1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:03 The following content is provided under a Creative 3 00:00:03 --> 00:00:04 Commons License. 4 00:00:04 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:09 offer high quality educational resources for free. 6 00:00:09 --> 00:00:12 To make a donation or to view additional materials from 7 00:00:12 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:26 PROFESSOR: Today I want to get started by correcting a mistake 10 00:00:26 --> 00:00:30 that I made last time. 11 00:00:30 --> 00:00:37 And this was mistaken terminology. 12 00:00:37 --> 00:00:41 I said that what we were computing, when we computed 13 00:00:41 --> 00:00:47 in this candy bar example was energy and not heat. 14 00:00:47 --> 00:00:49 But it's both. 15 00:00:49 --> 00:00:51 They're the same thing. 16 00:00:51 --> 00:00:55 And in fact, energy, heat and work are all the 17 00:00:55 --> 00:00:59 same thing in physics. 18 00:00:59 --> 00:01:06 I was foolishly considering the much more, what am I trying to 19 00:01:06 --> 00:01:09 say, the intuitive feeling of heat is just being the 20 00:01:09 --> 00:01:10 same as temperature. 21 00:01:10 --> 00:01:18 But in physics, usually heat is measured in calories and energy 22 00:01:18 --> 00:01:20 can be in lots of things. 23 00:01:20 --> 00:01:26 Maybe kilowatt-hours or ergs, these are various of the units. 24 00:01:26 --> 00:01:31 And work would be in things like foot-pounds. 25 00:01:31 --> 00:01:35 That is, lifting some weight some distance. 26 00:01:35 --> 00:01:39 And the amount of force you have to apply. 27 00:01:39 --> 00:01:42 And these all have conversions between them. 28 00:01:42 --> 00:01:45 They're all the same quantity, in different units. 29 00:01:45 --> 00:01:54 OK, so these are the same quantity. 30 00:01:54 --> 00:02:00 Different units. 31 00:02:00 --> 00:02:08 So that's about as much physics as we'll do for today. 32 00:02:08 --> 00:02:11 And sorry about that. 33 00:02:11 --> 00:02:18 Now, the example that I was starting to discuss last time 34 00:02:18 --> 00:02:22 and then I'm going to carry out today was this 35 00:02:22 --> 00:02:32 dartboard example. 36 00:02:32 --> 00:02:40 We have a dartboard, which is some kind of target. 37 00:02:40 --> 00:02:47 And we have a person, your little brother, who's 38 00:02:47 --> 00:02:48 standing over there. 39 00:02:48 --> 00:02:50 And somebody is throwing darts. 40 00:02:50 --> 00:02:55 And the question is, how likely is he to be hit. 41 00:02:55 --> 00:03:00 So I want to describe to you how we're going to make this 42 00:03:00 --> 00:03:02 problem into a math problem. 43 00:03:02 --> 00:03:03 Yep. 44 00:03:03 --> 00:03:04 STUDENT: [INAUDIBLE] 45 00:03:04 --> 00:03:06 PROFESSOR: What topic is this that we're going over. 46 00:03:06 --> 00:03:08 We're going over an example. 47 00:03:08 --> 00:03:11 Which is a dartboard example. 48 00:03:11 --> 00:03:16 And it has to do with probability. 49 00:03:16 --> 00:03:17 OK. 50 00:03:17 --> 00:03:33 So what is the probability that this guy, your little 51 00:03:33 --> 00:03:47 brother, gets hit by a dart. 52 00:03:47 --> 00:03:51 Now, we have to put some assumptions into this 53 00:03:51 --> 00:03:53 problem in order to make it a math problem. 54 00:03:53 --> 00:03:56 And I'm really going to try to make them pretty 55 00:03:56 --> 00:03:58 realistic assumptions. 56 00:03:58 --> 00:04:09 So the first assumption is that the number of hits is 57 00:04:09 --> 00:04:24 proportional to ce ^ - r ^2. 58 00:04:24 --> 00:04:27 So that's actually a kind of a normal distribution. 59 00:04:27 --> 00:04:30 That's the bell curve. 60 00:04:30 --> 00:04:34 But as a function of the radius. 61 00:04:34 --> 00:04:37 So this is the assumption that I'm going to 62 00:04:37 --> 00:04:39 make in this problem. 63 00:04:39 --> 00:04:46 And a problem in probability is a problem of the ratio 64 00:04:46 --> 00:04:48 of the part to the whole. 65 00:04:48 --> 00:04:53 So the part is where this little guy is standing. 66 00:04:53 --> 00:04:57 And the whole is all the possible places where 67 00:04:57 --> 00:04:58 the dart might hit. 68 00:04:58 --> 00:05:00 Which is maybe the whole blackboard or extending beyond, 69 00:05:00 --> 00:05:05 depending on how good an aim you think that this 70 00:05:05 --> 00:05:07 older child has. 71 00:05:07 --> 00:05:11 So, if you like, the part is, we'll start simply. 72 00:05:11 --> 00:05:14 I mean, this doesn't sweep all the way around. 73 00:05:14 --> 00:05:17 But we're going to talk about some section. 74 00:05:17 --> 00:05:18 Like this. 75 00:05:18 --> 00:05:22 Where this is some radius r1, and this other radius is 76 00:05:22 --> 00:05:24 some longer radius, r2. 77 00:05:24 --> 00:05:28 And the part that we'll first keep track of is 78 00:05:28 --> 00:05:29 everything around there. 79 00:05:29 --> 00:05:32 That's not very well-centered, but it's supposed to be 80 00:05:32 --> 00:05:35 two concentric circles. 81 00:05:35 --> 00:05:38 Maybe I should fix a bit so that it's a little 82 00:05:38 --> 00:05:39 bit easier to read here. 83 00:05:39 --> 00:05:41 So here we go. 84 00:05:41 --> 00:05:46 So here I have radius r1, and here I have radius r2. 85 00:05:46 --> 00:05:49 And then the region in between is what we're going to 86 00:05:49 --> 00:05:56 try to keep track of. 87 00:05:56 --> 00:05:59 So I claim that we'll be able to get, so this is what I'm 88 00:05:59 --> 00:06:04 calling a part, to start out with. 89 00:06:04 --> 00:06:08 And then we'll take a section of it to get the place where 90 00:06:08 --> 00:06:12 the person is standing. 91 00:06:12 --> 00:06:22 Now, I want to take a side view of e ^ - r ^2. 92 00:06:22 --> 00:06:23 The function that we're talking about. 93 00:06:23 --> 00:06:28 Again, that's the bell curve. 94 00:06:28 --> 00:06:31 And it sort of looks like this. 95 00:06:31 --> 00:06:36 This is the top value, this is now the r axis. 96 00:06:36 --> 00:06:40 And this is up, or at least, so you should think of this in 97 00:06:40 --> 00:06:43 terms of the fact that the horizontal here is the 98 00:06:43 --> 00:06:46 plane of the dartboard. 99 00:06:46 --> 00:06:50 And the vertical is measuring how likely it is that there 100 00:06:50 --> 00:06:52 will be darts piling up here. 101 00:06:52 --> 00:06:56 So if they were balls tumbling down or something else 102 00:06:56 --> 00:06:58 falling on, many of them would pile up here. 103 00:06:58 --> 00:07:02 Many fewer of them would be piling up farther away. 104 00:07:02 --> 00:07:06 And the chunk that we're keeping track of is the 105 00:07:06 --> 00:07:09 chunk between r1 and r2. 