1 00:00:00,000 --> 00:00:02,230 The following content is provided under a Creative 2 00:00:02,230 --> 00:00:02,896 Commons license. 3 00:00:02,896 --> 00:00:06,110 Your support will help MIT OpenCourseWare 4 00:00:06,110 --> 00:00:09,950 continue to offer high quality educational resources for free. 5 00:00:09,950 --> 00:00:12,650 To make a donation, or to view additional materials 6 00:00:12,650 --> 00:00:16,160 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,160 --> 00:00:21,670 at ocw.mit.edu. 8 00:00:21,670 --> 00:00:24,040 PROFESSOR: Today we're going to hold off just 9 00:00:24,040 --> 00:00:27,250 a little bit on boiling water. 10 00:00:27,250 --> 00:00:31,920 And talk about another application of integrals, 11 00:00:31,920 --> 00:00:36,030 and we'll get to the witches' cauldron in the middle. 12 00:00:36,030 --> 00:00:45,230 The thing that I'd like to start with today is average value. 13 00:00:45,230 --> 00:00:47,877 This is something that I mentioned a little bit earlier, 14 00:00:47,877 --> 00:00:49,460 and there was a misprint on the board, 15 00:00:49,460 --> 00:00:54,980 so I want to make sure that we have the definitions straight. 16 00:00:54,980 --> 00:00:56,770 And also the reasoning straight. 17 00:00:56,770 --> 00:00:58,970 This is one of the most important applications 18 00:00:58,970 --> 00:01:03,840 of integrals, one of the most important examples. 19 00:01:03,840 --> 00:01:08,300 If you take the average of a bunch of numbers, 20 00:01:08,300 --> 00:01:10,420 that looks like this. 21 00:01:10,420 --> 00:01:15,500 And we can view this as sampling a function, 22 00:01:15,500 --> 00:01:17,570 as we would with the Riemann sum. 23 00:01:17,570 --> 00:01:23,180 And what I said last week was that this 24 00:01:23,180 --> 00:01:28,020 tends to this expression here, which is 25 00:01:28,020 --> 00:01:32,960 called the continuous average. 26 00:01:32,960 --> 00:01:43,550 So this guy is the continuous average. 27 00:01:43,550 --> 00:01:49,270 Or just the average of f. 28 00:01:49,270 --> 00:01:53,710 And I want to explain that, just to make sure that we're all 29 00:01:53,710 --> 00:01:56,380 on the same page. 30 00:01:56,380 --> 00:01:58,930 In general, if you have a function 31 00:01:58,930 --> 00:02:01,220 and you want to interpret the integral, 32 00:02:01,220 --> 00:02:04,270 our first interpretation was that it's something 33 00:02:04,270 --> 00:02:09,860 like the area under the curve. 34 00:02:09,860 --> 00:02:15,440 But average value is another reasonable interpretation. 35 00:02:15,440 --> 00:02:19,230 Namely, if you take equally spaced 36 00:02:19,230 --> 00:02:24,590 points here, starting with x_0, x_1, x_2, 37 00:02:24,590 --> 00:02:29,940 all the way up to x_n, which is the left point b, 38 00:02:29,940 --> 00:02:32,320 and then we have values y_1, which 39 00:02:32,320 --> 00:02:38,750 is f(x_1); y_2, which is f(x_2); all the way up 40 00:02:38,750 --> 00:02:40,130 to y_n, which is f(x_n). 41 00:02:43,040 --> 00:02:47,260 And again, the spacing here that we're talking about is (b-a) / 42 00:02:47,260 --> 00:02:47,760 n. 43 00:02:50,690 --> 00:02:52,540 So remember that spacing, that's going to be 44 00:02:52,540 --> 00:02:57,100 the connection that we'll draw. 45 00:02:57,100 --> 00:03:08,070 Then the Riemann sum is y_1 through y_n, the sum of y_1 46 00:03:08,070 --> 00:03:12,080 through y_n, multiplied by delta x. 47 00:03:12,080 --> 00:03:16,370 And that's what tends, as delta x goes to 0, to the integral. 48 00:03:23,400 --> 00:03:26,060 The only change in point of view if I 49 00:03:26,060 --> 00:03:28,450 want to write this limiting property, which 50 00:03:28,450 --> 00:03:33,780 is right above here, the only change between here 51 00:03:33,780 --> 00:03:36,810 and here is that I want to divide by the length 52 00:03:36,810 --> 00:03:38,850 of the interval. b - a. 53 00:03:38,850 --> 00:03:42,000 So I will divide by b - a here. 54 00:03:42,000 --> 00:03:48,230 And divide by b - a over here. 55 00:03:48,230 --> 00:03:53,510 And then I'll just check what this thing actually is. 56 00:03:53,510 --> 00:03:57,560 Delta x / (b-a), what is that factor? 57 00:03:57,560 --> 00:04:01,650 Well, if we look over here to what delta x is, if you 58 00:04:01,650 --> 00:04:07,040 divide by b - a, it's 1 / n. 59 00:04:07,040 --> 00:04:10,210 So the factor delta x / (b-a) is 1 / n. 60 00:04:10,210 --> 00:04:12,660 That's what I put over here, the sum of y_1 61 00:04:12,660 --> 00:04:15,210 through y_n divided by n. 62 00:04:15,210 --> 00:04:22,760 And as this tends to 0, it's the same as n going to infinity. 63 00:04:22,760 --> 00:04:25,610 Those are the same things. 64 00:04:25,610 --> 00:04:27,170 The average value and the integral 65 00:04:27,170 --> 00:04:28,460 are very closely related. 66 00:04:28,460 --> 00:04:30,790 There's only this difference that we're dividing 67 00:04:30,790 --> 00:04:36,440 by the length of the interval. 68 00:04:36,440 --> 00:04:38,120 I want to give an example which is 69 00:04:38,120 --> 00:04:40,330 an incredibly simpleminded one, but it'll 70 00:04:40,330 --> 00:04:41,800 come into play later on. 71 00:04:41,800 --> 00:04:47,040 So let's take the example of a constant. 72 00:04:47,040 --> 00:04:50,730 And this is, I hope-- will make you slightly less confused 73 00:04:50,730 --> 00:04:52,540 about what I just wrote. 