1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,610 Commons license. 3 00:00:03,610 --> 00:00:05,980 Your support will help MIT OpenCourseWare 4 00:00:05,980 --> 00:00:09,930 continue to offer high quality educational resources for free. 5 00:00:09,930 --> 00:00:12,550 To make a donation or to view additional materials 6 00:00:12,550 --> 00:00:16,180 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,180 --> 00:00:21,970 at ocw.mit.edu. 8 00:00:21,970 --> 00:00:25,330 PROFESSOR: Today I want to get started 9 00:00:25,330 --> 00:00:30,340 by correcting a mistake that I made last time. 10 00:00:30,340 --> 00:00:37,410 And this was mistaken terminology. 11 00:00:37,410 --> 00:00:41,290 I said that what we were computing, 12 00:00:41,290 --> 00:00:44,480 when we computed in this candy bar example, 13 00:00:44,480 --> 00:00:47,510 was energy and not heat. 14 00:00:47,510 --> 00:00:49,270 But it's both. 15 00:00:49,270 --> 00:00:51,010 They're the same thing. 16 00:00:51,010 --> 00:00:55,320 And in fact, energy, heat and work 17 00:00:55,320 --> 00:00:59,470 are all the same thing in physics. 18 00:00:59,470 --> 00:01:03,895 I was foolishly considering the much more 19 00:01:03,895 --> 00:01:08,350 - what am I trying to say - the intuitive feeling of heat 20 00:01:08,350 --> 00:01:10,720 as just being the same as temperature. 21 00:01:10,720 --> 00:01:17,890 But in physics, usually heat is measured in calories 22 00:01:17,890 --> 00:01:20,480 and energy can be in lots of things. 23 00:01:20,480 --> 00:01:26,610 Maybe kilowatt-hours or ergs, these are various of the units. 24 00:01:26,610 --> 00:01:31,840 And work would be in things like foot-pounds. 25 00:01:31,840 --> 00:01:35,500 That is, lifting some weight some distance. 26 00:01:35,500 --> 00:01:39,240 And the amount of force you have to apply. 27 00:01:39,240 --> 00:01:42,070 And these all have conversions between them. 28 00:01:42,070 --> 00:01:45,880 They're all the same quantity, in different units. 29 00:01:45,880 --> 00:01:54,030 OK, so these are the same quantity. 30 00:01:54,030 --> 00:02:00,850 Different units. 31 00:02:00,850 --> 00:02:08,850 So that's about as much physics as we'll do for today. 32 00:02:08,850 --> 00:02:11,290 And sorry about that. 33 00:02:11,290 --> 00:02:18,800 Now, the example that I was starting to discuss last time 34 00:02:18,800 --> 00:02:21,980 and that I'm going to carry out today 35 00:02:21,980 --> 00:02:32,390 was this dartboard example. 36 00:02:32,390 --> 00:02:40,160 We have a dartboard, which is some kind of target. 37 00:02:40,160 --> 00:02:46,380 And we have a person, your little brother, 38 00:02:46,380 --> 00:02:48,540 who's standing over there. 39 00:02:48,540 --> 00:02:50,680 And somebody is throwing darts. 40 00:02:50,680 --> 00:02:55,790 And the question is, how likely is he to be hit. 41 00:02:55,790 --> 00:02:59,950 So I want to describe to you how we're 42 00:02:59,950 --> 00:03:02,550 going to make this problem into a math problem. 43 00:03:02,550 --> 00:03:03,050 Yep. 44 00:03:03,050 --> 00:03:03,904 STUDENT: [INAUDIBLE] 45 00:03:03,904 --> 00:03:06,070 PROFESSOR: What topic is this that we're going over. 46 00:03:06,070 --> 00:03:08,250 We're going over an example. 47 00:03:08,250 --> 00:03:11,650 Which is a dartboard example. 48 00:03:11,650 --> 00:03:16,090 And it has to do with probability. 49 00:03:16,090 --> 00:03:17,370 OK. 50 00:03:17,370 --> 00:03:38,260 So what is the probability that this guy, your little brother, 51 00:03:38,260 --> 00:03:47,700 gets hit by a dart. 52 00:03:47,700 --> 00:03:52,020 Now, we have to put some assumptions into this problem 53 00:03:52,020 --> 00:03:53,630 in order to make it a math problem. 54 00:03:53,630 --> 00:03:55,940 And I'm really going to try to make them 55 00:03:55,940 --> 00:03:58,630 pretty realistic assumptions. 56 00:03:58,630 --> 00:04:09,580 So the first assumption is that the number of hits is 57 00:04:09,580 --> 00:04:21,520 proportional to some constant times e^(-r^2). 58 00:04:24,430 --> 00:04:27,910 So that's actually a kind of a normal distribution. 59 00:04:27,910 --> 00:04:30,560 That's the bell curve. 60 00:04:30,560 --> 00:04:34,250 But as a function of the radius. 61 00:04:34,250 --> 00:04:37,060 So this is the assumption that I'm 62 00:04:37,060 --> 00:04:39,990 going to make in this problem. 63 00:04:39,990 --> 00:04:44,970 And a problem in probability is a problem 64 00:04:44,970 --> 00:04:48,540 of the ratio of the part to the whole. 65 00:04:48,540 --> 00:04:53,580 So the part is where this little guy is standing. 66 00:04:53,580 --> 00:04:56,670 And the whole is all the possible places 67 00:04:56,670 --> 00:04:58,232 where the dart might hit. 68 00:04:58,232 --> 00:04:59,690 Which is maybe the whole blackboard 69 00:04:59,690 --> 00:05:02,670 or extending beyond, depending on how good an aim 70 00:05:02,670 --> 00:05:07,780 you think that this older child has. 71 00:05:07,780 --> 00:05:11,730 So, if you like, the part is-- We'll start simply. 72 00:05:11,730 --> 00:05:14,290 I mean, this doesn't sweep all the way around. 73 00:05:14,290 --> 00:05:17,170 But we're going to talk about some section. 74 00:05:17,170 --> 00:05:18,930 Like this. 75 00:05:18,930 --> 00:05:22,400 Where this is some radius r_1, and this other radius 76 00:05:22,400 --> 00:05:24,460 is some longer radius, r_2. 77 00:05:24,460 --> 00:05:27,830 And the part that we'll first keep track of 78 00:05:27,830 --> 00:05:29,470 is everything around there. 79 00:05:29,470 --> 00:05:31,470 That's not very well centered, but it's supposed 80 00:05:31,470 --> 00:05:35,060 to be two concentric circles. 81 00:05:35,060 --> 00:05:38,840 Maybe I should fix that a bit so that it's a little bit easier 82 00:05:38,840 --> 00:05:39,784 to read here. 83 00:05:39,784 --> 00:05:41,260 So here we go. 84 00:05:41,260 --> 00:05:46,700 So here I have radius r_1, and here I have radius r_2. 85 00:05:46,700 --> 00:05:49,610 And then the region in between is what we're 86 00:05:49,610 --> 00:05:56,050 going to try to keep track of. 87 00:05:56,050 --> 00:05:59,170 So I claim that we'll be able to get-- 88 00:05:59,170 --> 00:06:04,100 So this is what I'm calling the part, to start out with. 89 00:06:04,100 --> 00:06:05,570 And then we'll take a section of it 90 00:06:05,570 --> 00:06:12,220 to get the place where the person is standing. 91 00:06:12,220 --> 00:06:22,157 Now, I want to take a side view of e^(-r^2). 92 00:06:22,157 --> 00:06:23,740 The function that we're talking about. 93 00:06:23,740 --> 00:06:28,650 Again, that's the bell curve. 94 00:06:28,650 --> 00:06:31,900 And it sort of looks like this. 95 00:06:31,900 --> 00:06:36,090 This is the top value, this is now the r-axis. 96 00:06:36,090 --> 00:06:39,490 And this is up, or at least-- So you 97 00:06:39,490 --> 00:06:41,540 should think of this in terms of the fact 98 00:06:41,540 --> 00:06:46,900 that the horizontal here is the plane of the dartboard. 99 00:06:46,900 --> 00:06:49,390 And the vertical is measuring how likely 100 00:06:49,390 --> 00:06:52,460 it is that there will be darts piling up here. 101 00:06:52,460 --> 00:06:55,780 So if they were balls tumbling down 102 00:06:55,780 --> 00:06:57,690 or something else falling on, many of them 103 00:06:57,690 --> 00:06:58,530 would pile up here. 104 00:06:58,530 --> 00:07:02,890 Many fewer of them would be piling up farther away. 105 00:07:02,890 --> 00:07:06,430 And the chunk that we're keeping track of 106 00:07:06,430 --> 00:07:09,200 is the chunk between r_1 and r_2. 