106 00:07:09 --> 00:07:11 That's the corresponding region here. 107 00:07:11 --> 00:07:15 And in order to calculate this part, we have to calculate 108 00:07:15 --> 00:07:20 this volume of revolution. 109 00:07:20 --> 00:07:21 Sweeping around. 110 00:07:21 --> 00:07:24 Because really, in disguise, this is a ring. 111 00:07:24 --> 00:07:26 This is a side view, it's really a ring. 112 00:07:26 --> 00:07:33 Because we're rotating it around this axis here. 113 00:07:33 --> 00:07:35 So we're trying to figure out what the total 114 00:07:35 --> 00:07:36 volume of that ring is. 115 00:07:36 --> 00:07:40 And that's going to be our weighting, our likelihood for 116 00:07:40 --> 00:07:44 whether the hits are occurring in this section or in this 117 00:07:44 --> 00:07:51 ring versus the rest. 118 00:07:51 --> 00:07:54 To set this up, I remind you we're going to use 119 00:07:54 --> 00:07:57 the method of shells. 120 00:07:57 --> 00:08:03 That's really the only one that's going to work here. 121 00:08:03 --> 00:08:07 And we want to integrate between r1 and r2. 122 00:08:07 --> 00:08:12 And the range here is that because this is, if you like, 123 00:08:12 --> 00:08:13 a solid of revolutions. 124 00:08:13 --> 00:08:17 So the variable r is the same as what we used to call x. 125 00:08:17 --> 00:08:19 And it's ranging between r1 and r2, and then we're 126 00:08:19 --> 00:08:20 sweeping it around. 127 00:08:20 --> 00:08:26 And the circumference of a little piece, so at a 128 00:08:26 --> 00:08:35 fixed distance r, here, the circumference is 129 00:08:35 --> 00:08:37 going to be 2 pi r. 130 00:08:37 --> 00:08:40 So 2 pi r is the circumference. 131 00:08:40 --> 00:08:41 And then the height is the height of that 132 00:08:41 --> 00:08:42 green stem there. 133 00:08:42 --> 00:08:45 That's e ^ - r ^2. 134 00:08:45 --> 00:08:50 And then we're multiplying by the thickness, which is dr. So 135 00:08:50 --> 00:08:53 the thickness of the green is dr. The height of this little 136 00:08:53 --> 00:09:00 green guy is e ^ - r^2, and the circumference is e pi times the 137 00:09:00 --> 00:09:03 radius, when we sweep the circle around. 138 00:09:03 --> 00:09:04 Question. 139 00:09:04 --> 00:09:21 STUDENT: [INAUDIBLE] 140 00:09:21 --> 00:09:25 PROFESSOR: So the question is, why is we're interested 141 00:09:25 --> 00:09:28 in not this pink area. 142 00:09:28 --> 00:09:32 And the reason is an interpretation of what 143 00:09:32 --> 00:09:33 I meant by this. 144 00:09:33 --> 00:09:38 What I meant by this is that if you wanted to add up what the 145 00:09:38 --> 00:09:42 likelihood is that this thing will be here versus here, I 146 00:09:42 --> 00:09:45 want it to be, really, the proportions are the number 147 00:09:45 --> 00:09:49 of hits times the, if you like, I wanted it d area. 148 00:09:49 --> 00:09:51 That's really what I meant here. 149 00:09:51 --> 00:09:55 The number of hits in a little chunk. 150 00:09:56 --> 00:09:59 So little, maybe I'll call it delta a. 151 00:09:59 --> 00:10:01 A little chuck. 152 00:10:01 --> 00:10:04 Is proportional to the chunk times that. 153 00:10:04 --> 00:10:05 So there's already an area factor. 154 00:10:05 --> 00:10:06 And there's a height. 155 00:10:06 --> 00:10:09 So there are a total of three dimensions involved. 156 00:10:09 --> 00:10:13 There's the area and then this height. 157 00:10:13 --> 00:10:19 So it's a matter of the, what I was given, what I intended 158 00:10:19 --> 00:10:21 to say the problem is. 159 00:10:21 --> 00:10:22 Yeah, another question. 160 00:10:22 --> 00:10:25 STUDENT: [INAUDIBLE] 161 00:10:25 --> 00:10:26 PROFESSOR: Yeah, yeah. 162 00:10:26 --> 00:10:27 Exactly. 163 00:10:27 --> 00:10:30 The height is c times that, whatever this c is. 164 00:10:30 --> 00:10:32 In fact, we don't know what the c is, but because we're going 165 00:10:32 --> 00:10:35 to have a part and a whole, we'll divide, the c 166 00:10:35 --> 00:10:37 will always cancel. 167 00:10:37 --> 00:10:39 So I'm throwing the c out. 168 00:10:39 --> 00:10:42 I don't know what it is, and in the end it won't matter. 169 00:10:42 --> 00:10:44 That's a very important question. 170 00:10:44 --> 00:10:44 Yes. 171 00:10:44 --> 00:10:49 STUDENT: [INAUDIBLE] 172 00:10:49 --> 00:10:50 PROFESSOR: Say it again? 173 00:10:50 --> 00:11:02 STUDENT: PROFESSOR: 174 00:11:02 --> 00:11:03 PROFESSOR: So what I mean is the number of 175 00:11:03 --> 00:11:05 hits in some chunk. 176 00:11:05 --> 00:11:10 That is, suppose you imagine, the question is, what does 177 00:11:10 --> 00:11:11 this left-hand side mean. 178 00:11:11 --> 00:11:12 That right? 179 00:11:12 --> 00:11:17 Is that the question that's you're asking? 180 00:11:17 --> 00:11:21 When I try to understand what the distribution of 181 00:11:21 --> 00:11:25 dartboard hits is, I should imagine my dartboard. 182 00:11:25 --> 00:11:28 And very often there'll be a whole bunch of 183 00:11:28 --> 00:11:29 holes in some places. 184 00:11:29 --> 00:11:30 And fewer holes else. 185 00:11:30 --> 00:11:33 I'm trying to figure out what the whole distribution 186 00:11:33 --> 00:11:35 of those marks is. 187 00:11:35 --> 00:11:38 And so some places will have more hits on them 188 00:11:38 --> 00:11:41 and some places will have fewer hits on them. 189 00:11:41 --> 00:11:43 And so what I want to measure is, on average, 190 00:11:43 --> 00:11:44 the number of hits. 191 00:11:44 --> 00:11:47 So this would really be some, this constant of 192 00:11:47 --> 00:11:49 proportionality is ambiguous because it depends on 193 00:11:49 --> 00:11:50 how many times you try. 194 00:11:50 --> 00:11:52 If you throw a thousand times, it'll be much 195 00:11:52 --> 00:11:54 more densely packed. 196 00:11:54 --> 00:11:58 And if you have only a hundred times it'll be fewer. 197 00:11:58 --> 00:12:01 So that's where this constant comes in. 198 00:12:01 --> 00:12:04 But given that you have a certain number of times that 199 00:12:04 --> 00:12:07 you tried, say, a thousand times, there will be a whole 200 00:12:07 --> 00:12:09 bunch more piled in the middle. 201 00:12:09 --> 00:12:12 And fewer and fewer as you get farther away. 202 00:12:12 --> 00:12:15 Assuming that the person's aim is reasonable. 203 00:12:15 --> 00:12:17 So that's what we're saying. 204 00:12:17 --> 00:12:19 So we're thinking in terms of the person's always 205 00:12:19 --> 00:12:20 aiming for the center. 206 00:12:20 --> 00:12:22 So it's most likely that the person will hit the center. 