54.350 And of course there are various heights. 00:44:54.350 --> 00:44:57.020 This starts out at y_0, this is y_2, 00:44:57.020 --> 00:45:03.490 and this middle segment is y_1. 00:45:03.490 --> 00:45:08.200 Now, the approximating curve that we're going to use 00:45:08.200 --> 00:45:14.690 is a parabola. 00:45:14.690 --> 00:45:17.400 That is, we're going to fit a parabola through these three 00:45:17.400 --> 00:45:25.240 points. 00:45:25.240 --> 00:45:28.500 And then we're going to use that as the approximating area. 00:45:28.500 --> 00:45:32.360 Now, it doesn't look like-- This looks like it misses. 00:45:32.360 --> 00:45:34.262 But actually, most functions mostly 00:45:34.262 --> 00:45:35.720 wiggle either one way or the other. 00:45:35.720 --> 00:45:36.490 They don't switch. 00:45:36.490 --> 00:45:38.800 They don't have inflection points. 00:45:38.800 --> 00:45:41.320 So, this is a lousy, at this scale. 00:45:41.320 --> 00:45:42.890 But when we get to a smaller scale, 00:45:42.890 --> 00:45:45.110 this becomes really fantastic. 00:45:45.110 --> 00:45:47.430 As an approximation. 00:45:47.430 --> 00:45:52.400 Now, I need to tell you what the arithmetic is. 00:45:52.400 --> 00:45:56.980 And in order to save time, it's on your problem 00:45:56.980 --> 00:46:00.080 set what the actual formula is. 00:46:00.080 --> 00:46:05.440 But I'm going to tell you how to think about it. 00:46:05.440 --> 00:46:09.530 I want you to think about it as follows. 00:46:09.530 --> 00:46:20.530 So the area under the parabola is 00:46:20.530 --> 00:46:30.800 going to be a base times some kind of average height. 00:46:30.800 --> 00:46:33.770 And the base here, you can already see. 00:46:33.770 --> 00:46:36.020 It's 2 delta x. 00:46:36.020 --> 00:46:39.430 The base is 2 delta x. 00:46:39.430 --> 00:46:40.920 Now, the average height is weird. 00:46:40.920 --> 00:46:43.070 You have to work out what it is for a parabola, 00:46:43.070 --> 00:46:46.420 depending on those three numbers, y_0, y_1, and y_2. 00:46:46.420 --> 00:46:48.760 And it turns out to be the following formula. 00:46:48.760 --> 00:46:50.440 It has to be an average, but it's 00:46:50.440 --> 00:46:51.910 an interesting weighted average. 00:46:51.910 --> 00:46:53.587 So this was the punchline, if you like. 00:46:53.587 --> 00:46:56.170 Is that they are such things as interesting weighted averages. 00:46:56.170 --> 00:46:58.680 This one's very simple, it just involves three numbers. 00:46:58.680 --> 00:46:59.980 But it's still interesting. 00:46:59.980 --> 00:47:00.771 It's the following. 00:47:00.771 --> 00:47:07.260 It turns out to be (y_0 + 4y_1 + y_2) / 6. 00:47:07.260 --> 00:47:08.160 Why divided by 6? 00:47:08.160 --> 00:47:10.730 Well, it's supposed to be an average. 00:47:10.730 --> 00:47:13.640 So the total is 1 + 4 + 1 of these things. 00:47:13.640 --> 00:47:16.090 And 6 is in the denominator. 00:47:16.090 --> 00:47:19.790 So it emphasizes the middle more than the sides. 00:47:19.790 --> 00:47:23.440 And that's what happens with a parabola. 00:47:23.440 --> 00:47:30.580 So this is a computation which is on your homework. 00:47:30.580 --> 00:47:34.440 And now we can put this together for the full Simpson's Rule 00:47:34.440 --> 00:47:36.190 formula. 00:47:36.190 --> 00:47:51.770 Which I'll put up over here. 00:47:51.770 --> 00:47:56.350 We have here 2 delta x, and we divide by 6. 00:47:56.350 --> 00:48:03.470 And then we have (y_0 + 4y_1 + y_2) / 6 plus-- 00:48:03.470 --> 00:48:05.390 That's the first chunk. 00:48:05.390 --> 00:48:09.130 Now, the second chunk, maybe I'll just put it in here, 00:48:09.130 --> 00:48:11.820 starts-- This is x_2. 00:48:11.820 --> 00:48:14.050 This is x_0. 00:48:14.050 --> 00:48:15.620 And it goes all the way to x_4. 00:48:15.620 --> 00:48:18.800 So x_2, x_3, x_4. 00:48:18.800 --> 00:48:21.920 So the next one involves the indices 2, 3 and 4. 00:48:21.920 --> 00:48:31.580 So this is (y_2 + 4y_3 + y_4) / 6. 00:48:31.580 --> 00:48:34.454 Oh, oh, oh, oh, no. 00:48:34.454 --> 00:48:35.870 I think I'll get rid of these 6's. 00:48:35.870 --> 00:48:37.990 I have too many 6's. 00:48:37.990 --> 00:48:39.850 Alright. 00:48:39.850 --> 00:48:41.057 Let's get rid of them here. 00:48:41.057 --> 00:48:41.890 Let's take them out. 00:48:41.890 --> 00:48:44.040 Put them out here. 00:48:44.040 --> 00:48:47.870 Thank you. 00:48:47.870 --> 00:48:51.650 All the way to the end, which is y_(n-2) plus 2y_(n-1)-- sorry, 00:48:51.650 --> 00:48:53.810 plus 4y_(n-1) plus y_n. 00:48:57.050 --> 00:49:01.580 I was about to divide by 6, but you saved me. 00:49:01.580 --> 00:49:04.350 So here are all the chunks. 00:49:04.350 --> 00:49:06.960 Now, what does this pattern come out to be? 00:49:06.960 --> 00:49:10.740 This comes out to be the following. 00:49:10.740 --> 00:49:16.950 This is 1, 4, 1, added to 1, 4, 1 added to 1, 4, 1. 00:49:16.950 --> 00:49:17.760 You add them up. 00:49:17.760 --> 00:49:22.140 You get 1, 4, and then there's a repeat, so you get a 2 and a 4, 00:49:22.140 --> 00:49:24.170 and a 2 and a 4 and a 1. 00:49:24.170 --> 00:49:29.250 So the pattern is that it starts out with 1's on the far ends. 00:49:29.250 --> 00:49:30.910 And then 4's next in. 00:49:30.910 --> 00:49:35.230 And then it alternates 2's and 4's in between. 00:49:35.230 --> 00:49:45.440 So the full pattern of Simpson's Rule is delta x / 3, 00:49:45.440 --> 00:49:48.720 I have now succeeded in canceling this 2 with this 6 00:49:48.720 --> 00:49:50.740 and getting out that factor of 2. 00:49:50.740 --> 00:50:00.966 And then here I have y0 + 4 y1 + 2 y2 + 4 y3 + ... 00:50:00.966 --> 00:50:03.520 It keeps on going and keeps on going and keeps on going. 00:50:03.520 --> 00:50:10.820 And in the end it's 2 y_(n - 2) + 4 y_(n - 1) + y_n. 00:50:10.820 --> 00:50:14.860 So again, 1 and a 4 to start. 00:50:14.860 --> 00:50:16.410 4 and a 1 to end. 00:50:16.410 --> 00:50:20.170 And then alternating 2's and 4's in the middle. 00:50:20.170 --> 00:50:25.650 And this weird weighted average is way better. 00:50:25.650 --> 00:50:28.890 As I will show you next time.