107 00:07:09,200 --> 00:07:11,590 That's the corresponding region here. 108 00:07:11,590 --> 00:07:14,180 And in order to calculate this part, 109 00:07:14,180 --> 00:07:20,500 we have to calculate this volume of revolution. 110 00:07:20,500 --> 00:07:21,350 Sweeping around. 111 00:07:21,350 --> 00:07:24,160 Because really, in disguise, this is a ring. 112 00:07:24,160 --> 00:07:26,480 This is a side view, it's really a ring. 113 00:07:26,480 --> 00:07:33,100 Because we're rotating it around this axis here. 114 00:07:33,100 --> 00:07:34,990 So we're trying to figure out what 115 00:07:34,990 --> 00:07:36,800 the total volume of that ring is. 116 00:07:36,800 --> 00:07:39,930 And that's going to be our weighting, our likelihood 117 00:07:39,930 --> 00:07:41,390 for whether the hits are occurring 118 00:07:41,390 --> 00:07:51,240 in this section, or in this ring, versus the rest. 119 00:07:51,240 --> 00:07:53,750 To set this up, I remind you we're going 120 00:07:53,750 --> 00:07:57,570 to use the method of shells. 121 00:07:57,570 --> 00:08:03,030 That's really the only one that's going to work here. 122 00:08:03,030 --> 00:08:07,990 And we want to integrate between r_1 and r_2. 123 00:08:07,990 --> 00:08:12,280 And the range here is that-- Because this is, if you like, 124 00:08:12,280 --> 00:08:13,470 a solid of revolution. 125 00:08:13,470 --> 00:08:17,250 So the variable r is the same as what we used to call x. 126 00:08:17,250 --> 00:08:18,924 And it's ranging between r_1 and r_2, 127 00:08:18,924 --> 00:08:20,340 and then we're sweeping it around. 128 00:08:20,340 --> 00:08:25,920 And the circumference of a little piece, 129 00:08:25,920 --> 00:08:35,040 so at a fixed distance r here, the circumference 130 00:08:35,040 --> 00:08:37,950 is going to be 2 pi r. 131 00:08:37,950 --> 00:08:40,240 So 2 pi r is the circumference. 132 00:08:40,240 --> 00:08:42,710 And then the height is the height of that green stem 133 00:08:42,710 --> 00:08:43,210 there. 134 00:08:43,210 --> 00:08:45,460 That's e^(-r^2). 135 00:08:45,460 --> 00:08:50,460 And then we're multiplying by the thickness, which is dr. 136 00:08:50,460 --> 00:08:52,530 So the thickness of the green is dr. 137 00:08:52,530 --> 00:08:57,860 The height of this little green guy is e^(-r^2), 138 00:08:57,860 --> 00:09:01,310 and the circumference is 2 pi times the radius, 139 00:09:01,310 --> 00:09:03,701 when we sweep the circle around. 140 00:09:03,701 --> 00:09:04,200 Question. 141 00:09:04,200 --> 00:09:21,970 STUDENT: [INAUDIBLE] 142 00:09:21,970 --> 00:09:23,850 PROFESSOR: So the question is, why 143 00:09:23,850 --> 00:09:28,020 is what we're interested in not this pink area. 144 00:09:28,020 --> 00:09:31,810 And the reason is an interpretation 145 00:09:31,810 --> 00:09:33,430 of what I meant by this. 146 00:09:33,430 --> 00:09:38,310 What I meant by this is that if you wanted to add up 147 00:09:38,310 --> 00:09:42,190 what the likelihood is that this thing will be here versus here, 148 00:09:42,190 --> 00:09:44,940 I want it to be, really, the proportions 149 00:09:44,940 --> 00:09:47,780 are the number of hits times the, if you like, 150 00:09:47,780 --> 00:09:49,590 I wanted it d area. 151 00:09:49,590 --> 00:09:51,850 That's really what I meant here. 152 00:09:51,850 --> 00:09:56,740 The number of hits in a little chunk. 153 00:09:56,740 --> 00:10:01,370 So little, maybe I'll call it delta A. A little chunk. 154 00:10:01,370 --> 00:10:04,480 Is proportional to the chunk times that. 155 00:10:04,480 --> 00:10:05,920 So there's already an area factor. 156 00:10:05,920 --> 00:10:06,920 And there's a height. 157 00:10:06,920 --> 00:10:09,580 So there are a total of three dimensions involved. 158 00:10:09,580 --> 00:10:13,120 There's the area and then this height. 159 00:10:13,120 --> 00:10:18,640 So it's a matter of the-- what I was given, what I 160 00:10:18,640 --> 00:10:21,720 intended to say the problem is. 161 00:10:21,720 --> 00:10:22,720 Yeah, another question. 162 00:10:22,720 --> 00:10:25,084 STUDENT: [INAUDIBLE] 163 00:10:25,084 --> 00:10:26,000 PROFESSOR: Yeah, yeah. 164 00:10:26,000 --> 00:10:27,060 Exactly. 165 00:10:27,060 --> 00:10:30,570 The height is c times that, whatever this c is. 166 00:10:30,570 --> 00:10:32,220 In fact, we don't know what the c is, 167 00:10:32,220 --> 00:10:34,460 but because we're going to have a part and a whole, 168 00:10:34,460 --> 00:10:37,490 we'll divide, the c will always cancel. 169 00:10:37,490 --> 00:10:39,240 So I'm throwing the c out. 170 00:10:39,240 --> 00:10:42,750 I don't know what it is, and in the end it won't matter. 171 00:10:42,750 --> 00:10:44,330 That's a very important question. 172 00:10:44,330 --> 00:10:44,830 Yes. 173 00:10:44,830 --> 00:10:49,180 STUDENT: [INAUDIBLE] 174 00:10:49,180 --> 00:10:50,180 PROFESSOR: Say it again? 175 00:10:50,180 --> 00:11:02,040 STUDENT: PROFESSOR: 176 00:11:02,040 --> 00:11:05,920 PROFESSOR: So what I mean is the number of hits in some chunk. 177 00:11:05,920 --> 00:11:09,520 That is, suppose you imagine, the question 178 00:11:09,520 --> 00:11:11,720 is, what does this left-hand side mean. 179 00:11:11,720 --> 00:11:12,250 That right? 180 00:11:12,250 --> 00:11:17,690 Is that the question that's you're asking? 181 00:11:17,690 --> 00:11:21,040 When I try to understand what the distribution 182 00:11:21,040 --> 00:11:25,420 of dartboard hits is, I should imagine my dartboard. 183 00:11:25,420 --> 00:11:28,400 And very often there'll be a whole bunch 184 00:11:28,400 --> 00:11:29,700 of holes in some places. 185 00:11:29,700 --> 00:11:30,810 And fewer holes else. 186 00:11:30,810 --> 00:11:33,010 I'm trying to figure out what the whole distribution 187 00:11:33,010 --> 00:11:35,590 of those marks is. 188 00:11:35,590 --> 00:11:38,410 And so some places will have more hits on them 189 00:11:38,410 --> 00:11:41,060 and some places will have fewer hits on them. 190 00:11:41,060 --> 00:11:43,980 And so what I want to measure is, on average, 191 00:11:43,980 --> 00:11:44,910 the number of hits. 192 00:11:44,910 --> 00:11:47,387 So this would really be some-- This constant 193 00:11:47,387 --> 00:11:49,470 of proportionality is ambiguous because it depends 194 00:11:49,470 --> 00:11:50,553 on how many times you try. 195 00:11:50,553 --> 00:11:52,470 If you throw a thousand times, it'll 196 00:11:52,470 --> 00:11:54,320 be much more densely packed. 197 00:11:54,320 --> 00:11:58,780 And if you have only a hundred times it'll be fewer. 198 00:11:58,780 --> 00:12:01,850 So that's where this constant comes in. 199 00:12:01,850 --> 00:12:04,610 But given that you have a certain number of times 200 00:12:04,610 --> 00:12:06,790 that you tried, say, a thousand times, 201 00:12:06,790 --> 00:12:09,500 there will be a whole bunch more piled in the middle. 202 00:12:09,500 --> 00:12:12,020 And fewer and fewer as you get farther away. 203 00:12:12,020 --> 00:12:15,800 Assuming that the person's aim is reasonable. 204 00:12:15,800 --> 00:12:17,024 So that's what we're saying. 205 00:12:17,024 --> 00:12:19,440 So we're thinking in terms of-- The person's always aiming 206 00:12:19,440 --> 00:12:20,065 for the center. 207 00:12:20,065 --> 00:12:22,660 So it's most likely that the person will hit the center. 