207 00:12:22 --> 00:12:25 But on the other hand, it's a fallible person, so 208 00:12:25 --> 00:12:26 the person may miss. 209 00:12:26 --> 00:12:29 And so it's less and less likely as you go farther out. 210 00:12:29 --> 00:12:31 And we're just counting how many times this gets hit, how 211 00:12:31 --> 00:12:33 many time this and so on. 212 00:12:33 --> 00:12:34 In proportion to the area. 213 00:12:34 --> 00:12:40 STUDENT: [INAUDIBLE] 214 00:12:40 --> 00:12:42 PROFESSOR: Yeah. r1 and r2 are arbitrary. 215 00:12:42 --> 00:12:44 We're going to make this calculation in general. 216 00:12:44 --> 00:12:46 We're going to calculate what the likelihood is that we 217 00:12:46 --> 00:12:50 hit any possible band. 218 00:12:50 --> 00:12:53 And I want to leave those as just letters for now. 219 00:12:53 --> 00:12:54 The r1 and the r2. 220 00:12:54 --> 00:12:57 Because I want to be able to try various different 221 00:12:57 --> 00:12:57 possibilities. 222 00:12:57 --> 00:13:03 STUDENT: [INAUDIBLE] 223 00:13:03 --> 00:13:06 PROFESSOR: Say it again., why do we have to take volume. 224 00:13:06 --> 00:13:08 So this is what we were addressing before. 225 00:13:08 --> 00:13:13 It's a volume because it's number of hit per unit area. 226 00:13:13 --> 00:13:17 So there's a height, that is, number of hits. 227 00:13:17 --> 00:13:20 And then there's an area and the product of those is, so 228 00:13:20 --> 00:13:23 this is, if you like, a histogram of the 229 00:13:23 --> 00:13:25 number of hits. 230 00:13:25 --> 00:13:26 But this should be measured per area. 231 00:13:26 --> 00:13:29 Not per length of r. 232 00:13:29 --> 00:13:32 Because on the real diagram, it's going all the way around. 233 00:13:32 --> 00:13:36 There's a lot more area to this red band than just 234 00:13:36 --> 00:13:41 the distance implies. 235 00:13:41 --> 00:13:42 OK. 236 00:13:42 --> 00:13:48 So, having discussed the setup, this is a pretty standard setup 237 00:13:48 --> 00:13:49 -- oh, one more question. 238 00:13:49 --> 00:13:49 Yes. 239 00:13:49 --> 00:14:13 STUDENT: [INAUDIBLE] 240 00:14:13 --> 00:14:14 PROFESSOR: Yeah. 241 00:14:14 --> 00:14:17 So the question is, why is this a realistic. 242 00:14:17 --> 00:14:21 Why is this choice of function here e to the minus r squared 243 00:14:21 --> 00:14:25 a realistic choice of function for the darts. 244 00:14:25 --> 00:14:30 So I can answer this with an analogy. 245 00:14:30 --> 00:14:34 When people were asking themselves where the V-2 246 00:14:34 --> 00:14:39 rockets from Germany hit London, they used this model. 247 00:14:39 --> 00:14:43 It turned out to be the one which was the most accurate. 248 00:14:43 --> 00:14:48 So that gives you an idea that this is actually real. 249 00:14:48 --> 00:14:53 The question, this makes it look like people are masters. 250 00:14:53 --> 00:14:57 That is, that they'll all hit in the center more 251 00:14:57 --> 00:14:58 often than elsewhere. 252 00:14:58 --> 00:15:03 But that's actually somewhat deceptive. 253 00:15:03 --> 00:15:08 There's a difference between the mode, that is, the most 254 00:15:08 --> 00:15:13 likely spot, and what happens on average. 255 00:15:13 --> 00:15:17 So in other words, the single most likely spot is the center. 256 00:15:17 --> 00:15:20 But there's rather little area in here, and in fact the 257 00:15:20 --> 00:15:24 likelihood of hitting that is some little tiny bit. 258 00:15:24 --> 00:15:25 In here. 259 00:15:25 --> 00:15:29 In fact, you're much more likely to be out here. 260 00:15:29 --> 00:15:32 So if you take the total of the volume, you'll see that much of 261 00:15:32 --> 00:15:34 the volume is contributed from out here. 262 00:15:34 --> 00:15:37 And in fact, the person hits rather rarely near the center. 263 00:15:37 --> 00:15:42 So this is not a ridiculous thing to do. 264 00:15:42 --> 00:15:45 If you think of it in terms of somebody's aiming at the center 265 00:15:45 --> 00:15:47 but there's some random thing which is throwing the person 266 00:15:47 --> 00:15:51 off, then there is likely to be to left or to the right, or 267 00:15:51 --> 00:15:54 they might even get lucky and all those errors cancel 268 00:15:54 --> 00:15:56 themselves and they happen to hit pretty much 269 00:15:56 --> 00:15:59 near the center. 270 00:15:59 --> 00:16:00 Yeah, another question. 271 00:16:00 --> 00:16:05 STUDENT: [INAUDIBLE] 272 00:16:05 --> 00:16:06 PROFESSOR: How does the little brother come into play? 273 00:16:06 --> 00:16:09 The little brother is going to come into play as follows. 274 00:16:09 --> 00:16:11 I'll tell you in advance. 275 00:16:11 --> 00:16:15 So the thing is, the little brother was not so stupid as to 276 00:16:15 --> 00:16:19 stand in front of the target. 277 00:16:19 --> 00:16:22 I know. 278 00:16:22 --> 00:16:26 He stood about twice the radius of the target away. 279 00:16:26 --> 00:16:29 And so, we're going to approximate the location 280 00:16:29 --> 00:16:32 by some sector here. 281 00:16:32 --> 00:16:35 Which is just going to be some chunk of one of these things. 282 00:16:35 --> 00:16:38 We'll just break off a piece of it. 283 00:16:38 --> 00:16:39 And that's how we're going to capture. 284 00:16:39 --> 00:16:42 So the point is, the target is here. 285 00:16:42 --> 00:16:46 But there is the possibility that the seven-year-old who's 286 00:16:46 --> 00:16:48 throwing the darts actually missed the target. 287 00:16:48 --> 00:16:50 That actually happens a lot. 288 00:16:50 --> 00:16:54 So, does that answer your question? 289 00:16:54 --> 00:16:57 Alright. are we ready now to to do? 290 00:16:57 --> 00:16:57 One more question. 291 00:16:57 --> 00:17:05 STUDENT: [INAUDIBLE] 292 00:17:05 --> 00:17:08 PROFESSOR: I'm giving you this property here. 293 00:17:08 --> 00:17:11 I'm telling you that this is what's called a mathematical 294 00:17:11 --> 00:17:13 model, when you give somebody something like this. 295 00:17:13 --> 00:17:16 In fact, that requires further justification. 296 00:17:16 --> 00:17:19 It's an interesting issue. 297 00:17:19 --> 00:17:19 Yeah. 298 00:17:19 --> 00:17:26 STUDENT: [INAUDIBLE] 299 00:17:26 --> 00:17:27 PROFESSOR: I'm giving it to you for now. 300 00:17:27 --> 00:17:30 And it's something which really has to be justified. 301 00:17:30 --> 00:17:32 In certain circumstances it is justified. 302 00:17:32 --> 00:17:37 But, OK. 303 00:17:37 --> 00:17:43 So anyway, here's our part. 304 00:17:43 --> 00:17:48 This is going to be our chunk, for now, that we're going to 305 00:17:48 --> 00:17:50 estimate the the importance. 