74 00:04:52,540 --> 00:04:54,220 As well as making you think that this 75 00:04:54,220 --> 00:04:57,370 is as simpleminded and reasonable as it should be. 76 00:04:57,370 --> 00:05:00,310 If I check what the average value of this constant is, 77 00:05:00,310 --> 00:05:04,680 it's given by this relatively complicated formula here. 78 00:05:04,680 --> 00:05:07,170 That is, I have to integrate the function c. 79 00:05:07,170 --> 00:05:09,570 Well, it's just the constant c. 80 00:05:09,570 --> 00:05:12,320 And however you do this, as an antiderivative or as thinking 81 00:05:12,320 --> 00:05:14,730 of it as a rectangle, the answer that you're going to get 82 00:05:14,730 --> 00:05:18,310 is c here. 83 00:05:18,310 --> 00:05:19,490 So work that out. 84 00:05:19,490 --> 00:05:20,730 The answer is c. 85 00:05:20,730 --> 00:05:23,780 And so the fact that the average of c 86 00:05:23,780 --> 00:05:28,480 is equal to c, which had better be the case for averages, 87 00:05:28,480 --> 00:05:31,760 explains the denominator. 88 00:05:31,760 --> 00:05:36,690 Explains the 1 / (b-a) there. 89 00:05:36,690 --> 00:05:39,107 That's cooked up exactly so that the average of a constant 90 00:05:39,107 --> 00:05:40,273 is what it's supposed to be. 91 00:05:40,273 --> 00:05:42,250 Otherwise we have the wrong normalizing factor. 92 00:05:42,250 --> 00:05:45,450 We've clearly got a piece of nonsense on our hands. 93 00:05:45,450 --> 00:05:50,470 And incidentally, it also explains the 1/n 94 00:05:50,470 --> 00:05:53,460 in the very first formula that I wrote down. 95 00:05:53,460 --> 00:05:55,360 The reason why this is called the average, 96 00:05:55,360 --> 00:05:57,580 or one reason why it's the right thing, 97 00:05:57,580 --> 00:06:00,177 is that if you took the same constant c, for y 98 00:06:00,177 --> 00:06:02,510 all the way across there n times, if you divide it by n, 99 00:06:02,510 --> 00:06:03,930 you get back c. 100 00:06:03,930 --> 00:06:05,490 That's what we mean by average value 101 00:06:05,490 --> 00:06:11,370 and that's why the n is there. 102 00:06:11,370 --> 00:06:14,960 So that was an easy example. 103 00:06:14,960 --> 00:06:17,610 Now none of the examples that we are going to give 104 00:06:17,610 --> 00:06:20,000 are going to be all that complicated. 105 00:06:20,000 --> 00:06:23,460 But they will get sort of steadily more sophisticated. 106 00:06:23,460 --> 00:06:32,610 The second example is going to be the average height 107 00:06:32,610 --> 00:06:47,860 of a point on a semicircle. 108 00:06:47,860 --> 00:06:52,800 And maybe I'll draw a picture of the semicircle first here. 109 00:06:52,800 --> 00:06:57,250 And we'll just make it the standard circle, the unit 110 00:06:57,250 --> 00:06:58,840 circle. 111 00:06:58,840 --> 00:07:04,890 So maybe I should have called it a unit semicircle. 112 00:07:04,890 --> 00:07:07,680 This is the point negative 1, this is the point 1. 113 00:07:07,680 --> 00:07:09,890 And we're picking a point over here 114 00:07:09,890 --> 00:07:15,540 and we're going to take the typical, or the average, height 115 00:07:15,540 --> 00:07:17,210 here. 116 00:07:17,210 --> 00:07:18,550 Integrating with respect to dx. 117 00:07:18,550 --> 00:07:21,770 So sort of continuously with respect to dx. 118 00:07:21,770 --> 00:07:25,030 Well, what is that? 119 00:07:25,030 --> 00:07:29,380 Well, according to the rule, it's 1/(b-a) times - sorry, 120 00:07:29,380 --> 00:07:30,820 it's up here in the box. 121 00:07:30,820 --> 00:07:34,080 1/(b-a), the integral from a to b, f(x) dx. 122 00:07:34,080 --> 00:07:39,000 That's 1 / (+1 - (-1)). 123 00:07:39,000 --> 00:07:45,240 The integral from - 1 to 1, square root of 1 - x^2, dx. 124 00:07:45,240 --> 00:07:52,026 Right, because the height is y is equal to-- this is y is 125 00:07:52,026 --> 00:07:57,260 equal to the square root of 1 - x^2. 126 00:07:57,260 --> 00:08:02,650 And to evaluate this is not as difficult as it seems. 127 00:08:02,650 --> 00:08:06,490 This is 1/2 times this quantity here, 128 00:08:06,490 --> 00:08:08,940 which we can interpret as an area. 129 00:08:08,940 --> 00:08:12,520 It's the area of the semicircle. 130 00:08:12,520 --> 00:08:19,410 So this is the area of the semicircle, 131 00:08:19,410 --> 00:08:22,730 which we know to be half the area of the circle. 132 00:08:22,730 --> 00:08:26,700 So it's pi/2. 133 00:08:26,700 --> 00:08:29,200 And so the answer, here the average height, is pi/4. 134 00:08:35,500 --> 00:08:38,670 Now, later in the class and actually not in this unit, 135 00:08:38,670 --> 00:08:40,330 we'll actually be able to calculate 136 00:08:40,330 --> 00:08:42,424 the antiderivative of this. 137 00:08:42,424 --> 00:08:45,090 So in other words, we'll be able to calculate this analytically. 138 00:08:45,090 --> 00:08:47,090 For right now we just have the geometric reason 139 00:08:47,090 --> 00:08:51,190 why the value of this is pi/2. 140 00:08:51,190 --> 00:08:53,070 And we'll do that in the fourth unit 141 00:08:53,070 --> 00:08:57,140 when we do a lot of techniques of integration. 142 00:08:57,140 --> 00:08:58,290 So here's an example. 143 00:08:58,290 --> 00:09:00,260 Turns out, the average height of this is pi/4. 144 00:09:03,770 --> 00:09:06,700 Now, the next example that I want to give 145 00:09:06,700 --> 00:09:09,440 introduces a little bit of confusion. 146 00:09:09,440 --> 00:09:13,370 And I'm not going to resolve this confusion completely, 147 00:09:13,370 --> 00:09:15,870 but I'm going to try to get you used to it. 148 00:09:15,870 --> 00:09:22,770 I'm going to take the average height again. 149 00:09:22,770 --> 00:09:33,690 But now, with respect to arc length. 150 00:09:33,690 --> 00:09:37,250 Which is usually denoted theta. 151 00:09:37,250 --> 00:09:43,080 Now, this brings up an extremely important feature of averages. 152 00:09:43,080 --> 00:09:45,560 Which is that you have to specify 153 00:09:45,560 --> 00:09:48,740 the variable with respect to which the average is 154 00:09:48,740 --> 00:09:51,290 taking place. 