208 00:12:22,660 --> 00:12:24,820 But on the other hand, it's a fallible person, 209 00:12:24,820 --> 00:12:26,440 so the person may miss. 210 00:12:26,440 --> 00:12:29,272 And so it's less and less likely as you go farther out. 211 00:12:29,272 --> 00:12:31,480 And we're just counting how many times this gets hit, 212 00:12:31,480 --> 00:12:33,220 how many time this and so on. 213 00:12:33,220 --> 00:12:34,830 In proportion to the area. 214 00:12:34,830 --> 00:12:40,460 STUDENT: [INAUDIBLE] 215 00:12:40,460 --> 00:12:42,252 PROFESSOR: Yeah. r_1 and r_2 are arbitrary. 216 00:12:42,252 --> 00:12:44,251 We're going to make this calculation in general. 217 00:12:44,251 --> 00:12:46,240 We're going to calculate what the likelihood is 218 00:12:46,240 --> 00:12:50,290 that we hit any possible band. 219 00:12:50,290 --> 00:12:53,010 And I want to leave those as just letters for now. 220 00:12:53,010 --> 00:12:54,990 The r_1 and the r_2. 221 00:12:54,990 --> 00:12:57,110 Because I want to be able to try various different 222 00:12:57,110 --> 00:12:57,990 possibilities. 223 00:12:57,990 --> 00:13:03,730 STUDENT: [INAUDIBLE] 224 00:13:03,730 --> 00:13:06,310 PROFESSOR: Say it again, why do we have to take volume? 225 00:13:06,310 --> 00:13:08,830 So this is what we were addressing before. 226 00:13:08,830 --> 00:13:13,900 It's a volume because it's number of hit per unit area. 227 00:13:13,900 --> 00:13:17,590 So there's a height, that is, number of hits. 228 00:13:17,590 --> 00:13:19,820 And then there's an area and the product of those 229 00:13:19,820 --> 00:13:22,610 is-- So this is, if you like, a histogram 230 00:13:22,610 --> 00:13:25,100 of the number of hits. 231 00:13:25,100 --> 00:13:26,910 But this should be measured per area. 232 00:13:26,910 --> 00:13:29,400 Not per length of r. 233 00:13:29,400 --> 00:13:32,020 Because on the real diagram, it's going all the way around. 234 00:13:32,020 --> 00:13:35,760 There's a lot more area to this red band 235 00:13:35,760 --> 00:13:41,150 than just the distance implies. 236 00:13:41,150 --> 00:13:42,940 OK. 237 00:13:42,940 --> 00:13:47,460 So, having discussed the setup, this 238 00:13:47,460 --> 00:13:49,770 is a pretty standard setup-- Oh, one more question. 239 00:13:49,770 --> 00:13:50,270 Yes. 240 00:13:50,270 --> 00:14:13,564 STUDENT: [INAUDIBLE] 241 00:14:13,564 --> 00:14:14,230 PROFESSOR: Yeah. 242 00:14:14,230 --> 00:14:17,700 So the question is, why is this a realistic. 243 00:14:17,700 --> 00:14:21,260 Why is this choice of function here, e^(-r^2), 244 00:14:21,260 --> 00:14:25,670 a realistic choice of function for the darts. 245 00:14:25,670 --> 00:14:30,480 So I can answer this with an analogy. 246 00:14:30,480 --> 00:14:32,800 When people were asking themselves 247 00:14:32,800 --> 00:14:38,120 where the V-2 rockets from Germany hit London, 248 00:14:38,120 --> 00:14:39,170 they used this model. 249 00:14:39,170 --> 00:14:43,100 It turned out to be the one which was the most accurate. 250 00:14:43,100 --> 00:14:48,630 So that gives you an idea that this is actually real. 251 00:14:48,630 --> 00:14:53,670 The question, this makes it look like people are masters. 252 00:14:53,670 --> 00:14:56,940 That is, that they'll all hit in the center 253 00:14:56,940 --> 00:14:58,870 more often than elsewhere. 254 00:14:58,870 --> 00:15:03,960 But that's actually somewhat deceptive. 255 00:15:03,960 --> 00:15:08,040 There's a difference between the mode, 256 00:15:08,040 --> 00:15:13,510 that is, the most likely spot, and what happens on average. 257 00:15:13,510 --> 00:15:17,960 So in other words, the single most likely spot is the center. 258 00:15:17,960 --> 00:15:20,720 But there's rather little area in here, and in fact 259 00:15:20,720 --> 00:15:24,860 the likelihood of hitting that is some little tiny bit. 260 00:15:24,860 --> 00:15:25,620 In here. 261 00:15:25,620 --> 00:15:29,850 In fact, you're much more likely to be out here. 262 00:15:29,850 --> 00:15:31,539 So if you take the total of the volume, 263 00:15:31,539 --> 00:15:32,955 you'll see that much of the volume 264 00:15:32,955 --> 00:15:34,260 is contributed from out here. 265 00:15:34,260 --> 00:15:37,780 And in fact, the person hits rather rarely near the center. 266 00:15:37,780 --> 00:15:42,737 So this is not a ridiculous thing to do. 267 00:15:42,737 --> 00:15:45,320 If you think of it in terms of somebody's aiming at the center 268 00:15:45,320 --> 00:15:46,950 but there's some random thing which 269 00:15:46,950 --> 00:15:49,770 is throwing the person off, then there 270 00:15:49,770 --> 00:15:51,710 is likely to be to left or to the right, 271 00:15:51,710 --> 00:15:54,080 or they might even get lucky and all those errors 272 00:15:54,080 --> 00:15:55,470 cancel themselves and they happen 273 00:15:55,470 --> 00:15:59,090 to hit pretty much near the center. 274 00:15:59,090 --> 00:16:00,240 Yeah, another question. 275 00:16:00,240 --> 00:16:06,266 STUDENT: [INAUDIBLE] PROFESSOR: How does the little brother 276 00:16:06,266 --> 00:16:06,890 come into play? 277 00:16:06,890 --> 00:16:09,390 The little brother is going to come into play as follows. 278 00:16:09,390 --> 00:16:11,300 I'll tell you in advance. 279 00:16:11,300 --> 00:16:15,500 So the thing is, the little brother was not so stupid 280 00:16:15,500 --> 00:16:19,760 as to stand in front of the target. 281 00:16:19,760 --> 00:16:22,220 I know. 282 00:16:22,220 --> 00:16:26,800 He stood about twice the radius of the target away. 283 00:16:26,800 --> 00:16:29,960 And so, we're going to approximate the location 284 00:16:29,960 --> 00:16:32,460 by some sector here. 285 00:16:32,460 --> 00:16:35,920 Which is just going to be some chunk of one of these things. 286 00:16:35,920 --> 00:16:38,010 We'll just break off a piece of it. 287 00:16:38,010 --> 00:16:39,850 And that's how we're going to capture. 288 00:16:39,850 --> 00:16:42,030 So the point is, the target is here. 289 00:16:42,030 --> 00:16:46,000 But there is the possibility that the seven-year-old 290 00:16:46,000 --> 00:16:48,550 who's throwing the darts actually missed the target. 291 00:16:48,550 --> 00:16:50,780 That actually happens a lot. 292 00:16:50,780 --> 00:16:54,660 So, does that answer your question? 293 00:16:54,660 --> 00:16:55,530 Alright. 294 00:16:55,530 --> 00:16:57,170 Are we ready now to to do this? 295 00:16:57,170 --> 00:16:57,950 One more question. 296 00:16:57,950 --> 00:17:05,250 STUDENT: [INAUDIBLE] 297 00:17:05,250 --> 00:17:08,320 PROFESSOR: I'm giving you this property here. 298 00:17:08,320 --> 00:17:10,110 I'm telling you that-- This is what's 299 00:17:10,110 --> 00:17:12,540 called a mathematical model, when you give 300 00:17:12,540 --> 00:17:13,900 somebody something like this. 301 00:17:13,900 --> 00:17:16,450 In fact, that requires further justification. 302 00:17:16,450 --> 00:17:19,120 It's an interesting issue. 303 00:17:19,120 --> 00:17:19,620 Yeah. 304 00:17:19,620 --> 00:17:26,084 STUDENT: [INAUDIBLE] 305 00:17:26,084 --> 00:17:27,750 PROFESSOR: I'm giving it to you for now. 306 00:17:27,750 --> 00:17:30,120 And it's something which really has to be justified. 307 00:17:30,120 --> 00:17:32,080 In certain circumstances it is justified. 