306 00:17:50 --> 00:17:56 The relative importance of. 307 00:17:56 --> 00:18:00 And now, this is something whose antiderivative we can 308 00:18:00 --> 00:18:03 just do by substitution or by guessing. 309 00:18:03 --> 00:18:07 It's just - pi e ^ - r^2. 310 00:18:07 --> 00:18:11 If you differentiate that, you get a - 2r, which cancels 311 00:18:11 --> 00:18:12 the minus sign here. 312 00:18:12 --> 00:18:15 So you get 2pi re ^ - r ^2. 313 00:18:15 --> 00:18:18 So that's the antiderivative. 314 00:18:18 --> 00:18:24 And we're evaluating it at r1 and r2. 315 00:18:24 --> 00:18:28 So with the minus sign that's going to get reversed. 316 00:18:28 --> 00:18:34 The answer is going to be pi ( e ^ - r1^2, that's the bottom 317 00:18:34 --> 00:18:39 one. - e ^ - r2 ^2), that's the top. 318 00:18:39 --> 00:18:45 So this is what our part gives us. 319 00:18:45 --> 00:18:48 And, more technically, if you wanted to multiply through by 320 00:18:48 --> 00:18:50 c, it would be c times this. 321 00:18:50 --> 00:18:55 I'll say that in just a second. 322 00:18:55 --> 00:18:58 OK, now I want to work. 323 00:18:58 --> 00:19:04 So, if you like, the part is equal to maybe even I'll call 324 00:19:04 --> 00:19:12 it c pi (e^ -r1 ^2 - e ^ - r2 ^2). 325 00:19:12 --> 00:19:14 That's what it really is if I put in this factor. 326 00:19:14 --> 00:19:17 So now there's no prejudice as to how many attempts we make. 327 00:19:17 --> 00:19:20 Whether it was a thousand attempts or a million 328 00:19:20 --> 00:19:22 attempts at the target. 329 00:19:22 --> 00:19:26 Now, the most important second feature here of these kinds of 330 00:19:26 --> 00:19:29 modeling problems is, there is always some kind 331 00:19:29 --> 00:19:30 of idealization. 332 00:19:30 --> 00:19:33 And the next thing that I want to discuss with you is the 333 00:19:33 --> 00:19:36 interpretation of the whole. 334 00:19:36 --> 00:19:41 That is, what's the family of all possibilities. 335 00:19:41 --> 00:19:47 And in this case, what I'm going to claim is that the 336 00:19:47 --> 00:19:52 reasonable way to think of the whole is it's that r can range 337 00:19:52 --> 00:19:55 all the way from 0 to infinity. 338 00:19:55 --> 00:19:57 Now, you may not like this. 339 00:19:57 --> 00:20:01 But these are maybe my first and third favorite number, my 340 00:20:01 --> 00:20:03 second favorite number being 1. 341 00:20:03 --> 00:20:07 So infinity is a really useful concept. 342 00:20:07 --> 00:20:12 Of course, it's nonsense in the context of the darts. 343 00:20:12 --> 00:20:16 Because if you think of the basement wall where the kid 344 00:20:16 --> 00:20:20 might miss a target, he'd probably hit the wall. 345 00:20:20 --> 00:20:22 He's probably not going to hit one of the walls to the right, 346 00:20:22 --> 00:20:25 and anyway he's certainly not going to hit over there. 347 00:20:25 --> 00:20:29 So there's something artificial about sending the possibility 348 00:20:29 --> 00:20:32 of hits all the way out to infinity. 349 00:20:32 --> 00:20:37 On the other hand, the shape of this curve is such that the 350 00:20:37 --> 00:20:39 real tail ends here, because of this exponential 351 00:20:39 --> 00:20:40 decrease, are tiny. 352 00:20:40 --> 00:20:42 And that's negligible. 353 00:20:42 --> 00:20:46 And the point is that actually the value, if you go all the 354 00:20:46 --> 00:20:50 way out to infinity, is the easiest value to calculate. 355 00:20:50 --> 00:20:52 So by doing this, I'm idealizing the problem but I'm 356 00:20:52 --> 00:20:55 actually making the numbers come out much more cleanly. 357 00:20:55 --> 00:20:58 And this is just always done in mathematics. 358 00:20:58 --> 00:21:00 That's what we did when we went from differences to 359 00:21:00 --> 00:21:02 differentials, to differentiation and 360 00:21:02 --> 00:21:04 infinitesimals. 361 00:21:04 --> 00:21:06 So we like that. 362 00:21:06 --> 00:21:08 Because it makes things easier, not because it 363 00:21:08 --> 00:21:10 makes things harder. 364 00:21:10 --> 00:21:12 So we're just going to pretend the whole is 365 00:21:12 --> 00:21:13 from 0 to infinity. 366 00:21:13 --> 00:21:15 And now let's just see what it is. 367 00:21:15 --> 00:21:21 It's c pi times the starting place is e^ - 0 ^2. 368 00:21:21 --> 00:21:24 That's r1, right, this is the role that r1 plays is this, 369 00:21:24 --> 00:21:26 and the r2 is this value. 370 00:21:26 --> 00:21:30 Minus e ^ - infinity ^2. 371 00:21:30 --> 00:21:33 Which is that negligibly small number, 0. 372 00:21:33 --> 00:21:36 So this is just c pi. 373 00:21:36 --> 00:21:39 Because this number is 1, and this other number is 0. 374 00:21:39 --> 00:21:48 This is just (1 - 0), in the parentheses there. 375 00:21:48 --> 00:21:52 And now I can tell you from these two numbers what 376 00:21:52 --> 00:21:54 the probability is. 377 00:21:54 --> 00:21:59 The probability that we landed on the target in a radius 378 00:21:59 --> 00:22:05 between r1 and r2, so that's this annulus here, is the ratio 379 00:22:05 --> 00:22:12 of the part to the whole. 380 00:22:12 --> 00:22:16 Which in this case just cancels the c pi. 381 00:22:16 --> 00:22:21 So it's (e ^ - r1 ^2 - e ^ - r2^2). 382 00:22:21 --> 00:22:25 There's the formula for the probability. 383 00:22:25 --> 00:22:28 So the c canceled and the pi canceled. 384 00:22:28 --> 00:22:32 It's all gone. 385 00:22:32 --> 00:22:34 Now, again, let me just emphasize the way this 386 00:22:34 --> 00:22:35 formula is supposed to work. 387 00:22:35 --> 00:22:43 The total probability of every possibility here is supposed 388 00:22:43 --> 00:22:45 to be set up to be 1. 389 00:22:45 --> 00:22:48 This is some fraction of 1. 390 00:22:48 --> 00:22:51 If you like, it's a percent. 391 00:22:51 --> 00:22:51 Yes. 392 00:22:51 --> 00:22:57 STUDENT: [INAUDIBLE] 393 00:22:57 --> 00:22:59 PROFESSOR: This one is giving, the question is, doesn't this 394 00:22:59 --> 00:23:00 just give the probability of the ring? 395 00:23:00 --> 00:23:04 This gives you the probability of the ring, but this is 396 00:23:04 --> 00:23:05 a very, very wide ring. 397 00:23:05 --> 00:23:08 This is a ring starting with 0, nothing, on the inside. 398 00:23:08 --> 00:23:09 And then going all the way out. 399 00:23:09 --> 00:23:11 So that's everything. 400 00:23:11 --> 00:23:13 So this corresponds to everything. 401 00:23:13 --> 00:23:22 This corresponds to a ring. 402 00:23:22 --> 00:23:24 So now, let's see. 403 00:23:24 --> 00:23:26 Where do I want to go from here. 404 00:23:26 --> 00:23:29 So in order to make progress here, I still have to give you 405 00:23:29 --> 00:23:31 one more piece of information. 