155 00:09:51,290 --> 00:09:54,620 And the answer will be different depending on the variable. 156 00:09:54,620 --> 00:09:56,060 So it's not going to be the same. 157 00:09:56,060 --> 00:09:58,130 Wow, can't spell the word length here. 158 00:09:58,130 --> 00:10:02,230 Just like the plural of witches the last time. 159 00:10:02,230 --> 00:10:03,150 We'll work on that. 160 00:10:03,150 --> 00:10:11,410 We'll fix all of our, that's an ancient Gaelic word, I think. 161 00:10:11,410 --> 00:10:16,560 Lengh. 162 00:10:16,560 --> 00:10:22,190 So now, let me show you that it's not quite the same here. 163 00:10:22,190 --> 00:10:24,660 It's especially exaggerated if maybe I 164 00:10:24,660 --> 00:10:30,440 shift this little interval dx over to the right-hand end. 165 00:10:30,440 --> 00:10:33,230 And you can see that the little portion that corresponds 166 00:10:33,230 --> 00:10:35,630 to it, which is the d theta piece, 167 00:10:35,630 --> 00:10:39,130 has a different length from the dx piece. 168 00:10:39,130 --> 00:10:40,690 And indeed, as you come down here, 169 00:10:40,690 --> 00:10:43,760 these very short portions of dx length 170 00:10:43,760 --> 00:10:47,770 have much longer portions of theta length. 171 00:10:47,770 --> 00:10:51,570 So that the average that we're taking when we do it 172 00:10:51,570 --> 00:10:54,670 with respect to theta is going to emphasize 173 00:10:54,670 --> 00:10:56,060 the low values more. 174 00:10:56,060 --> 00:10:58,090 They're going to be more exaggerated. 175 00:10:58,090 --> 00:11:02,322 And the average should be lower than the average 176 00:11:02,322 --> 00:11:03,030 that we got here. 177 00:11:03,030 --> 00:11:04,880 So we should expect a different number. 178 00:11:04,880 --> 00:11:06,726 And it's not going to be / 4, it's 179 00:11:06,726 --> 00:11:07,850 going to be something else. 180 00:11:07,850 --> 00:11:12,170 Whatever it is, it should be smaller than pi / 4. 181 00:11:12,170 --> 00:11:14,200 Now, let's set up the integral. 182 00:11:14,200 --> 00:11:17,100 The integral follows the same rule. 183 00:11:17,100 --> 00:11:19,760 It's just 1 over the length of the interval times 184 00:11:19,760 --> 00:11:24,390 the integral over the interval of the function. 185 00:11:24,390 --> 00:11:27,480 That's the integral, but now where does theta range? 186 00:11:27,480 --> 00:11:32,430 This time, theta goes from 0 to pi. 187 00:11:32,430 --> 00:11:34,990 So the integral is from 0 to pi. 188 00:11:34,990 --> 00:11:40,140 And the thing we divide by is pi. 189 00:11:40,140 --> 00:11:42,590 And the integration requires us to know 190 00:11:42,590 --> 00:11:44,160 the formula for the height. 191 00:11:44,160 --> 00:11:47,410 Which is sin theta. 192 00:11:47,410 --> 00:11:48,680 In terms of theta, of course. 193 00:11:48,680 --> 00:11:50,960 It's the same as square root of 1 - x^2, 194 00:11:50,960 --> 00:11:53,360 but it's expressed in terms of theta. 195 00:11:53,360 --> 00:11:55,200 So it's this. 196 00:11:55,200 --> 00:11:58,340 And here's our average. 197 00:11:58,340 --> 00:12:01,210 I'll put this up here. 198 00:12:01,210 --> 00:12:09,010 So that's the formula for the height. 199 00:12:09,010 --> 00:12:10,810 So let's work it out. 200 00:12:10,810 --> 00:12:12,724 This one, we have the advantage of being 201 00:12:12,724 --> 00:12:15,140 able to work out because we know the antiderivative of sin 202 00:12:15,140 --> 00:12:16,920 theta. 203 00:12:16,920 --> 00:12:25,530 It happens with this factor of pi, it's -cos theta. 204 00:12:25,530 --> 00:12:35,730 And so, that's -1/pi cos pi-- sorry. (cos pi - cos 0). 205 00:12:35,730 --> 00:12:42,820 Which is -1/pi (-2), which is 2 / pi. 206 00:12:47,340 --> 00:12:51,020 And sure enough, if you check it, 207 00:12:51,020 --> 00:13:02,250 you'll see that 2 / pi < pi / 4, because pi^2 > 8. 208 00:13:02,250 --> 00:13:02,980 Yeah, question. 209 00:13:02,980 --> 00:13:06,610 STUDENT: [INAUDIBLE] 210 00:13:06,610 --> 00:13:09,700 PROFESSOR: The question is how do I get sin theta. 211 00:13:09,700 --> 00:13:16,080 And the answer is, on this diagram, if theta is over here, 212 00:13:16,080 --> 00:13:19,950 then this height is this, and this is the angle theta, 213 00:13:19,950 --> 00:13:22,790 then the height is the sine. 214 00:13:22,790 --> 00:13:26,600 OK. 215 00:13:26,600 --> 00:13:27,320 Another question. 216 00:13:27,320 --> 00:13:32,160 STUDENT: [INAUDIBLE] 217 00:13:32,160 --> 00:13:41,240 PROFESSOR: The question is, what is the first one, 218 00:13:41,240 --> 00:13:43,230 the first one is an average of height, 219 00:13:43,230 --> 00:13:51,310 of a point on a semicircle and this one is with respect to x. 220 00:13:51,310 --> 00:13:52,890 So what this reveals is that it's 221 00:13:52,890 --> 00:13:55,490 ambiguous to say what the average value of something is, 222 00:13:55,490 --> 00:13:59,240 unless you've explained what the underlying averaging 223 00:13:59,240 --> 00:14:00,200 variable is. 224 00:14:00,200 --> 00:14:08,140 STUDENT: [INAUDIBLE] 225 00:14:08,140 --> 00:14:10,800 PROFESSOR: The next question is how should you 226 00:14:10,800 --> 00:14:13,860 interpret this value. 227 00:14:13,860 --> 00:14:17,250 That is, what came out of this calculation? 228 00:14:17,250 --> 00:14:22,460 And the answer is only sort of embedded 229 00:14:22,460 --> 00:14:26,050 in this calculation itself. 230 00:14:26,050 --> 00:14:27,690 So here's a way of thinking of it 231 00:14:27,690 --> 00:14:29,730 which is anticipating our next subject. 232 00:14:29,730 --> 00:14:31,520 Which is probability. 