308 00:17:32,080 --> 00:17:37,620 But, OK. 309 00:17:37,620 --> 00:17:43,800 So anyway, here's our part. 310 00:17:43,800 --> 00:17:46,200 This is going to be our chunk, for now, 311 00:17:46,200 --> 00:17:50,760 that we're going to estimate the importance, 312 00:17:50,760 --> 00:17:56,210 the relative importance, of. 313 00:17:56,210 --> 00:18:01,460 And now, this is something whose antiderivative we can just do 314 00:18:01,460 --> 00:18:03,600 by substitution or by guessing. 315 00:18:03,600 --> 00:18:07,870 It's just -pi e^(-r^2). 316 00:18:07,870 --> 00:18:10,650 If you differentiate that, you get a -2r, which 317 00:18:10,650 --> 00:18:12,730 cancels the minus sign here. 318 00:18:12,730 --> 00:18:15,370 So you get 2pi r e^(-r^2). 319 00:18:15,370 --> 00:18:18,400 So that's the antiderivative. 320 00:18:18,400 --> 00:18:24,710 And we're evaluating it at r_1 and r_2. 321 00:18:24,710 --> 00:18:28,490 So with the minus sign that's going to get reversed. 322 00:18:28,490 --> 00:18:33,350 The answer is going to be pi times e^(-r_1^2), 323 00:18:33,350 --> 00:18:39,090 that's the bottom one, minus e^(r_2^2), that's the top. 324 00:18:39,090 --> 00:18:45,777 So this is what our part gives us. 325 00:18:45,777 --> 00:18:48,110 And, more technically, if you wanted to multiply through 326 00:18:48,110 --> 00:18:50,350 by c, it would be c times this. 327 00:18:50,350 --> 00:18:55,140 I'll say that in just a second. 328 00:18:55,140 --> 00:18:59,280 OK, now I want to work-- So, if you like, 329 00:18:59,280 --> 00:19:06,490 the part is equal to, maybe even I'll call it c pi, 330 00:19:06,490 --> 00:19:07,740 times e^(-r_1^2) - e^(-r_2^2). 331 00:19:12,429 --> 00:19:14,470 That's what it really is if I put in this factor. 332 00:19:14,470 --> 00:19:17,690 So now there's no prejudice as to how many attempts we make. 333 00:19:17,690 --> 00:19:20,490 Whether it was a thousand attempts or a million 334 00:19:20,490 --> 00:19:22,740 attempts at the target. 335 00:19:22,740 --> 00:19:25,480 Now, the most important second feature 336 00:19:25,480 --> 00:19:27,250 here of these kinds of modeling problems 337 00:19:27,250 --> 00:19:30,560 is, there is always some kind of idealization. 338 00:19:30,560 --> 00:19:33,070 And the next thing that I want to discuss with you 339 00:19:33,070 --> 00:19:36,940 is the interpretation of the whole. 340 00:19:36,940 --> 00:19:41,870 That is, what's the family of all possibilities. 341 00:19:41,870 --> 00:19:45,790 And in this case, what I'm going to claim 342 00:19:45,790 --> 00:19:50,160 is that the reasonable way to think of the whole 343 00:19:50,160 --> 00:19:55,940 is it's that r can range all the way from 0 to infinity. 344 00:19:55,940 --> 00:19:57,210 Now, you may not like this. 345 00:19:57,210 --> 00:20:01,730 But these are maybe my first and third favorite number, 346 00:20:01,730 --> 00:20:03,810 my second favorite number being 1. 347 00:20:03,810 --> 00:20:07,930 So infinity is a really useful concept. 348 00:20:07,930 --> 00:20:12,010 Of course, it's nonsense in the context of the darts. 349 00:20:12,010 --> 00:20:14,770 Because if you think of the basement wall 350 00:20:14,770 --> 00:20:17,140 where the kid might miss a target, 351 00:20:17,140 --> 00:20:19,619 he'd probably hit the wall. 352 00:20:19,619 --> 00:20:22,160 He's probably not going to hit one of the walls to the right, 353 00:20:22,160 --> 00:20:25,080 and anyway he's certainly not going to hit over there. 354 00:20:25,080 --> 00:20:28,870 So there's something artificial about sending 355 00:20:28,870 --> 00:20:32,390 the possibility of hits all the way out to infinity. 356 00:20:32,390 --> 00:20:34,850 On the other hand, the shape of this curve 357 00:20:34,850 --> 00:20:38,310 is such that the real tail ends here, 358 00:20:38,310 --> 00:20:40,990 because of this exponential decrease, are tiny. 359 00:20:40,990 --> 00:20:42,630 And that's negligible. 360 00:20:42,630 --> 00:20:45,700 And the point is that actually the value, 361 00:20:45,700 --> 00:20:47,550 if you go all the way out to infinity, 362 00:20:47,550 --> 00:20:50,020 is the easiest value to calculate. 363 00:20:50,020 --> 00:20:52,697 So by doing this, I'm idealizing the problem 364 00:20:52,697 --> 00:20:54,530 but I'm actually making the numbers come out 365 00:20:54,530 --> 00:20:55,890 much more cleanly. 366 00:20:55,890 --> 00:20:58,180 And this is just always done in mathematics. 367 00:20:58,180 --> 00:21:00,290 That's what we did when we went from differences 368 00:21:00,290 --> 00:21:02,460 to differentials, to differentiation 369 00:21:02,460 --> 00:21:04,640 and infinitesimals. 370 00:21:04,640 --> 00:21:06,282 So we like that. 371 00:21:06,282 --> 00:21:08,740 Because it makes things easier, not because it makes things 372 00:21:08,740 --> 00:21:10,320 harder. 373 00:21:10,320 --> 00:21:12,240 So we're just going to pretend the whole is 374 00:21:12,240 --> 00:21:13,450 from 0 to infinity. 375 00:21:13,450 --> 00:21:15,550 And now let's just see what it is. 376 00:21:15,550 --> 00:21:21,330 It's c pi times, the starting place is e^(-0^2). 377 00:21:21,330 --> 00:21:24,630 That's r_1, right, this is the role that r_1 plays is this, 378 00:21:24,630 --> 00:21:26,374 and the r_2 is this value. 379 00:21:26,374 --> 00:21:27,290 Minus e^(-infinity^2). 380 00:21:30,180 --> 00:21:33,940 Which is that negligibly small number, 0. 381 00:21:33,940 --> 00:21:36,750 So this is just c pi. 382 00:21:36,750 --> 00:21:39,700 Because this number is 1, and this other number is 0. 383 00:21:39,700 --> 00:21:48,420 This is just (1 - 0), in the parentheses there. 384 00:21:48,420 --> 00:21:52,300 And now I can tell you from these two numbers 385 00:21:52,300 --> 00:21:54,990 what the probability is. 386 00:21:54,990 --> 00:21:58,450 The probability that we landed on the target 387 00:21:58,450 --> 00:22:04,250 in a radius between r_1 and r_2, so that's this annulus here, 388 00:22:04,250 --> 00:22:12,350 is the ratio of the part to the whole. 389 00:22:12,350 --> 00:22:16,060 Which in this case just cancels the c pi. 390 00:22:16,060 --> 00:22:21,280 So it's e^(-r_1^2) - e^(-r_2^2). 391 00:22:21,280 --> 00:22:25,520 There's the formula for the probability. 392 00:22:25,520 --> 00:22:28,100 So the c canceled and the pi canceled. 393 00:22:28,100 --> 00:22:32,170 It's all gone. 394 00:22:32,170 --> 00:22:34,040 Now, again, let me just emphasize 395 00:22:34,040 --> 00:22:35,750 the way this formula is supposed to work. 396 00:22:35,750 --> 00:22:42,210 The total probability of every possibility 397 00:22:42,210 --> 00:22:45,900 here is supposed to be set up to be 1. 398 00:22:45,900 --> 00:22:48,530 This is some fraction of 1. 399 00:22:48,530 --> 00:22:51,191 If you like, it's a percent. 400 00:22:51,191 --> 00:22:51,690 Yes. 401 00:22:51,690 --> 00:22:57,149 STUDENT: [INAUDIBLE] 402 00:22:57,149 --> 00:22:59,690 PROFESSOR: This one is giving-- The question is, doesn't this 403 00:22:59,690 --> 00:23:00,780 just give the probability of the ring? 404 00:23:00,780 --> 00:23:02,620 This gives you the probability of the ring, 405 00:23:02,620 --> 00:23:05,510 but this is a very, very wide ring. 406 00:23:05,510 --> 00:23:08,479 This is a ring starting with 0, nothing, on the inside. 407 00:23:08,479 --> 00:23:09,770 And then going all the way out. 408 00:23:09,770 --> 00:23:11,880 So that's everything. 409 00:23:11,880 --> 00:23:13,380 So this corresponds to everything. 