406 00:23:31 --> 00:23:35 And this is, again, supposed to be realistic. 407 00:23:35 --> 00:23:40 When I was three years old and my brother's friend Ralph 408 00:23:40 --> 00:23:43 was seven, I watched him throwing darts a lot. 409 00:23:43 --> 00:23:54 And I would say that for Ralph, so for Ralph, at age seven, 410 00:23:54 --> 00:23:57 anyway, later on he got a little better at it. 411 00:23:57 --> 00:24:00 But Ralph at age seven, the probability that he hit 412 00:24:00 --> 00:24:08 the target was about 1/2. 413 00:24:08 --> 00:24:09 Right? 414 00:24:09 --> 00:24:12 So he hit the target about half the time. 415 00:24:12 --> 00:24:15 And the other times, there was cement on the walls of the 416 00:24:15 --> 00:24:17 basement, it wasn't that bad. 417 00:24:17 --> 00:24:19 Just bounced off. 418 00:24:19 --> 00:24:21 That also meant that the points got a little 419 00:24:21 --> 00:24:22 blunter as time went on. 420 00:24:22 --> 00:24:26 So it was a little less dangerous when they hit. 421 00:24:26 --> 00:24:26 Alright. 422 00:24:26 --> 00:24:33 So now, so here's the extra assumption that I want to make. 423 00:24:33 --> 00:24:41 So a is going to be the radius of the target. 424 00:24:41 --> 00:24:45 Now, the other realistic assumption that I want to 425 00:24:45 --> 00:24:49 make is where this little kid would be standing. 426 00:24:49 --> 00:24:52 And now, here, I want to get very specific and just do the 427 00:24:52 --> 00:24:54 competition in one case. 428 00:24:54 --> 00:25:00 We're going to imagine the target is here. 429 00:25:00 --> 00:25:02 And the kid is standing, say, between, so we'll 430 00:25:02 --> 00:25:04 just do a section of this. 431 00:25:04 --> 00:25:13 This is between 3 o'clock and 5 o'clock. 432 00:25:13 --> 00:25:15 There's more of him, but it's lower down and 433 00:25:15 --> 00:25:17 maybe negligible here. 434 00:25:17 --> 00:25:20 So this section is the part, the chunk that 435 00:25:20 --> 00:25:23 we want to see about. 436 00:25:23 --> 00:25:31 And this is a, and then this distance here is 2a. 437 00:25:31 --> 00:25:35 And then the longest distance here is 3a. 438 00:25:35 --> 00:25:39 So the longest distance is 3a. 439 00:25:39 --> 00:25:45 So in other words, what I'm saying is that the probability, 440 00:25:45 --> 00:25:58 if you're standing too close, the chance Ralph hits younger 441 00:25:58 --> 00:26:06 brother is about 1/6, right? 442 00:26:06 --> 00:26:12 Because 2/12 = 1/6. 443 00:26:12 --> 00:26:22 1/6 of the probability that we're between a and 3a. 444 00:26:22 --> 00:26:29 That's the number that we're looking for. 445 00:26:29 --> 00:26:29 Another question. 446 00:26:29 --> 00:26:33 STUDENT: [INAUDIBLE] 447 00:26:33 --> 00:26:36 PROFESSOR: The 2/12 came from the fact that we assumed. 448 00:26:36 --> 00:26:39 So we made a very, very bold assumption here. 449 00:26:39 --> 00:26:42 We assumed that this human being, who is actually 450 00:26:42 --> 00:26:46 standing, the floor is about down here. 451 00:26:46 --> 00:26:48 Maybe he wasn't that tall. 452 00:26:48 --> 00:26:51 But anyway, so he's really a little bigger than this. 453 00:26:51 --> 00:26:54 That the part of him that was close to the target covered 454 00:26:54 --> 00:27:00 about this section here, between radius 2a and 3a. 455 00:27:00 --> 00:27:03 As you'll see, actually from the computation, because the 456 00:27:03 --> 00:27:08 likelihood drops off pretty quickly, whatever of him was 457 00:27:08 --> 00:27:11 standing outside there wouldn't have mattered anyway. 458 00:27:11 --> 00:27:13 So we're just worried about the part that's closest 459 00:27:13 --> 00:27:19 to the target here. 460 00:27:19 --> 00:27:19 STUDENT: [INAUDIBLE] 461 00:27:19 --> 00:27:20 PROFESSOR: Why is it out of 12? 462 00:27:20 --> 00:27:22 Because I made it a clock. 463 00:27:22 --> 00:27:25 And I made it from 3 o'clock to 5 o'clock, so it's 2 of 464 00:27:25 --> 00:27:28 the 12 hours of a clock. 465 00:27:28 --> 00:27:31 It's just a way of me making a section that 466 00:27:31 --> 00:27:36 you can visibly see. 467 00:27:36 --> 00:27:40 So now, so here's what we're trying to calculate. 468 00:27:40 --> 00:27:43 And in order to figure this out, I need one 469 00:27:43 --> 00:27:47 more item. here. 470 00:27:47 --> 00:27:49 So maybe I'll leave myself a little bit of room. 471 00:27:49 --> 00:27:52 I have to figure out something about what our 472 00:27:52 --> 00:27:53 information gave us. 473 00:27:53 --> 00:27:59 Which is that the probability, sorry, this 474 00:27:59 --> 00:28:01 probability was 1/2. 475 00:28:01 --> 00:28:03 So let's remember what this is. 476 00:28:03 --> 00:28:09 This is going to be e ^ - 0 ^2 - e ^ - a ^2. 477 00:28:09 --> 00:28:11 That's what's this probability is. 478 00:28:11 --> 00:28:16 And that's equal to 1/2. 479 00:28:16 --> 00:28:21 So that means that 1 - e ^ - a ^2 = 1/2. 480 00:28:21 --> 00:28:26 Which means that e ^ - a ^2 = 1/2. 481 00:28:26 --> 00:28:29 I'm not going to calculate a, this is the part about the 482 00:28:29 --> 00:28:32 information about a that I want to use. 483 00:28:32 --> 00:28:33 That's what I'll use. 484 00:28:33 --> 00:28:36 And now I'm going to calculate this other probability here. 485 00:28:36 --> 00:28:41 So the probability right up there is this. 486 00:28:41 --> 00:28:50 And that's going to be the same as e ^ - (2a)^2 - e ^ - (3a)^2. 487 00:28:51 --> 00:28:54 That's what we calculated. 488 00:28:54 --> 00:28:59 And now I want to use some of the arithmetic of exponents. 489 00:28:59 --> 00:29:01 This is (e ^ - a ^2)^4. 490 00:29:01 --> 00:29:05 491 00:29:05 --> 00:29:10 Because it's really (2a)^2 = 4. 492 00:29:10 --> 00:29:15 The quantity is 4a^2, and then I bring that exponent out. - 493 00:29:15 --> 00:29:25 (e ^ - a ^2) ^ 9 that's 3 ^2. 494 00:29:25 --> 00:29:31 And so this comes out to be (1/2) ^ 4 - (1/2) ^9. 495 00:29:31 --> 00:29:36 496 00:29:36 --> 00:29:39 Which is approximately 1/16. 497 00:29:39 --> 00:29:40 This is negligible. 498 00:29:40 --> 00:29:41 This part here. 499 00:29:41 --> 00:29:43 And this is actually why these tails, as you go out to 500 00:29:43 --> 00:29:48 infinity, don't really matter that much. 501 00:29:48 --> 00:29:49 So this is a much smaller number. 502 00:29:49 --> 00:29:53 So the probability of the whole band is 1/16. 503 00:29:53 --> 00:29:55 And now I can answer the question up here. 504 00:29:55 --> 00:30:01 This is approximately 1/6 * 1/16. 