233 00:14:31,520 --> 00:14:35,510 Which is, suppose you picked a number at random 234 00:14:35,510 --> 00:14:36,660 in this interval. 235 00:14:36,660 --> 00:14:39,440 With equal likelihood, one place and another. 236 00:14:39,440 --> 00:14:42,080 And then you saw what height was above that. 237 00:14:42,080 --> 00:14:45,272 That would be the interpretation of this first average value. 238 00:14:45,272 --> 00:14:46,980 And the second one is, I picked something 239 00:14:46,980 --> 00:14:50,110 at random on this circle. 240 00:14:50,110 --> 00:14:53,240 And equally likely, any possible point on this circle 241 00:14:53,240 --> 00:14:54,970 according to its length. 242 00:14:54,970 --> 00:14:58,180 And then I ask what the height of that point is. 243 00:14:58,180 --> 00:15:04,670 And those are just different things. 244 00:15:04,670 --> 00:15:05,460 Another question. 245 00:15:05,460 --> 00:15:09,510 STUDENT: [INAUDIBLE] 246 00:15:09,510 --> 00:15:12,340 PROFESSOR: cos pi, shouldn't it be 0? 247 00:15:12,340 --> 00:15:19,440 No. cos of-- it's -1. cos pi is -1. 248 00:15:19,440 --> 00:15:25,110 Cosine, sorry. 249 00:15:25,110 --> 00:15:29,240 No, cos 0 = 1. cos pi = -1. 250 00:15:29,240 --> 00:15:30,476 And so they cancel. 251 00:15:30,476 --> 00:15:31,600 That is, they don't cancel. 252 00:15:31,600 --> 00:15:35,880 It's -1 - 1, which is -2. 253 00:15:35,880 --> 00:15:37,830 Key point. 254 00:15:37,830 --> 00:15:38,330 Yeah. 255 00:15:38,330 --> 00:15:49,680 STUDENT: [INAUDIBLE] 256 00:15:49,680 --> 00:15:51,260 PROFESSOR: All right, let me repeat. 257 00:15:51,260 --> 00:15:55,150 So the question was to repeat the reasoning 258 00:15:55,150 --> 00:16:00,580 by which I guessed in advance that probably this 259 00:16:00,580 --> 00:16:04,220 was going to be the relationship between the average value 260 00:16:04,220 --> 00:16:06,460 with respect to arc length versus the average value 261 00:16:06,460 --> 00:16:10,790 with respect to this horizontal distance. 262 00:16:10,790 --> 00:16:15,405 And it had to do with the previous way this diagram was 263 00:16:15,405 --> 00:16:16,980 drawn. 264 00:16:16,980 --> 00:16:21,690 Which is comparing an interval in dx 265 00:16:21,690 --> 00:16:25,800 with an interval in theta. 266 00:16:25,800 --> 00:16:27,780 A little section in theta. 267 00:16:27,780 --> 00:16:32,530 And when you're near the top, they're nearly this same. 268 00:16:32,530 --> 00:16:34,424 That is, it's more or less balanced. 269 00:16:34,424 --> 00:16:36,340 It's a little curved here, a little different. 270 00:16:36,340 --> 00:16:39,010 But here it becomes very exaggerated. 271 00:16:39,010 --> 00:16:42,390 The d theta lengths are much longer than the dx lengths. 272 00:16:42,390 --> 00:16:45,990 Which means that importance given 273 00:16:45,990 --> 00:16:49,750 by the theta variable to these parts of the circle 274 00:16:49,750 --> 00:16:52,930 is larger, relative to these parts. 275 00:16:52,930 --> 00:16:55,830 Whereas if you look at this section versus this section 276 00:16:55,830 --> 00:16:57,660 for the dx, they give equal weights 277 00:16:57,660 --> 00:16:59,090 to these two equal lengths. 278 00:16:59,090 --> 00:17:01,660 But here, with respect to theta, this is relatively short 279 00:17:01,660 --> 00:17:03,090 and this is much larger. 280 00:17:03,090 --> 00:17:05,150 So, as I say, the theta variable's 281 00:17:05,150 --> 00:17:09,140 emphasizing the lower parts of the semicircle more. 282 00:17:09,140 --> 00:17:10,930 That's because this length is shorter 283 00:17:10,930 --> 00:17:12,340 and this length is longer. 284 00:17:12,340 --> 00:17:16,130 Whereas these two are the same. 285 00:17:16,130 --> 00:17:20,110 It's a balancing act of the relative weights. 286 00:17:20,110 --> 00:17:23,190 I'm going to say that again in a different way, and maybe 287 00:17:23,190 --> 00:17:31,357 this will-- The lower part is more important for theta. 288 00:17:31,357 --> 00:17:32,190 STUDENT: [INAUDIBLE] 289 00:17:32,190 --> 00:17:33,595 PROFESSOR: So the question is, but shouldn't it 290 00:17:33,595 --> 00:17:35,920 have a bigger value because it's a longer length. 291 00:17:35,920 --> 00:17:37,620 Never with averages. 292 00:17:37,620 --> 00:17:39,540 Whatever the length is, we're always dividing. 293 00:17:39,540 --> 00:17:42,760 We're always compensating by the total. 294 00:17:42,760 --> 00:17:45,410 We have the integral from 0 to pi, but we're dividing by pi. 295 00:17:45,410 --> 00:17:49,760 Here we had the integral from -1 to 1, but we're dividing by 2. 296 00:17:49,760 --> 00:17:52,220 So we divide by something different each time. 297 00:17:52,220 --> 00:17:53,850 And this is very, very important. 298 00:17:53,850 --> 00:17:55,410 It's that the average of a constant 299 00:17:55,410 --> 00:17:58,070 is that same constant regardless of which one we did. 300 00:17:58,070 --> 00:18:00,070 So if it were a constant, we would always 301 00:18:00,070 --> 00:18:01,270 compensate for the length. 302 00:18:01,270 --> 00:18:03,780 So the length never matters. 303 00:18:03,780 --> 00:18:09,160 If it's the integral from 0 to 1,000,000, or 100, let's say, 304 00:18:09,160 --> 00:18:12,036 1/100 c dx, it's just the same. 305 00:18:12,036 --> 00:18:14,160 It's always that, it doesn't matter how long it is. 306 00:18:14,160 --> 00:18:17,405 Because we compensate. 307 00:18:17,405 --> 00:18:19,405 That's really the difference between an integral 308 00:18:19,405 --> 00:18:24,750 and an average, is that we're dividing by the total. 309 00:18:24,750 --> 00:18:28,320 Now I want to introduce another notion, which is actually 310 00:18:28,320 --> 00:18:31,561 what's underlying these two examples that I just wrote 311 00:18:31,561 --> 00:18:32,060 down. 312 00:18:32,060 --> 00:18:35,210 And this is by far the one which you should emphasize 313 00:18:35,210 --> 00:18:39,470 the most in your thoughts. 