410 00:23:13,380 --> 00:23:22,650 This corresponds to a ring. 411 00:23:22,650 --> 00:23:24,990 So now, let's see. 412 00:23:24,990 --> 00:23:26,390 Where do I want to go from here. 413 00:23:26,390 --> 00:23:28,550 So in order to make progress here, 414 00:23:28,550 --> 00:23:31,230 I still have to give you one more piece of information. 415 00:23:31,230 --> 00:23:35,650 And this is, again, supposed to be realistic. 416 00:23:35,650 --> 00:23:40,350 When I was three years old and my brother's friend Ralph 417 00:23:40,350 --> 00:23:43,920 was seven, I watched him throwing darts a lot. 418 00:23:43,920 --> 00:23:54,200 And I would say that for Ralph, so for Ralph, at age seven, 419 00:23:54,200 --> 00:23:57,080 anyway, later on he got a little better at it. 420 00:23:57,080 --> 00:23:59,950 But Ralph at age seven, the probability that he 421 00:23:59,950 --> 00:24:08,440 hit the target was about 1/2. 422 00:24:08,440 --> 00:24:09,660 Right? 423 00:24:09,660 --> 00:24:12,040 So he hit the target about half the time. 424 00:24:12,040 --> 00:24:14,560 And the other times, there was cement 425 00:24:14,560 --> 00:24:17,870 on the walls of the basement, it wasn't that bad. 426 00:24:17,870 --> 00:24:19,250 Just bounced off. 427 00:24:19,250 --> 00:24:22,000 That also meant that the points got a little blunter as time 428 00:24:22,000 --> 00:24:22,560 went on. 429 00:24:22,560 --> 00:24:26,150 So it was a little less dangerous when they hit. 430 00:24:26,150 --> 00:24:26,650 Alright. 431 00:24:26,650 --> 00:24:33,210 So now, so here's the extra assumption that I want to make. 432 00:24:33,210 --> 00:24:41,260 So a is going to be the radius of the target. 433 00:24:41,260 --> 00:24:45,660 Now, the other realistic assumption that I want to make 434 00:24:45,660 --> 00:24:49,170 is where this little kid would be standing. 435 00:24:49,170 --> 00:24:51,650 And now, here, I want to get very specific 436 00:24:51,650 --> 00:24:54,280 and just do the computation in one case. 437 00:24:54,280 --> 00:25:00,430 We're going to imagine the target is here. 438 00:25:00,430 --> 00:25:03,200 And the kid is standing, say, between-- So we'll just 439 00:25:03,200 --> 00:25:04,430 do a section of this. 440 00:25:04,430 --> 00:25:13,130 This is between 3 o'clock and 5 o'clock. 441 00:25:13,130 --> 00:25:16,740 There's more of him, but it's lower down and maybe negligible 442 00:25:16,740 --> 00:25:17,350 here. 443 00:25:17,350 --> 00:25:20,400 So this section is the part, the chunk 444 00:25:20,400 --> 00:25:23,330 that we want to see about. 445 00:25:23,330 --> 00:25:31,210 And this is a, and then this distance here is 2a. 446 00:25:31,210 --> 00:25:35,330 And then the longest distance here is 3a. 447 00:25:35,330 --> 00:25:39,000 So the longest distance is 3a. 448 00:25:39,000 --> 00:25:40,870 So in other words, what I'm saying 449 00:25:40,870 --> 00:25:50,180 is that the probability, if you're standing too close, 450 00:25:50,180 --> 00:26:06,660 the chance Ralph hits younger brother is about 1/6-- Right? 451 00:26:06,660 --> 00:26:12,820 Because 2/12 = 1/6. 452 00:26:12,820 --> 00:26:22,020 1/6 of the probability that we're between a and 3a. 453 00:26:22,020 --> 00:26:29,052 That's the number that we're looking for. 454 00:26:29,052 --> 00:26:29,760 Another question. 455 00:26:29,760 --> 00:26:33,900 STUDENT: [INAUDIBLE] 456 00:26:33,900 --> 00:26:36,680 PROFESSOR: The 2/12 came from the fact that we assumed. 457 00:26:36,680 --> 00:26:39,250 So we made a very, very bold assumption here. 458 00:26:39,250 --> 00:26:41,530 We assumed that this human being, 459 00:26:41,530 --> 00:26:46,760 who is actually standing-- The floor is about down here. 460 00:26:46,760 --> 00:26:48,460 Maybe he wasn't that tall. 461 00:26:48,460 --> 00:26:51,560 But anyway, so he's really a little bigger than this. 462 00:26:51,560 --> 00:26:53,850 That the part of him that was close to the target 463 00:26:53,850 --> 00:27:00,500 covered about this section here, between radius 2a and 3a. 464 00:27:00,500 --> 00:27:02,740 As you'll see, actually from the computation, 465 00:27:02,740 --> 00:27:07,290 because the likelihood drops off pretty quickly, whatever 466 00:27:07,290 --> 00:27:09,140 of him was standing outside there 467 00:27:09,140 --> 00:27:11,269 wouldn't have mattered anyway. 468 00:27:11,269 --> 00:27:13,060 So we're just worried about the part that's 469 00:27:13,060 --> 00:27:18,677 closest to the target here. 470 00:27:18,677 --> 00:27:19,510 STUDENT: [INAUDIBLE] 471 00:27:19,510 --> 00:27:20,801 PROFESSOR: Why is it out of 12? 472 00:27:20,801 --> 00:27:22,650 Because I made it a clock. 473 00:27:22,650 --> 00:27:25,110 And I made it from 3 o'clock to 5 o'clock, 474 00:27:25,110 --> 00:27:28,790 so it's 2 of the 12 hours of a clock. 475 00:27:28,790 --> 00:27:32,640 It's just a way of me making a section that you can visibly 476 00:27:32,640 --> 00:27:36,400 see. 477 00:27:36,400 --> 00:27:40,060 So now, so here's what we're trying to calculate. 478 00:27:40,060 --> 00:27:47,790 And in order to figure this out, I need one more item. here. 479 00:27:47,790 --> 00:27:49,870 So maybe I'll leave myself a little bit of room. 480 00:27:49,870 --> 00:27:52,010 I have to figure out something about what 481 00:27:52,010 --> 00:27:53,830 our information gave us. 482 00:27:53,830 --> 00:28:00,380 Which is that the probability, sorry, this probability 483 00:28:00,380 --> 00:28:01,690 was 1/2. 484 00:28:01,690 --> 00:28:03,580 So let's remember what this is. 485 00:28:03,580 --> 00:28:09,470 This is going to be e^(-0^2) - e^(-a^2). 486 00:28:09,470 --> 00:28:11,180 That's what's this probability is. 487 00:28:11,180 --> 00:28:16,210 And that's equal to 1/2. 488 00:28:16,210 --> 00:28:21,330 So that means that - 1 - e^(-a^2). = 1/2. 489 00:28:21,330 --> 00:28:26,620 Which means that e^(-a^2) = 1/2. 490 00:28:26,620 --> 00:28:28,220 I'm not going to calculate a, this 491 00:28:28,220 --> 00:28:30,050 is the part about the information about a 492 00:28:30,050 --> 00:28:32,480 that I want to use. 493 00:28:32,480 --> 00:28:33,520 That's what I'll use. 494 00:28:33,520 --> 00:28:36,250 And now I'm going to calculate this other probability here. 495 00:28:36,250 --> 00:28:41,590 So the probability right up there is this. 496 00:28:41,590 --> 00:28:47,390 And that's going to be the same as e^(-(2a)^2) - e^(-(3a)^2). 497 00:28:51,610 --> 00:28:54,650 That's what we calculated. 498 00:28:54,650 --> 00:28:59,200 And now I want to use some of the arithmetic of exponents. 499 00:28:59,200 --> 00:29:01,380 This is (e^(-a^2))^2. 500 00:29:05,180 --> 00:29:11,890 Because it's really (2a)^2 is 4-- the quantity is 4a^2, 501 00:29:11,890 --> 00:29:14,580 and then I bring that exponent out. 502 00:29:14,580 --> 00:29:19,740 Minus e^(-a^2) to the 9th power, that's 3^2. 503 00:29:25,040 --> 00:29:28,560 And so this comes out to be (1/2)^4 - (1/2)^9. 504 00:29:36,190 --> 00:29:39,340 Which is approximately 1/16. 505 00:29:39,340 --> 00:29:40,380 This is negligible. 506 00:29:40,380 --> 00:29:41,180 This part here. 507 00:29:41,180 --> 00:29:43,110 And this is actually why these tails, 508 00:29:43,110 --> 00:29:48,130 as you go out to infinity, don't really matter that much. 509 00:29:48,130 --> 00:29:49,880 So this is a much smaller number. 510 00:29:49,880 --> 00:29:53,390 So the probability of the whole band is 1/16. 511 00:29:53,390 --> 00:29:55,720 And now I can answer the question up here. 512 00:29:55,720 --> 00:30:01,110 This is approximately 1/6 * 1/16. 