505 00:30:01 --> 00:30:07 Which is about 1/100, or about 1%. 506 00:30:07 --> 00:30:14 So if I stood there for 100 attempts, then my chances of 507 00:30:14 --> 00:30:18 getting hit were pretty high. 508 00:30:18 --> 00:30:27 So that's the computation. 509 00:30:27 --> 00:30:34 That's a typical example of a problem in probability. 510 00:30:34 --> 00:30:37 And let me just make one more connection with 511 00:30:37 --> 00:30:42 what we did before. 512 00:30:42 --> 00:30:46 This is connected to weighted averages or integrals 513 00:30:46 --> 00:30:47 over weights. 514 00:30:47 --> 00:30:58 But the weight that's involved in this problem was w ( r) =, 515 00:30:58 --> 00:31:00 so let's just look at what happened in all of 516 00:31:00 --> 00:31:01 those integrals. 517 00:31:01 --> 00:31:04 What happened in all the integrals was, we had 518 00:31:04 --> 00:31:08 this factor here. 519 00:31:08 --> 00:31:09 2 pi r. 520 00:31:09 --> 00:31:23 And if I include this c, it was really 2 pi c r e ^ - r ^2. 521 00:31:23 --> 00:31:27 This was the weight that we were using. 522 00:31:27 --> 00:31:30 The relative importance of things. 523 00:31:30 --> 00:31:33 Now, this is not the same as e ^ - r ^2 that 524 00:31:33 --> 00:31:34 we started out with. 525 00:31:34 --> 00:31:38 Because this is the one-dimensional histogram. 526 00:31:38 --> 00:31:40 And that has to do with the method of shells that gave us 527 00:31:40 --> 00:31:43 that extra factor of r here. 528 00:31:43 --> 00:31:46 So that also connects with the question at the beginning. 529 00:31:46 --> 00:31:49 Which had to do with this paradox, and it looks like 530 00:31:49 --> 00:31:51 these places in the middle are the most likely. 531 00:31:51 --> 00:31:54 But that's the plot of e ^ - r ^2. 532 00:31:54 --> 00:31:58 If you actually look at this plot here, you see that as r 533 00:31:58 --> 00:32:00 goes to 0, it's going to 0. 534 00:32:00 --> 00:32:03 This is a different graph here. 535 00:32:03 --> 00:32:07 And actually, so this is what's happening really. 536 00:32:07 --> 00:32:10 In terms of how likely it is that you'll get within a 537 00:32:10 --> 00:32:18 certain distance of the center of the target. 538 00:32:18 --> 00:32:20 Again, it's not the area under that curve that we're doing. 539 00:32:20 --> 00:32:27 It's that volume of revolution. 540 00:32:27 --> 00:32:29 We're going to change subjects now. 541 00:32:29 --> 00:32:30 OK, one more question. 542 00:32:30 --> 00:32:32 Yes. 543 00:32:32 --> 00:32:32 STUDENT: [INAUDIBLE] 544 00:32:32 --> 00:32:35 PROFESSOR: Yeah, that's supposed to be the 545 00:32:35 --> 00:32:36 graph of w (r). 546 00:32:36 --> 00:32:47 STUDENT: [INAUDIBLE] 547 00:32:47 --> 00:32:49 PROFESSOR: Well, so, the question is, wouldn't 548 00:32:49 --> 00:32:50 the importance of the center be greatest? 549 00:32:50 --> 00:32:53 It's a question of which variable you're using. 550 00:32:53 --> 00:32:57 According according to pure radius, it's not. 551 00:32:57 --> 00:32:59 It turns out that there are some bands in radius which are 552 00:32:59 --> 00:33:06 more important, more likely for hits than others. 553 00:33:06 --> 00:33:08 It really has to do with the fact that the center, or the 554 00:33:08 --> 00:33:10 core, of the target is really tiny. 555 00:33:10 --> 00:33:11 So it's harder to hit. 556 00:33:11 --> 00:33:16 Whereas a whole band around the outside has a lot more area. 557 00:33:16 --> 00:33:18 Many, many ways to hit that band. 558 00:33:18 --> 00:33:19 So it's a much larger area. 559 00:33:19 --> 00:33:24 So there's a competition there between those two things. 560 00:33:24 --> 00:33:25 So we're going to move on. 561 00:33:25 --> 00:33:28 And I want to talk now about a different kind 562 00:33:28 --> 00:33:30 of weighted average. 563 00:33:30 --> 00:33:36 These weighted averages are going to be much simpler. 564 00:33:36 --> 00:34:01 And they come up in what's called numerical integration. 565 00:34:01 --> 00:34:04 There are many, many methods of integrating numerically. 566 00:34:04 --> 00:34:06 And they're important because many, many integrals 567 00:34:06 --> 00:34:08 don't have formulas. 568 00:34:08 --> 00:34:13 And so you have to compute them with a calculator or a machine. 569 00:34:13 --> 00:34:24 So the first type that we've already done are Riemann sums. 570 00:34:24 --> 00:34:28 They turned out to be incredibly inefficient. 571 00:34:28 --> 00:34:30 They're lousy. 572 00:34:30 --> 00:34:32 The next rule that I'm going to describe is 573 00:34:32 --> 00:34:33 a little improvement. 574 00:34:33 --> 00:34:38 It's called the trapezoidal rule. 575 00:34:38 --> 00:34:41 And this one is much more reasonable then 576 00:34:41 --> 00:34:43 the Riemann sum. 577 00:34:43 --> 00:34:49 Unfortunately, it's actually also pretty lousy. 578 00:34:49 --> 00:34:53 There's another rule which is just a slightly trickier rule. 579 00:34:53 --> 00:34:56 And it's actually amazing that it exists. 580 00:34:56 --> 00:35:01 And it's called Simpson's Rule. 581 00:35:01 --> 00:35:06 And this one is actually pretty good. 582 00:35:06 --> 00:35:07 It's clever. 583 00:35:07 --> 00:35:12 So let's get started with these. 584 00:35:12 --> 00:35:15 And the way I'll get started is by reminding you what 585 00:35:15 --> 00:35:16 the Riemann sum is. 586 00:35:16 --> 00:35:18 So this is a good review, because you need to know 587 00:35:18 --> 00:35:19 all three of these. 588 00:35:19 --> 00:35:26 And you're going to want to see them all laid out in parallel. 589 00:35:26 --> 00:35:31 So here's the setup. 590 00:35:31 --> 00:35:33 OK, so here we go. 591 00:35:33 --> 00:35:36 We have our graph, we have our function it starts out 592 00:35:36 --> 00:35:38 at a, it ends up at b. 593 00:35:38 --> 00:35:40 Maybe goes on, but we're only paying attention 594 00:35:40 --> 00:35:41 to this interval. 595 00:35:41 --> 00:35:44 This is a function y = f(x). 596 00:35:44 --> 00:35:48 And we split it up when we do Riemann's sum. 597 00:35:48 --> 00:35:50 So this is 1, this is Riemann sums. 598 00:35:50 --> 00:35:54 We start out with a, which is a point we call x0, and then 599 00:35:54 --> 00:35:57 we go a certain distance. 600 00:35:57 --> 00:35:58 We go all the way over to b. 601 00:35:58 --> 00:36:04 And we subdivide this thing with these delta x's. 602 00:36:04 --> 00:36:08 Which are the step sizes. 603 00:36:08 --> 00:36:12 So we have these little steps of size delta x. 604 00:36:12 --> 00:36:16 Corresponding to these x values, we have y values. 