314 00:18:39,470 --> 00:18:42,480 Because it is much more flexible, 315 00:18:42,480 --> 00:18:49,470 and is much more typical of real life problems. 316 00:18:49,470 --> 00:18:53,920 So the idea of a weighted average is the following. 317 00:18:53,920 --> 00:18:57,580 You take the integral, say from a to b, of some function. 318 00:18:57,580 --> 00:19:02,550 But now you multiply by a weight. 319 00:19:02,550 --> 00:19:05,560 And you have to divide by the total. 320 00:19:05,560 --> 00:19:07,410 And what's the total going to be? 321 00:19:07,410 --> 00:19:11,690 It's the integral from a to b of this total weighting 322 00:19:11,690 --> 00:19:14,020 that we have. 323 00:19:14,020 --> 00:19:17,190 Now, why is this the correct notion? 324 00:19:17,190 --> 00:19:20,420 I'm going to explain it to you in two ways. 325 00:19:20,420 --> 00:19:24,800 The first is this very simpleminded thing 326 00:19:24,800 --> 00:19:30,480 that I wrote on the board there, with the constants. 327 00:19:30,480 --> 00:19:38,480 What we want is the average value of c to be c. 328 00:19:38,480 --> 00:19:40,820 Otherwise this makes no sense as an average. 329 00:19:40,820 --> 00:19:43,070 Now, let's just look at this definition here. 330 00:19:43,070 --> 00:19:44,660 And see that that's correct. 331 00:19:44,660 --> 00:19:49,920 If you integrate c, from a to b, w(x) dx, 332 00:19:49,920 --> 00:19:54,710 and you divide by the integral from a to b, w(x) dx, 333 00:19:54,710 --> 00:19:57,020 not surprisingly, the c factors out. 334 00:19:57,020 --> 00:19:59,010 It's a constant. 335 00:19:59,010 --> 00:20:03,840 So this is c times the integral a to b, w(x) dx, divided 336 00:20:03,840 --> 00:20:05,380 by the same thing. 337 00:20:05,380 --> 00:20:08,212 And that's why we picked it. 338 00:20:08,212 --> 00:20:10,170 We picked it so that these things would cancel. 339 00:20:10,170 --> 00:20:14,660 And this would give c. 340 00:20:14,660 --> 00:20:17,260 So in the previous case, this property 341 00:20:17,260 --> 00:20:20,140 explains the denominator. 342 00:20:20,140 --> 00:20:29,760 And again over here, it explains the denominator. 343 00:20:29,760 --> 00:20:32,510 And let me just give you one more explanation. 344 00:20:32,510 --> 00:20:38,340 Which is maybe a real-life-- pretend real-life example. 345 00:20:38,340 --> 00:20:43,060 You have a stock which you bought for $10 one year. 346 00:20:43,060 --> 00:20:46,740 And then six months later you brought some more for $20. 347 00:20:46,740 --> 00:20:50,360 And then you bought some more for $30. 348 00:20:50,360 --> 00:20:53,810 Now, what's the average price of your stock? 349 00:20:53,810 --> 00:20:58,370 Well, it depends on how many shares you bought. 350 00:20:58,370 --> 00:21:00,700 If you bought this many shares the first time, 351 00:21:00,700 --> 00:21:02,440 and this many shares the second time, 352 00:21:02,440 --> 00:21:04,310 and this many shares the third time, 353 00:21:04,310 --> 00:21:07,230 this is the total amount that you spent. 354 00:21:07,230 --> 00:21:13,870 And the average price is the total price divided 355 00:21:13,870 --> 00:21:17,120 by the total number of shares. 356 00:21:17,120 --> 00:21:23,150 And this is the discrete analog of this continuous averaging 357 00:21:23,150 --> 00:21:24,070 process here. 358 00:21:24,070 --> 00:21:28,260 The function f now, so I use w for weight, the function f now 359 00:21:28,260 --> 00:21:31,150 is the function whose values are 10, 20 and 30. 360 00:21:31,150 --> 00:21:34,830 And the weightings are the relative importance 361 00:21:34,830 --> 00:21:42,030 of the different purchases. 362 00:21:42,030 --> 00:21:51,586 So again, these w_i's are weights. 363 00:21:51,586 --> 00:21:52,710 There was another question. 364 00:21:52,710 --> 00:21:54,400 Out in the audience, at some point. 365 00:21:54,400 --> 00:21:55,300 Over here, yes. 366 00:21:55,300 --> 00:22:04,520 STUDENT: [INAUDIBLE] 367 00:22:04,520 --> 00:22:06,920 PROFESSOR: Very, very good point. 368 00:22:06,920 --> 00:22:15,710 So in this numerator here, the statement is-- in this example, 369 00:22:15,710 --> 00:22:17,020 we factored out c. 370 00:22:17,020 --> 00:22:20,710 But here we cannot factor out f(x). 371 00:22:20,710 --> 00:22:23,450 That's extremely important and that is the whole point. 372 00:22:23,450 --> 00:22:26,930 So, in other words, the weighted average is very interesting 373 00:22:26,930 --> 00:22:31,680 - you have to do two different integrals to figure it out 374 00:22:31,680 --> 00:22:32,380 in general. 375 00:22:32,380 --> 00:22:34,050 When it happens that this is c, it's 376 00:22:34,050 --> 00:22:35,430 an extremely boring integral. 377 00:22:35,430 --> 00:22:37,100 Which in fact because, it's an average, 378 00:22:37,100 --> 00:22:38,799 you don't even have to calculate at all. 379 00:22:38,799 --> 00:22:40,340 Factor it out and cancel these things 380 00:22:40,340 --> 00:22:43,530 and never bother to calculate these two numbers. 381 00:22:43,530 --> 00:22:46,580 So these massive numbers just cancel. 382 00:22:46,580 --> 00:22:48,880 So it's a very special property of a constant, 383 00:22:48,880 --> 00:22:55,650 that it factors out. 384 00:22:55,650 --> 00:22:59,960 That was our first discussion, and now with this example 385 00:22:59,960 --> 00:23:02,900 I'm going to go back to the heating up of the witches' 386 00:23:02,900 --> 00:23:06,190 cauldron and we'll use average value 387 00:23:06,190 --> 00:23:19,700 to illustrate the integral that we get in that context as well. 388 00:23:19,700 --> 00:23:20,910 I remind you, let's see. 389 00:23:20,910 --> 00:23:25,480 The situation with the witches' cauldron was this. 390 00:23:25,480 --> 00:23:40,130 The first important thing is that there were-- so this 391 00:23:40,130 --> 00:23:42,770 is the big cauldron here. 392 00:23:42,770 --> 00:23:47,480 This is the one whose height is 1 meter and whose 393 00:23:47,480 --> 00:23:53,190 width is 2 meters. 