513 00:30:01,110 --> 00:30:07,340 Which is about 1/100, or about 1%. 514 00:30:07,340 --> 00:30:13,340 So if I stood there for 100 attempts, 515 00:30:13,340 --> 00:30:18,720 then my chances of getting hit were pretty high. 516 00:30:18,720 --> 00:30:27,970 So that's the computation. 517 00:30:27,970 --> 00:30:34,940 That's a typical example of a problem in probability. 518 00:30:34,940 --> 00:30:36,860 And let me just make one more connection 519 00:30:36,860 --> 00:30:42,660 with what we did before. 520 00:30:42,660 --> 00:30:46,130 This is connected to weighted averages or integrals 521 00:30:46,130 --> 00:30:47,470 over weights. 522 00:30:47,470 --> 00:30:53,110 But the weight that's involved in this problem 523 00:30:53,110 --> 00:30:59,810 was w(r) is equal to-- So let's just look at what happened 524 00:30:59,810 --> 00:31:01,520 in all of those integrals. 525 00:31:01,520 --> 00:31:03,790 What happened in all the integrals was, we 526 00:31:03,790 --> 00:31:09,060 had this factor here, 2 pi r. 527 00:31:09,060 --> 00:31:23,330 And if I include this c, it was really 2 pi c r e^(-r^2). 528 00:31:23,330 --> 00:31:27,960 This was the weight that we were using. 529 00:31:27,960 --> 00:31:30,770 The relative importance of things. 530 00:31:30,770 --> 00:31:34,260 Now, this is not the same as the e^(-r^2) that we started out 531 00:31:34,260 --> 00:31:34,760 with. 532 00:31:34,760 --> 00:31:38,000 Because this is the one-dimensional histogram. 533 00:31:38,000 --> 00:31:40,850 And that has to do with the method of shells that gave us 534 00:31:40,850 --> 00:31:43,210 that extra factor of r here. 535 00:31:43,210 --> 00:31:45,090 So that also connects with the question 536 00:31:45,090 --> 00:31:47,610 at the beginning, which had to do with this paradox, 537 00:31:47,610 --> 00:31:50,420 that it looks like these places in the middle 538 00:31:50,420 --> 00:31:51,440 are the most likely. 539 00:31:51,440 --> 00:31:54,500 But that's the plot e^(-r^2). 540 00:31:54,500 --> 00:31:57,240 If you actually look at this plot here, 541 00:31:57,240 --> 00:32:00,330 you see that as r goes to 0, it's going to 0. 542 00:32:00,330 --> 00:32:03,010 This is a different graph here. 543 00:32:03,010 --> 00:32:07,720 And actually, so this is what's happening really. 544 00:32:07,720 --> 00:32:09,730 In terms of how likely it is that you'll 545 00:32:09,730 --> 00:32:18,392 get within a certain distance of the center of the target. 546 00:32:18,392 --> 00:32:20,850 Again, it's not the area under that curve that we're doing. 547 00:32:20,850 --> 00:32:27,940 It's that volume of revolution. 548 00:32:27,940 --> 00:32:29,494 We're going to change subjects now. 549 00:32:29,494 --> 00:32:30,410 OK, one more question. 550 00:32:30,410 --> 00:32:31,937 Yes. 551 00:32:31,937 --> 00:32:32,770 STUDENT: [INAUDIBLE] 552 00:32:32,770 --> 00:32:36,120 PROFESSOR: Yeah, that's supposed to be the graph of w(r). 553 00:32:36,120 --> 00:32:47,359 STUDENT: [INAUDIBLE] 554 00:32:47,359 --> 00:32:48,900 PROFESSOR: Well, so, the question is, 555 00:32:48,900 --> 00:32:50,983 wouldn't the importance of the center be greatest? 556 00:32:50,983 --> 00:32:53,790 It's a question of which variable you're using. 557 00:32:53,790 --> 00:32:57,057 According to pure radius, it's not. 558 00:32:57,057 --> 00:32:58,640 It turns out that there are some bands 559 00:32:58,640 --> 00:33:00,930 in radius which are more important, more likely 560 00:33:00,930 --> 00:33:06,206 for hits than others. 561 00:33:06,206 --> 00:33:07,580 It really has to do with the fact 562 00:33:07,580 --> 00:33:10,980 that the center, or the core, of the target is really tiny. 563 00:33:10,980 --> 00:33:11,950 So it's harder to hit. 564 00:33:11,950 --> 00:33:16,170 Whereas a whole band around the outside has a lot more area. 565 00:33:16,170 --> 00:33:18,500 Many, many ways to hit that band. 566 00:33:18,500 --> 00:33:19,810 So it's a much larger area. 567 00:33:19,810 --> 00:33:24,220 So there's a competition there between those two things. 568 00:33:24,220 --> 00:33:25,910 So we're going to move on. 569 00:33:25,910 --> 00:33:28,170 And I want to talk now about a different kind 570 00:33:28,170 --> 00:33:30,550 of weighted average. 571 00:33:30,550 --> 00:33:36,410 These weighted averages are going to be much simpler. 572 00:33:36,410 --> 00:34:01,260 And they come up in what's called numerical integration. 573 00:34:01,260 --> 00:34:04,597 There are many, many methods of integrating numerically. 574 00:34:04,597 --> 00:34:06,680 And they're important because many, many integrals 575 00:34:06,680 --> 00:34:08,510 don't have formulas. 576 00:34:08,510 --> 00:34:13,980 And so you have to compute them with a calculator or a machine. 577 00:34:13,980 --> 00:34:24,400 So the first type that we've already done are Riemann sums. 578 00:34:24,400 --> 00:34:28,170 They turned out to be incredibly inefficient. 579 00:34:28,170 --> 00:34:30,270 They're lousy. 580 00:34:30,270 --> 00:34:32,120 The next rule that I'm going to describe 581 00:34:32,120 --> 00:34:33,370 is a little improvement. 582 00:34:33,370 --> 00:34:38,130 It's called the trapezoidal rule. 583 00:34:38,130 --> 00:34:41,340 And this one is much more reasonable 584 00:34:41,340 --> 00:34:43,370 than the Riemann sum. 585 00:34:43,370 --> 00:34:49,140 Unfortunately, it's actually also pretty lousy. 586 00:34:49,140 --> 00:34:53,150 There's another rule which is just a slightly trickier rule. 587 00:34:53,150 --> 00:34:56,030 And it's actually amazing that it exists. 588 00:34:56,030 --> 00:35:01,750 And it's called Simpson's Rule. 589 00:35:01,750 --> 00:35:06,690 And this one is actually pretty good. 590 00:35:06,690 --> 00:35:07,770 It's clever. 591 00:35:07,770 --> 00:35:12,320 So let's get started with these. 592 00:35:12,320 --> 00:35:15,030 And the way I'll get started is by reminding you 593 00:35:15,030 --> 00:35:16,030 what the Riemann sum is. 594 00:35:16,030 --> 00:35:17,779 So this is a good review, because you need 595 00:35:17,779 --> 00:35:19,790 to know all three of these. 596 00:35:19,790 --> 00:35:26,540 And you're going to want to see them all laid out in parallel. 597 00:35:26,540 --> 00:35:31,800 So here's the setup. 598 00:35:31,800 --> 00:35:33,580 OK, so here we go. 599 00:35:33,580 --> 00:35:36,660 We have our graph, we have our function, it starts out at a, 600 00:35:36,660 --> 00:35:38,240 it ends up at b. 601 00:35:38,240 --> 00:35:40,160 Maybe goes on, but we're only paying attention 602 00:35:40,160 --> 00:35:41,380 to this interval. 603 00:35:41,380 --> 00:35:44,760 This is a function y = f(x). 604 00:35:44,760 --> 00:35:48,160 And we split it up when we do Riemann sums. 605 00:35:48,160 --> 00:35:50,450 So this is 1, this is Riemann sums. 606 00:35:50,450 --> 00:35:53,930 We start out with a, which is a point we call x_0, 607 00:35:53,930 --> 00:35:57,370 and then we go a certain distance. 608 00:35:57,370 --> 00:35:58,710 We go all the way over to b. 609 00:35:58,710 --> 00:36:04,150 And we subdivide this thing with these delta x's. 610 00:36:04,150 --> 00:36:08,130 Which are the step sizes. 611 00:36:08,130 --> 00:36:12,840 So we have these little steps of size delta x. 612 00:36:12,840 --> 00:36:18,860 Corresponding to these x values, we have y values. y_0 = f(x_0), 613 00:36:18,860 --> 00:36:21,090 that's the point above a. 614 00:36:21,090 --> 00:36:26,840 Then y_1 = f(x_1), that's the point above x_1. 