605 00:36:16 --> 00:36:21 y0 = f(x0), that's the point above a. 606 00:36:21 --> 00:36:26 Then y1 = f ( x1), that's the point above x1. 607 00:36:26 --> 00:36:33 And so forth, all the way up to yn, which = f (xn). 608 00:36:33 --> 00:36:36 In order to figure out the area, you really need to know 609 00:36:36 --> 00:36:37 something about the function. 610 00:36:37 --> 00:36:39 You need to be able to evaluate it. 611 00:36:39 --> 00:36:40 So that's what we've done. 612 00:36:40 --> 00:36:43 We've evaluated here at n + 1 points. 613 00:36:43 --> 00:36:46 Enumerated 0 through n. 614 00:36:46 --> 00:36:49 And those are the numbers out of which we're going to get 615 00:36:49 --> 00:36:53 all of our approximations to the integral. 616 00:36:53 --> 00:36:56 So somehow we want average these numbers. 617 00:36:56 --> 00:37:01 So here's our goal. 618 00:37:01 --> 00:37:13 Our goal is to average, or add, I'm using average 619 00:37:13 --> 00:37:14 very loosely here. 620 00:37:14 --> 00:37:18 But I was going to say add up these numbers to 621 00:37:18 --> 00:37:25 get an approximation. 622 00:37:25 --> 00:37:30 To average or add the y's. 623 00:37:30 --> 00:37:37 To get an approximation to the integral. 624 00:37:37 --> 00:37:43 Which we know is the area under the curve. 625 00:37:43 --> 00:37:49 So here's what the Riemann sum is. 626 00:37:49 --> 00:37:51 It's the following thing. 627 00:37:51 --> 00:37:56 You take (y0 + y1 + ... up to yn - 1) And you 628 00:37:56 --> 00:37:57 multiply by delta x. 629 00:37:57 --> 00:38:00 That's it. 630 00:38:00 --> 00:38:07 Now, this one is the one with left endpoints. 631 00:38:07 --> 00:38:09 So the left-hand sum. 632 00:38:09 --> 00:38:11 There's also a right one. 633 00:38:11 --> 00:38:13 Which is, if you start at the right-hand ends. 634 00:38:13 --> 00:38:17 And that will go from y1 to yn. 635 00:38:17 --> 00:38:22 OK, so this one is the right-hand. 636 00:38:22 --> 00:38:25 Right Riemann sum. 637 00:38:25 --> 00:38:33 Those are the two that we did before. 638 00:38:33 --> 00:38:35 Now I'm going to describe to you the next two. 639 00:38:35 --> 00:38:42 They have a similar pattern to them. 640 00:38:42 --> 00:38:51 And the one with trapezoids requires a picture. 641 00:38:51 --> 00:38:53 Here's a shape. 642 00:38:53 --> 00:39:00 And here's a bunch of values. 643 00:39:00 --> 00:39:08 And we're trying to estimate the size of these chunks. 644 00:39:08 --> 00:39:11 And now, instead of doing something stupid, which is to 645 00:39:11 --> 00:39:14 draw horizontal lines in rectangles, we're going to do 646 00:39:14 --> 00:39:16 something slightly more clever. 647 00:39:16 --> 00:39:21 Which is to draw straight lines that are diagonal. 648 00:39:21 --> 00:39:23 You see that many of them actually coincide probably 649 00:39:23 --> 00:39:25 pretty closely with what I drew there. 650 00:39:25 --> 00:39:29 Although if they're curved, they miss by a little bit. 651 00:39:29 --> 00:39:32 So this is called the trapezoidal rule. 652 00:39:32 --> 00:39:34 Because if you pick one of these shapes, say, this is 653 00:39:34 --> 00:39:40 y2 and this is y3, if you pick one of these shapes, 654 00:39:40 --> 00:39:43 this height here is y2. 655 00:39:43 --> 00:39:47 And this height is y3, and this base is delta x. 656 00:39:47 --> 00:39:52 This is a trapezoid. 657 00:39:52 --> 00:40:01 So this being a trapezoid, I can figure out its area. 658 00:40:01 --> 00:40:02 And what do I get? 659 00:40:02 --> 00:40:12 I get the base times the average height. 660 00:40:12 --> 00:40:15 If you think about if, you work out what happens when you do 661 00:40:15 --> 00:40:18 something with a straight line on top, like that, you'll 662 00:40:18 --> 00:40:21 get this average. 663 00:40:21 --> 00:40:31 So this is the average height of the trapezoid. 664 00:40:31 --> 00:40:34 And now I want to add up. 665 00:40:34 --> 00:40:41 I want to add them all up to get my formula for 666 00:40:41 --> 00:40:45 the trapezoidal rule. 667 00:40:45 --> 00:40:46 So what do I do? 668 00:40:46 --> 00:40:50 I have delta x times the first one. 669 00:40:50 --> 00:40:53 Which is y0 + y1 / 2. 670 00:40:53 --> 00:40:55 That's the first trapezoid. 671 00:40:55 --> 00:40:59 The next one is y1 + y2 / 2. 672 00:40:59 --> 00:41:01 And this keeps on going. 673 00:41:01 --> 00:41:07 And at the end, I have yn - 2 + yn - 1 / 2. 674 00:41:07 --> 00:41:14 And then last of all, I have yn - 1 + yn / 2. 675 00:41:14 --> 00:41:16 That's a very long formula here. 676 00:41:16 --> 00:41:21 We're going to simplify it quite a bit in just a second. 677 00:41:21 --> 00:41:22 What's this equal to? 678 00:41:22 --> 00:41:28 Well, notice that I get y0 / 2 to start out with. 679 00:41:28 --> 00:41:32 And now, y1 got mentioned twice. 680 00:41:32 --> 00:41:35 Each time with a factor of 1/2. 681 00:41:35 --> 00:41:39 So we get a whole y1 in here. 682 00:41:39 --> 00:41:41 And the same thing is going to be true of 683 00:41:41 --> 00:41:43 all the middle terms. 684 00:41:43 --> 00:41:48 You're going to get y2 and all the way up to yn - 1. 685 00:41:48 --> 00:41:53 But then, the last one is unmatched. yn is only 686 00:41:53 --> 00:41:59 1/2, only counts 1/2. 687 00:41:59 --> 00:42:17 So here is what's known as the trapezoidal rule. 688 00:42:17 --> 00:42:22 Now, I'd like to compare it for you to the Riemann sums, 689 00:42:22 --> 00:42:25 which are sitting just to the left here. 690 00:42:25 --> 00:42:28 Here's the left one, and here's the right one. 691 00:42:28 --> 00:42:32 If you take the average of the left and the right, that is, 692 00:42:32 --> 00:42:36 1/2 of this + 1/2 of that, there's an overlap. 693 00:42:36 --> 00:42:39 The y1 through yn things are listed in both. 694 00:42:39 --> 00:42:42 But the y0 only gets counted 1/2 and the yn only 695 00:42:42 --> 00:42:44 gets counted 1/2. 696 00:42:44 --> 00:42:47 So what this is, is this is the symmetric compromise between 697 00:42:47 --> 00:42:49 the two Riemann sums. 698 00:42:49 --> 00:42:56 This is actually equal to the left Riemann sum + the 699 00:42:56 --> 00:43:02 right Riemann sum / 2. 700 00:43:02 --> 00:43:10 It's the average of them. 701 00:43:10 --> 00:43:13 Now, this would be great and it does look like it's closer. 702 00:43:13 --> 00:43:16 But actually it's not as impressive as it looks. 703 00:43:16 --> 00:43:21 If you actually do it in practice, it's 704 00:43:21 --> 00:43:23 not very efficient. 705 00:43:23 --> 00:43:25 Although it's way better than a Riemann sum, it's 706 00:43:25 --> 00:43:27 still not good enough. 