394 00:23:53,190 --> 00:23:56,540 And it's a parabola of revolution here. 395 00:23:56,540 --> 00:24:06,500 And it had about approximately 1600 liters in it. 396 00:24:06,500 --> 00:24:14,640 And this curve was y = x^2. 397 00:24:14,640 --> 00:24:17,580 And the situation that I described 398 00:24:17,580 --> 00:24:24,900 at the end of last time was that the initial temperature 399 00:24:24,900 --> 00:24:28,560 was T = 0 degrees Celsius. 400 00:24:28,560 --> 00:24:34,250 And the final temperature, instead 401 00:24:34,250 --> 00:24:37,590 of being a constant temperature, we 402 00:24:37,590 --> 00:24:41,290 were heating this guy up from the bottom. 403 00:24:41,290 --> 00:24:48,340 And it was hotter on the bottom than on the top. 404 00:24:48,340 --> 00:24:50,370 And the final temperature was given 405 00:24:50,370 --> 00:24:57,880 by the formula T is equal to 100 minus 30 times the height y. 406 00:24:57,880 --> 00:25:04,090 So at y = 0, at the bottom, it's 100. 407 00:25:04,090 --> 00:25:10,730 And at the top, T = 70 degrees. 408 00:25:10,730 --> 00:25:14,610 OK, so this is the final configuration 409 00:25:14,610 --> 00:25:16,020 for the temperature. 410 00:25:16,020 --> 00:25:33,100 And the question was how much energy do we need. 411 00:25:33,100 --> 00:25:35,210 So, the first observation here, and this 412 00:25:35,210 --> 00:25:39,500 is the reason for giving this example, 413 00:25:39,500 --> 00:25:42,630 is that it's important to realize 414 00:25:42,630 --> 00:25:54,700 that you want to use the method of disks in this case. 415 00:25:54,700 --> 00:25:57,630 The reason-- So it doesn't have to do with, 416 00:25:57,630 --> 00:26:00,070 you shouldn't think of the disks first. 417 00:26:00,070 --> 00:26:05,010 But what you should think of is the horizontal. 418 00:26:05,010 --> 00:26:10,840 We must use horizontals because T is constant on horizontals. 419 00:26:10,840 --> 00:26:12,740 It's not constant on verticals. 420 00:26:12,740 --> 00:26:16,130 If we set things up with shells, as we did last time, 421 00:26:16,130 --> 00:26:18,790 to compute the volume of this, then T 422 00:26:18,790 --> 00:26:21,760 will vary along the shell. 423 00:26:21,760 --> 00:26:24,130 And we will still have an averaging problem, 424 00:26:24,130 --> 00:26:26,730 an integral problem along the vertical portion. 425 00:26:26,730 --> 00:26:30,340 But if we do it this way, T is constant on this whole level 426 00:26:30,340 --> 00:26:31,770 here. 427 00:26:31,770 --> 00:26:34,300 And so there's no more calculus involved 428 00:26:34,300 --> 00:26:36,760 in calculating what the contribution is of any given 429 00:26:36,760 --> 00:26:39,840 level. 430 00:26:39,840 --> 00:26:49,620 So T is constant on horizontals. 431 00:26:49,620 --> 00:26:52,620 Actually, in disguise, this is that same trick 432 00:26:52,620 --> 00:26:53,370 that we have here. 433 00:26:53,370 --> 00:26:55,596 We can factor constants out of integrals. 434 00:26:55,596 --> 00:26:56,970 You could view it as an integral, 435 00:26:56,970 --> 00:27:03,480 but the point is that it's more elementary than that. 436 00:27:03,480 --> 00:27:06,430 Now I have to set it up for you. 437 00:27:06,430 --> 00:27:08,920 And in order to do that, I need to remember 438 00:27:08,920 --> 00:27:10,470 what the equation is. 439 00:27:10,470 --> 00:27:12,960 Which is y = x^2. 440 00:27:12,960 --> 00:27:18,500 And the formula for the total amount of energy 441 00:27:18,500 --> 00:27:25,000 is going to be volume times the number of degrees. 442 00:27:25,000 --> 00:27:31,480 That's going to be equal to the energy that we need here. 443 00:27:31,480 --> 00:27:33,250 And so let's add it up. 444 00:27:33,250 --> 00:27:38,540 It's the integral from 0 to 1, and this is with respect to y. 445 00:27:38,540 --> 00:27:41,300 So the y level goes from 0 to 1. 446 00:27:41,300 --> 00:27:47,310 This top level is y = 1, this bottom level is y = 0. 447 00:27:47,310 --> 00:27:55,090 And the disk that we get, this is the point (x, y) here, 448 00:27:55,090 --> 00:27:56,220 is rotated around. 449 00:27:56,220 --> 00:28:01,230 And its radius is x. 450 00:28:01,230 --> 00:28:09,480 So the thickness is dy, and the area of the disk is pi x^2. 451 00:28:09,480 --> 00:28:11,430 And the thing that we're averaging 452 00:28:11,430 --> 00:28:14,550 is T. Well, we're not yet averaging, 453 00:28:14,550 --> 00:28:16,010 we're just integrating it. 454 00:28:16,010 --> 00:28:24,170 We're just adding up the total. 455 00:28:24,170 --> 00:28:29,080 Now I'm just going to plug in the various values for this. 456 00:28:29,080 --> 00:28:36,580 And what I'm going to get is T, again, is 100 - 30y. 457 00:28:36,580 --> 00:28:40,960 And this radius is measured up to this very end. 458 00:28:40,960 --> 00:28:42,440 So x^2 = y. 459 00:28:42,440 --> 00:28:45,410 So this is pi y dy. 460 00:28:45,410 --> 00:28:47,800 And this is the integral that we'll be able to evaluate. 461 00:28:47,800 --> 00:28:48,660 Yeah, question. 462 00:28:48,660 --> 00:28:50,250 STUDENT: [INAUDIBLE] 463 00:28:50,250 --> 00:29:00,070 PROFESSOR: All right. 