615 00:36:26,840 --> 00:36:33,120 And so forth, all the way up to y_n, which is f(x_n). 616 00:36:33,120 --> 00:36:35,650 In order to figure out the area, you really 617 00:36:35,650 --> 00:36:37,400 need to know something about the function. 618 00:36:37,400 --> 00:36:39,217 You need to be able to evaluate it. 619 00:36:39,217 --> 00:36:40,300 So that's what we've done. 620 00:36:40,300 --> 00:36:43,330 We've evaluated here at n + 1 points. 621 00:36:43,330 --> 00:36:46,150 Enumerated 0 through n. 622 00:36:46,150 --> 00:36:48,910 And those are the numbers out of which 623 00:36:48,910 --> 00:36:53,390 we're going to get all of our approximations to the integral. 624 00:36:53,390 --> 00:36:56,370 So somehow we want average these numbers. 625 00:36:56,370 --> 00:37:01,780 So here's our goal. 626 00:37:01,780 --> 00:37:14,450 Our goal is to average, or add, I'm using average very loosely 627 00:37:14,450 --> 00:37:14,950 here. 628 00:37:14,950 --> 00:37:18,830 But I was going to say add up these numbers 629 00:37:18,830 --> 00:37:25,750 to get an approximation. 630 00:37:25,750 --> 00:37:30,730 To average or add the y's. 631 00:37:30,730 --> 00:37:37,670 To get an approximation to the integral. 632 00:37:37,670 --> 00:37:43,410 Which we know is the area under the curve. 633 00:37:43,410 --> 00:37:49,730 So here's what the Riemann sum is. 634 00:37:49,730 --> 00:37:51,020 It's the following thing. 635 00:37:51,020 --> 00:37:56,030 You take y_0 plus y_1 plus... up to y_(n-1). 636 00:37:56,030 --> 00:37:57,520 And you multiply by delta x. 637 00:37:57,520 --> 00:38:00,040 That's it. 638 00:38:00,040 --> 00:38:07,010 Now, this one is the one with left endpoints. 639 00:38:07,010 --> 00:38:09,350 So the left-hand sum. 640 00:38:09,350 --> 00:38:11,120 There's also a right one. 641 00:38:11,120 --> 00:38:13,210 Which is, if you start at the right-hand ends. 642 00:38:13,210 --> 00:38:17,830 And that will go from y_1 to y_n. 643 00:38:17,830 --> 00:38:22,170 OK, so this one is the right-hand. 644 00:38:22,170 --> 00:38:25,170 Right Riemann sum. 645 00:38:25,170 --> 00:38:33,110 Those are the two that we did before. 646 00:38:33,110 --> 00:38:35,940 Now I'm going to describe to you the next two. 647 00:38:35,940 --> 00:38:42,410 They have a similar pattern to them. 648 00:38:42,410 --> 00:38:51,200 And the one with trapezoids requires a picture. 649 00:38:51,200 --> 00:38:53,110 Here's a shape. 650 00:38:53,110 --> 00:39:00,690 And here's a bunch of values. 651 00:39:00,690 --> 00:39:08,870 And we're trying to estimate the size of these chunks. 652 00:39:08,870 --> 00:39:10,950 And now, instead of doing something stupid, which 653 00:39:10,950 --> 00:39:14,050 is to draw horizontal lines in rectangles, 654 00:39:14,050 --> 00:39:16,320 we're going to do something slightly more clever. 655 00:39:16,320 --> 00:39:21,550 Which is to draw straight lines that are diagonal. 656 00:39:21,550 --> 00:39:23,177 You see that many of them actually 657 00:39:23,177 --> 00:39:25,510 coincide probably pretty closely with what I drew there. 658 00:39:25,510 --> 00:39:29,650 Although if they're curved, they miss by a little bit. 659 00:39:29,650 --> 00:39:32,270 So this is called the trapezoidal rule. 660 00:39:32,270 --> 00:39:35,660 Because if you pick one of these shapes, say this is y_2 661 00:39:35,660 --> 00:39:40,120 and this is y_3, if you pick one of these shapes, 662 00:39:40,120 --> 00:39:44,440 this height here is y_2, and this height is y_3, 663 00:39:44,440 --> 00:39:47,720 and this base is delta x. 664 00:39:47,720 --> 00:39:52,590 This is a trapezoid. 665 00:39:52,590 --> 00:40:01,500 So this being a trapezoid, I can figure out its area. 666 00:40:01,500 --> 00:40:02,760 And what do I get? 667 00:40:02,760 --> 00:40:12,020 I get the base times the average height. 668 00:40:12,020 --> 00:40:14,470 If you think about if, you work out 669 00:40:14,470 --> 00:40:16,060 what happens when you do something 670 00:40:16,060 --> 00:40:18,220 with a straight line on top, like that, 671 00:40:18,220 --> 00:40:21,080 you'll get this average. 672 00:40:21,080 --> 00:40:31,450 So this is the average height of the trapezoid. 673 00:40:31,450 --> 00:40:34,580 And now I want to add up. 674 00:40:34,580 --> 00:40:41,600 I want to add them all up to get my formula 675 00:40:41,600 --> 00:40:45,190 for the trapezoidal rule. 676 00:40:45,190 --> 00:40:46,090 So what do I do? 677 00:40:46,090 --> 00:40:50,210 I have delta x times the first one. 678 00:40:50,210 --> 00:40:53,040 Which is (y_0 + y_1) / 2. 679 00:40:53,040 --> 00:40:55,010 That's the first trapezoid. 680 00:40:55,010 --> 00:40:59,520 The next one is (y_1 + y_2) / 2. 681 00:40:59,520 --> 00:41:01,710 And this keeps on going. 682 00:41:01,710 --> 00:41:07,790 And at the end, I have (y_(n-2) + y_(n-1)) / 2. 683 00:41:07,790 --> 00:41:10,110 And then last of all, I have (y_(n-1) + y_n) / 2. 684 00:41:14,610 --> 00:41:16,560 That's a very long formula here. 685 00:41:16,560 --> 00:41:21,390 We're going to simplify it quite a bit in just a second. 686 00:41:21,390 --> 00:41:22,600 What's this equal to? 687 00:41:22,600 --> 00:41:28,820 Well, notice that I get y_0 / 2 to start out with. 688 00:41:28,820 --> 00:41:32,930 And now, y_1 got mentioned twice. 689 00:41:32,930 --> 00:41:35,400 Each time with a factor of 1/2. 690 00:41:35,400 --> 00:41:39,405 So we get a whole y_1 in here. 691 00:41:39,405 --> 00:41:40,780 And the same thing is going to be 692 00:41:40,780 --> 00:41:43,940 true of all the middle terms. 693 00:41:43,940 --> 00:41:48,580 You're going to get y_2 and all the way up to y_(n-1). 694 00:41:48,580 --> 00:41:53,750 But then, the last one is unmatched. y_n is only 1/2, 695 00:41:53,750 --> 00:41:59,790 only counts 1/2. 696 00:41:59,790 --> 00:42:17,590 So here is what's known as the trapezoidal rule. 697 00:42:17,590 --> 00:42:19,430 Now, I'd like to compare it for you 698 00:42:19,430 --> 00:42:25,480 to the Riemann sums, which are sitting just to the left here. 699 00:42:25,480 --> 00:42:28,900 Here's the left one, and here's the right one. 700 00:42:28,900 --> 00:42:32,570 If you take the average of the left and the right, 701 00:42:32,570 --> 00:42:34,960 that is, a half of this plus a half of that, 702 00:42:34,960 --> 00:42:36,370 there's an overlap. 703 00:42:36,370 --> 00:42:39,250 The y_1 through y_n things are listed in both. 704 00:42:39,250 --> 00:42:41,860 But the y_0 only gets counted 1/2 705 00:42:41,860 --> 00:42:44,170 and the y_n only gets counted 1/2. 706 00:42:44,170 --> 00:42:46,740 So what this is, is this is the symmetric compromise 707 00:42:46,740 --> 00:42:49,710 between the two Riemann sums. 708 00:42:49,710 --> 00:42:54,410 This is actually equal to the left Riemann 709 00:42:54,410 --> 00:43:02,020 sum plus the right Riemann sum divided by 2. 710 00:43:02,020 --> 00:43:10,600 It's the average of them. 711 00:43:10,600 --> 00:43:13,340 Now, this would be great and it does look like it's closer. 712 00:43:13,340 --> 00:43:16,500 But actually it's not as impressive as it looks. 713 00:43:16,500 --> 00:43:18,910 If you actually do it in practice, 714 00:43:18,910 --> 00:43:23,190 it's not very efficient. 715 00:43:23,190 --> 00:43:25,120 Although it's way better than a Riemann sum, 716 00:43:25,120 --> 00:43:27,560 it's still not good enough. 