707 00:43:27 --> 00:43:37 So now I need to describe to you the fancier rule. 708 00:43:37 --> 00:43:52 Which is known as Simpson's Rule. 709 00:43:52 --> 00:44:00 And so, this is, if you like, 3, Method 3. 710 00:44:00 --> 00:44:09 The idea is again to divide things into chunks. 711 00:44:09 --> 00:44:16 But now it always needs n to be even. 712 00:44:16 --> 00:44:19 In other words, we're going to deal not with just one 713 00:44:19 --> 00:44:24 box, we're going to deal with pairs of boxes. 714 00:44:24 --> 00:44:28 Here's delta x, and here's delta x again. 715 00:44:28 --> 00:44:34 And we're going to study the area of this piece here. 716 00:44:34 --> 00:44:39 So let me focus just on that part. 717 00:44:39 --> 00:44:50 Let's reproduce it over here. 718 00:44:50 --> 00:44:52 And here's the delta x, here's delta x. 719 00:44:52 --> 00:44:54 And of course there are various heights. 720 00:44:54 --> 00:44:57 This starts out at y0, this is y2, and this 721 00:44:57 --> 00:45:03 middle segment is y1. 722 00:45:03 --> 00:45:06 Now, the approximating curve that we're going 723 00:45:06 --> 00:45:14 to use is a parabola. 724 00:45:14 --> 00:45:16 That is, we're going to fit a parabola through 725 00:45:16 --> 00:45:25 these three points. 726 00:45:25 --> 00:45:28 And then we're going to use that as the approximating area. 727 00:45:28 --> 00:45:32 Now, it doesn't look like, this looks like it misses. 728 00:45:32 --> 00:45:34 But actually, most functions mostly wiggle either 729 00:45:34 --> 00:45:35 one way or the other. 730 00:45:35 --> 00:45:36 They don't switch. 731 00:45:36 --> 00:45:38 They don't have inflection points. 732 00:45:38 --> 00:45:41 So, this is a lousy, at this scale. 733 00:45:41 --> 00:45:43 But when we get to a smaller scale, this becomes 734 00:45:43 --> 00:45:45 really fantastic. 735 00:45:45 --> 00:45:47 As an approximation. 736 00:45:47 --> 00:45:52 Now, I need to tell you what the arithmetic is. 737 00:45:52 --> 00:45:57 And in order to save time, it's on your problem set 738 00:45:57 --> 00:46:00 what the actual formula is. 739 00:46:00 --> 00:46:05 But I'm going to tell you how to think about it. 740 00:46:05 --> 00:46:09 I want you to think about it as follows. 741 00:46:09 --> 00:46:24 So the area under the parabola is going to be a base times 742 00:46:24 --> 00:46:30 some kind of average height. 743 00:46:30 --> 00:46:33 And the base here, you can already see. 744 00:46:33 --> 00:46:36 It's 2 delta x. 745 00:46:36 --> 00:46:39 The base is 2 delta x. 746 00:46:39 --> 00:46:40 Now, the average height is weird. 747 00:46:40 --> 00:46:43 You have to work out what it is for a parabola, depending on 748 00:46:43 --> 00:46:46 those three numbers. y0, y1, and y2. 749 00:46:46 --> 00:46:48 And it turns out to be the following formula. 750 00:46:48 --> 00:46:51 It has to be an average, but it's an interesting 751 00:46:51 --> 00:46:51 weighted average. 752 00:46:51 --> 00:46:54 So this was the punchline, if you like. 753 00:46:54 --> 00:46:55 Is that there were such things as interesting 754 00:46:55 --> 00:46:56 weighted averages. 755 00:46:56 --> 00:46:58 This one's very simple, it just involves three numbers. 756 00:46:58 --> 00:46:59 But it's still interesting. 757 00:46:59 --> 00:47:00 It's the following. 758 00:47:00 --> 00:47:07 It turns out to be y0 + 4 y1 + y2 / 6. 759 00:47:07 --> 00:47:08 Why divided by 6? 760 00:47:08 --> 00:47:10 Well, it's supposed to be an average. 761 00:47:10 --> 00:47:13 So the total is 1 + 4 + 1 of these things. 762 00:47:13 --> 00:47:16 And 6 is in the denominator. 763 00:47:16 --> 00:47:19 So it emphasizes the middle more than the sides. 764 00:47:19 --> 00:47:23 And that's what happens with a parabola. 765 00:47:23 --> 00:47:30 So this is a computation which is on your homework. 766 00:47:30 --> 00:47:33 And now we can put this together for the full 767 00:47:33 --> 00:47:36 Simpson's Rule formula. 768 00:47:36 --> 00:47:51 Which I'll put up over here. 769 00:47:51 --> 00:47:56 We have here 2 delta x, and we divide by 6. 770 00:47:56 --> 00:48:02 And then we have y0 + 4 y1 + y2 / 6 +... 771 00:48:03 --> 00:48:05 that's the first chunk. 772 00:48:05 --> 00:48:08 Now, the second chunk, maybe I'll just put it in 773 00:48:08 --> 00:48:11 here, starts this is x2. 774 00:48:11 --> 00:48:14 This is x0. 775 00:48:14 --> 00:48:15 And it goes all the way to x4. 776 00:48:15 --> 00:48:18 So x2, x3, x4. 777 00:48:18 --> 00:48:21 So the next one involves the indices 2, 3 and 4. 778 00:48:21 --> 00:48:31 So this is y2 + 4 y3 + y4 / 6. 779 00:48:31 --> 00:48:34 Oh, oh, oh, oh, no. 780 00:48:34 --> 00:48:35 I think I'll get rid of these 6's. 781 00:48:35 --> 00:48:37 I have too many 6's. 782 00:48:37 --> 00:48:39 Alright. 783 00:48:39 --> 00:48:41 Let's get rid of them here. 784 00:48:41 --> 00:48:41 Let's take them out. 785 00:48:41 --> 00:48:44 Put them out here. 786 00:48:44 --> 00:48:47 Thank you. 787 00:48:47 --> 00:48:53 All the way to the end, which is y(n - 2) + 2 y(n - 1) - 788 00:48:53 --> 00:48:57 sorry, + 4 y(n - 1) + yn. 789 00:48:57 --> 00:49:01 I was about to divide by 6, but you saved me. 790 00:49:01 --> 00:49:04 So here are all the chunks. 791 00:49:04 --> 00:49:06 Now, what does this pattern come out to be? 792 00:49:06 --> 00:49:10 This comes out to be the following. 793 00:49:10 --> 00:49:16 This is 1, 4, 1, added to 1, 4, 1 added to 1, 4, 1. 794 00:49:16 --> 00:49:17 You add them up. 795 00:49:17 --> 00:49:22 You 1, 4, and then there's a repeat, so you get a 2 and a 796 00:49:22 --> 00:49:24 4, and a 2 and a 4 and a 1. 797 00:49:24 --> 00:49:27 So the pattern is that it starts out with 798 00:49:27 --> 00:49:29 1's on the far ends. 799 00:49:29 --> 00:49:30 And then 4's next in. 800 00:49:30 --> 00:49:35 And then it alternates 2's and 4's in between. 801 00:49:35 --> 00:49:44 So the full pattern of Simpson's Rule is delta x / 802 00:49:44 --> 00:49:47 3, I have now succeeded in canceling this 2 with this 803 00:49:47 --> 00:49:50 6 and getting out that factor of 2. 804 00:49:50 --> 00:50:01 And then here I have y0 + 4 y1 + 2 y2 + 4 y3 + ... it keeps on 805 00:50:01 --> 00:50:03 going and keeps on going and keeps on going. 806 00:50:03 --> 00:50:10 And in the end it's 2 y(n - 2) + 4 y(n - 1) + yn. 807 00:50:10 --> 00:50:14 So again, 1 and a 4 to start. 808 00:50:14 --> 00:50:16 1 and a 4 and a 1 to end. 809 00:50:16 --> 00:50:20 And then alternating 2's and 4's in the middle And this 810 00:50:20 --> 00:50:25 weird weighted average is way better. 811 00:50:25 --> 00:50:28 As I will show you next time. 812 00:50:28 --> 00:50:29