464 00:29:00,070 --> 00:29:05,710 Well, let's carry this out. 465 00:29:05,710 --> 00:29:09,160 Let's finish off the calculation here. 466 00:29:09,160 --> 00:29:10,230 Let's see. 467 00:29:10,230 --> 00:29:16,380 This is equal to, what it it equal to? 468 00:29:16,380 --> 00:29:19,580 Well, I'll put it over here. 469 00:29:19,580 --> 00:29:26,590 It's equal to 50 pi y ^2 minus-- right, 470 00:29:26,590 --> 00:29:30,010 because this is 100 pi y, and then there's a 30, 471 00:29:30,010 --> 00:29:37,480 this is 100 pi y - 30 pi y^2, and I have to take 472 00:29:37,480 --> 00:29:38,780 the antiderivative of that. 473 00:29:38,780 --> 00:29:48,120 So I get 50 pi y^2, and I get 10 pi y^3, evaluated at 0 and 1. 474 00:29:48,120 --> 00:29:57,930 And that is 40 pi. 475 00:29:57,930 --> 00:30:03,510 Now, I spent a tremendous amount of time 476 00:30:03,510 --> 00:30:07,900 last time focusing on units. 477 00:30:07,900 --> 00:30:11,250 Because I want to tell you how to get a realistic number out 478 00:30:11,250 --> 00:30:12,080 of this. 479 00:30:12,080 --> 00:30:16,400 And there's a subtle point here that I pointed out 480 00:30:16,400 --> 00:30:19,160 last time that had to do with changing meters to centimeters. 481 00:30:19,160 --> 00:30:22,570 I claim that I've treated those correctly. 482 00:30:22,570 --> 00:30:25,970 So, what we have here is that the answer 483 00:30:25,970 --> 00:30:34,800 is in degrees, that is Celsius, times cubic meters. 484 00:30:34,800 --> 00:30:36,570 These are the correct units. 485 00:30:36,570 --> 00:30:43,160 And now, I can translate this into-- Celsius is spelled 486 00:30:43,160 --> 00:30:44,896 with a C. That's interesting. 487 00:30:44,896 --> 00:30:46,630 Celsius. 488 00:30:46,630 --> 00:30:50,850 I can translate this into units that you're more familiar with. 489 00:30:50,850 --> 00:30:57,620 So let's try 40 pi degrees times m^3, 490 00:30:57,620 --> 00:30:59,710 and then do the conversion factors. 491 00:30:59,710 --> 00:31:08,840 First of all there's one calorie per degree times a milliliter. 492 00:31:08,840 --> 00:31:11,150 That's one conversion. 493 00:31:11,150 --> 00:31:14,600 And then let's see. 494 00:31:14,600 --> 00:31:18,580 I'm going to have to translate from centimeters so I have here 495 00:31:18,580 --> 00:31:22,350 (100 cm / m)^3. 496 00:31:25,110 --> 00:31:30,550 So these are the two conversion factors that I need. 497 00:31:30,550 --> 00:31:38,690 And so, I get 40 pi 10^6, that's 100^3. 498 00:31:38,690 --> 00:31:46,550 And this is in calories. 499 00:31:46,550 --> 00:31:48,470 So how much is this? 500 00:31:48,470 --> 00:31:51,180 Well, it's a little better, maybe, 501 00:31:51,180 --> 00:31:57,777 to do it in 40 pi * 1,000 kilocalories, 502 00:31:57,777 --> 00:31:59,610 because these are the ones that you actually 503 00:31:59,610 --> 00:32:05,710 see on your nutrition labels of foods. 504 00:32:05,710 --> 00:32:12,600 And so this number is around 125 or so. 505 00:32:12,600 --> 00:32:15,960 Let's see, is that about right? 506 00:32:15,960 --> 00:32:17,930 Let's make sure I've got these numbers right. 507 00:32:17,930 --> 00:32:20,550 Yeah, this is about 125. 508 00:32:20,550 --> 00:32:22,780 40 times pi. 509 00:32:22,780 --> 00:32:32,220 And so one candy bar-- This is a Halloween example, so. 510 00:32:32,220 --> 00:32:38,460 One candy bar is about 250 kilocalories. 511 00:32:38,460 --> 00:32:44,620 So this is half a candy bar. 512 00:32:44,620 --> 00:32:57,230 So the answer to our question is that it takes 500 candy bars 513 00:32:57,230 --> 00:33:02,900 to heat up this thing. 514 00:33:02,900 --> 00:33:07,570 OK, so that's our example. 515 00:33:07,570 --> 00:33:08,630 Now, yeah. 516 00:33:08,630 --> 00:33:09,160 Question. 517 00:33:09,160 --> 00:33:13,710 STUDENT: [INAUDIBLE] 518 00:33:13,710 --> 00:33:16,980 PROFESSOR: What does the integral give us? 519 00:33:16,980 --> 00:33:22,971 This integral is-- the integral represents the following 520 00:33:22,971 --> 00:33:23,470 things. 521 00:33:23,470 --> 00:33:26,330 So the question is, what does this integral give us. 522 00:33:26,330 --> 00:33:27,640 So here's the integral. 523 00:33:27,640 --> 00:33:30,640 Here it is, rewritten so that it can be calculated. 524 00:33:30,640 --> 00:33:34,170 And what this integral is giving us is the following thing. 525 00:33:34,170 --> 00:33:36,160 You have to imagine the following idea. 526 00:33:36,160 --> 00:33:39,220 You've got a little chunk of water in here. 527 00:33:39,220 --> 00:33:41,690 And you're going to raise is from 0 degrees 528 00:33:41,690 --> 00:33:46,840 all the way up to whatever the target temperature is. 529 00:33:46,840 --> 00:33:50,810 And so that little milliliter of water, if you like, 530 00:33:50,810 --> 00:33:53,700 has to be raised from 0 to some number which 531 00:33:53,700 --> 00:33:56,730 is a function of the height. 532 00:33:56,730 --> 00:33:59,950 It's something between 70 and 100 degrees. 533 00:33:59,950 --> 00:34:02,790 And the one right above it also has 534 00:34:02,790 --> 00:34:05,240 to be raised to a temperature, although a slightly 535 00:34:05,240 --> 00:34:06,280 different temperature. 536 00:34:06,280 --> 00:34:08,840 And what we're doing with the integral is we're adding up 537 00:34:08,840 --> 00:34:15,647 all of those degrees and the calorie represents how much it 538 00:34:15,647 --> 00:34:17,230 takes, one calorie represents how much 539 00:34:17,230 --> 00:34:21,980 it takes to raise by 1 degree 1 milliliter of water. 540 00:34:21,980 --> 00:34:26,620 One cubic centimeter of water. 541 00:34:26,620 --> 00:34:31,610 That's the definition of a calorie. 542 00:34:31,610 --> 00:34:32,990 And we're adding it up. 543 00:34:32,990 --> 00:34:35,430 So in other words, each of these cubes is one thing. 