717 00:43:27,560 --> 00:43:37,250 So now I need to describe to you the fancier rule. 718 00:43:37,250 --> 00:43:52,010 Which is known as Simpson's Rule. 719 00:43:52,010 --> 00:44:00,160 And so, this is, if you like, 3, Method 3. 720 00:44:00,160 --> 00:44:09,770 The idea is again to divide things into chunks. 721 00:44:09,770 --> 00:44:16,690 But now it always needs n to be even. 722 00:44:16,690 --> 00:44:20,750 In other words, we're going to deal not with just one box, 723 00:44:20,750 --> 00:44:24,800 we're going to deal with pairs of boxes. 724 00:44:24,800 --> 00:44:28,370 Here's delta x, and here's delta x again. 725 00:44:28,370 --> 00:44:34,720 And we're going to study the area of this piece here. 726 00:44:34,720 --> 00:44:39,270 So let me focus just on that part. 727 00:44:39,270 --> 00:44:50,700 Let's reproduce it over here. 728 00:44:50,700 --> 00:44:52,684 And here's the delta x, here's delta x. 729 00:44:52,684 --> 00:44:54,350 And of course there are various heights. 730 00:44:54,350 --> 00:44:57,020 This starts out at y_0, this is y_2, 731 00:44:57,020 --> 00:45:03,490 and this middle segment is y_1. 732 00:45:03,490 --> 00:45:08,200 Now, the approximating curve that we're going to use 733 00:45:08,200 --> 00:45:14,690 is a parabola. 734 00:45:14,690 --> 00:45:17,400 That is, we're going to fit a parabola through these three 735 00:45:17,400 --> 00:45:25,240 points. 736 00:45:25,240 --> 00:45:28,500 And then we're going to use that as the approximating area. 737 00:45:28,500 --> 00:45:32,360 Now, it doesn't look like-- This looks like it misses. 738 00:45:32,360 --> 00:45:34,262 But actually, most functions mostly 739 00:45:34,262 --> 00:45:35,720 wiggle either one way or the other. 740 00:45:35,720 --> 00:45:36,490 They don't switch. 741 00:45:36,490 --> 00:45:38,800 They don't have inflection points. 742 00:45:38,800 --> 00:45:41,320 So, this is a lousy, at this scale. 743 00:45:41,320 --> 00:45:42,890 But when we get to a smaller scale, 744 00:45:42,890 --> 00:45:45,110 this becomes really fantastic. 745 00:45:45,110 --> 00:45:47,430 As an approximation. 746 00:45:47,430 --> 00:45:52,400 Now, I need to tell you what the arithmetic is. 747 00:45:52,400 --> 00:45:56,980 And in order to save time, it's on your problem 748 00:45:56,980 --> 00:46:00,080 set what the actual formula is. 749 00:46:00,080 --> 00:46:05,440 But I'm going to tell you how to think about it. 750 00:46:05,440 --> 00:46:09,530 I want you to think about it as follows. 751 00:46:09,530 --> 00:46:20,530 So the area under the parabola is 752 00:46:20,530 --> 00:46:30,800 going to be a base times some kind of average height. 753 00:46:30,800 --> 00:46:33,770 And the base here, you can already see. 754 00:46:33,770 --> 00:46:36,020 It's 2 delta x. 755 00:46:36,020 --> 00:46:39,430 The base is 2 delta x. 756 00:46:39,430 --> 00:46:40,920 Now, the average height is weird. 757 00:46:40,920 --> 00:46:43,070 You have to work out what it is for a parabola, 758 00:46:43,070 --> 00:46:46,420 depending on those three numbers, y_0, y_1, and y_2. 759 00:46:46,420 --> 00:46:48,760 And it turns out to be the following formula. 760 00:46:48,760 --> 00:46:50,440 It has to be an average, but it's 761 00:46:50,440 --> 00:46:51,910 an interesting weighted average. 762 00:46:51,910 --> 00:46:53,587 So this was the punchline, if you like. 763 00:46:53,587 --> 00:46:56,170 Is that they are such things as interesting weighted averages. 764 00:46:56,170 --> 00:46:58,680 This one's very simple, it just involves three numbers. 765 00:46:58,680 --> 00:46:59,980 But it's still interesting. 766 00:46:59,980 --> 00:47:00,771 It's the following. 767 00:47:00,771 --> 00:47:07,260 It turns out to be (y_0 + 4y_1 + y_2) / 6. 768 00:47:07,260 --> 00:47:08,160 Why divided by 6? 769 00:47:08,160 --> 00:47:10,730 Well, it's supposed to be an average. 770 00:47:10,730 --> 00:47:13,640 So the total is 1 + 4 + 1 of these things. 771 00:47:13,640 --> 00:47:16,090 And 6 is in the denominator. 772 00:47:16,090 --> 00:47:19,790 So it emphasizes the middle more than the sides. 773 00:47:19,790 --> 00:47:23,440 And that's what happens with a parabola. 774 00:47:23,440 --> 00:47:30,580 So this is a computation which is on your homework. 775 00:47:30,580 --> 00:47:34,440 And now we can put this together for the full Simpson's Rule 776 00:47:34,440 --> 00:47:36,190 formula. 777 00:47:36,190 --> 00:47:51,770 Which I'll put up over here. 778 00:47:51,770 --> 00:47:56,350 We have here 2 delta x, and we divide by 6. 779 00:47:56,350 --> 00:48:03,470 And then we have (y_0 + 4y_1 + y_2) / 6 plus-- 780 00:48:03,470 --> 00:48:05,390 That's the first chunk. 781 00:48:05,390 --> 00:48:09,130 Now, the second chunk, maybe I'll just put it in here, 782 00:48:09,130 --> 00:48:11,820 starts-- This is x_2. 783 00:48:11,820 --> 00:48:14,050 This is x_0. 784 00:48:14,050 --> 00:48:15,620 And it goes all the way to x_4. 785 00:48:15,620 --> 00:48:18,800 So x_2, x_3, x_4. 786 00:48:18,800 --> 00:48:21,920 So the next one involves the indices 2, 3 and 4. 787 00:48:21,920 --> 00:48:31,580 So this is (y_2 + 4y_3 + y_4) / 6. 788 00:48:31,580 --> 00:48:34,454 Oh, oh, oh, oh, no. 789 00:48:34,454 --> 00:48:35,870 I think I'll get rid of these 6's. 790 00:48:35,870 --> 00:48:37,990 I have too many 6's. 791 00:48:37,990 --> 00:48:39,850 Alright. 792 00:48:39,850 --> 00:48:41,057 Let's get rid of them here. 793 00:48:41,057 --> 00:48:41,890 Let's take them out. 794 00:48:41,890 --> 00:48:44,040 Put them out here. 795 00:48:44,040 --> 00:48:47,870 Thank you. 796 00:48:47,870 --> 00:48:51,650 All the way to the end, which is y_(n-2) plus 2y_(n-1)-- sorry, 797 00:48:51,650 --> 00:48:53,810 plus 4y_(n-1) plus y_n. 798 00:48:57,050 --> 00:49:01,580 I was about to divide by 6, but you saved me. 799 00:49:01,580 --> 00:49:04,350 So here are all the chunks. 800 00:49:04,350 --> 00:49:06,960 Now, what does this pattern come out to be? 801 00:49:06,960 --> 00:49:10,740 This comes out to be the following. 802 00:49:10,740 --> 00:49:16,950 This is 1, 4, 1, added to 1, 4, 1 added to 1, 4, 1. 803 00:49:16,950 --> 00:49:17,760 You add them up. 804 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, 805 00:49:22,140 --> 00:49:24,170 and a 2 and a 4 and a 1. 806 00:49:24,170 --> 00:49:29,250 So the pattern is that it starts out with 1's on the far ends. 807 00:49:29,250 --> 00:49:30,910 And then 4's next in. 808 00:49:30,910 --> 00:49:35,230 And then it alternates 2's and 4's in between. 809 00:49:35,230 --> 00:49:45,440 So the full pattern of Simpson's Rule is delta x / 3, 810 00:49:45,440 --> 00:49:48,720 I have now succeeded in canceling this 2 with this 6 811 00:49:48,720 --> 00:49:50,740 and getting out that factor of 2. 812 00:49:50,740 --> 00:50:00,966 And then here I have y0 + 4 y1 + 2 y2 + 4 y3 + ... 813 00:50:00,966 --> 00:50:03,520 It keeps on going and keeps on going and keeps on going. 814 00:50:03,520 --> 00:50:10,820 And in the end it's 2 y_(n - 2) + 4 y_(n - 1) + y_n. 815 00:50:10,820 --> 00:50:14,860 So again, 1 and a 4 to start. 816 00:50:14,860 --> 00:50:16,410 4 and a 1 to end. 817 00:50:16,410 --> 00:50:20,170 And then alternating 2's and 4's in the middle. 818 00:50:20,170 --> 00:50:25,650 And this weird weighted average is way better. 819 00:50:25,650 --> 00:50:28,890 As I will show you next time.