544 00:34:35,430 --> 00:34:38,010 And now we have to add it up over this massive thing, which 545 00:34:38,010 --> 00:34:40,150 is 1600 liters. 546 00:34:40,150 --> 00:34:42,140 And we have a lot of different little cubes. 547 00:34:42,140 --> 00:34:43,420 And that's what we did. 548 00:34:43,420 --> 00:34:45,320 When we glommed them all together. 549 00:34:45,320 --> 00:34:48,800 That's what the integral is doing for us. 550 00:34:48,800 --> 00:34:54,200 Other questions. 551 00:34:54,200 --> 00:34:57,400 Now I want to connect this with weighted averages 552 00:34:57,400 --> 00:34:58,800 before we go on. 553 00:34:58,800 --> 00:35:03,640 Because that was the reason why I did weighted averages first. 554 00:35:03,640 --> 00:35:14,810 I'm going to compute also the average final temperature. 555 00:35:14,810 --> 00:35:17,502 So, final because this is the interesting one, 556 00:35:17,502 --> 00:35:19,460 the average starting temperature's very boring, 557 00:35:19,460 --> 00:35:21,030 it's 0. 558 00:35:21,030 --> 00:35:26,090 The average final temperature is-- individually 559 00:35:26,090 --> 00:35:27,840 the temperatures are different. 560 00:35:27,840 --> 00:35:29,770 And the answer here is it's the integral 561 00:35:29,770 --> 00:35:37,930 from 0 to 1 of T pi y dy divided by the integral 562 00:35:37,930 --> 00:35:42,040 from 0 to 1 of pi y dy. 563 00:35:42,040 --> 00:35:43,970 So this is the total temperature, 564 00:35:43,970 --> 00:35:46,820 weighted appropriately to the volume of water that's 565 00:35:46,820 --> 00:35:49,180 involved at that temperature, divided 566 00:35:49,180 --> 00:35:52,880 by the total volume of water. 567 00:35:52,880 --> 00:35:55,130 And we computed these two numbers. 568 00:35:55,130 --> 00:35:58,350 The number in the numerator is what we call 40 pi. 569 00:35:58,350 --> 00:36:00,420 And the number in the denominator, actually this 570 00:36:00,420 --> 00:36:02,870 is easier than what we did last time with shells; 571 00:36:02,870 --> 00:36:04,800 you can just look at this and see that it's 572 00:36:04,800 --> 00:36:06,160 the area under a triangle. 573 00:36:06,160 --> 00:36:08,500 It's pi / 2. 574 00:36:08,500 --> 00:36:11,760 And so the answer here is 80 degrees. 575 00:36:11,760 --> 00:36:14,910 This is the average temperature. 576 00:36:14,910 --> 00:36:17,670 Note that this is a weighted average. 577 00:36:17,670 --> 00:36:22,810 The weighting here is different according to the height. 578 00:36:22,810 --> 00:36:28,190 The weighting factor is pi y. 579 00:36:28,190 --> 00:36:30,010 That's the weighting factor. 580 00:36:30,010 --> 00:36:32,270 And that's not surprising. 581 00:36:32,270 --> 00:36:35,630 When y is small, there's less volume down here. 582 00:36:35,630 --> 00:36:38,510 Up above, those are more important volumes, 583 00:36:38,510 --> 00:36:41,720 because there's more water up at the top of the cauldron 584 00:36:41,720 --> 00:36:43,960 than there is down at the bottom of the cauldron. 585 00:36:43,960 --> 00:36:46,480 If you compare this to the ordinary average, 586 00:36:46,480 --> 00:36:48,570 if you take the maximum temperature 587 00:36:48,570 --> 00:36:52,630 plus the minimum temperature, divided by 2, 588 00:36:52,630 --> 00:36:56,470 that would be (100 + 70) / 2. 589 00:36:56,470 --> 00:36:59,830 You would get 85 degrees. 590 00:36:59,830 --> 00:37:01,610 And that's bigger. 591 00:37:01,610 --> 00:37:02,440 Why? 592 00:37:02,440 --> 00:37:05,030 Because the cooler water is on top. 593 00:37:05,030 --> 00:37:08,530 And the actual average, the correct weighted average, 594 00:37:08,530 --> 00:37:11,800 is lower than this fake average. 595 00:37:11,800 --> 00:37:15,520 Which is not the true average in this context. 596 00:37:15,520 --> 00:37:17,780 All right so the weighting is that the thing 597 00:37:17,780 --> 00:37:33,160 is getting fatter near the top. 598 00:37:33,160 --> 00:37:38,980 So now I'm going to do another example of weighted average. 599 00:37:38,980 --> 00:37:46,020 And this example is also very much worth your while. 600 00:37:46,020 --> 00:37:48,940 It's the other incredibly important one 601 00:37:48,940 --> 00:37:51,960 in interpreting integrals. 602 00:37:51,960 --> 00:37:56,790 And it's a very, very simple example of a function f. 603 00:37:56,790 --> 00:37:59,240 The weightings will be different, 604 00:37:59,240 --> 00:38:03,320 but the function f, will be of a very particular kind. 605 00:38:03,320 --> 00:38:07,450 Namely, the function f will be practically a constant. 606 00:38:07,450 --> 00:38:08,480 But not quite. 607 00:38:08,480 --> 00:38:10,090 It's going to be a constant on one 608 00:38:10,090 --> 00:38:13,880 interval, and then 0 on the rest. 609 00:38:13,880 --> 00:38:16,950 So we'll do those weighted averages now. 610 00:38:16,950 --> 00:38:34,900 And this subject is called probability. 611 00:38:34,900 --> 00:38:39,130 In probability, what we do, so I'm just 612 00:38:39,130 --> 00:38:43,910 going to give some examples here. 613 00:38:43,910 --> 00:38:54,620 I'm going to pick a point in quotation marks - at random. 614 00:38:54,620 --> 00:39:00,900 In the region y < x < 1 - x^2. 615 00:39:00,900 --> 00:39:05,060 That's this shape here. 616 00:39:05,060 --> 00:39:08,770 Well, let's draw it right down here. 617 00:39:08,770 --> 00:39:09,310 For now. 618 00:39:09,310 --> 00:39:10,380 So, somewhere in here. 619 00:39:10,380 --> 00:39:13,310 Some point, (x, y). 620 00:39:13,310 --> 00:39:18,335 And then I need to tell you, according to what 621 00:39:18,335 --> 00:39:20,330 this random really means. 622 00:39:20,330 --> 00:39:31,780 This is proportional to area, if you like. 623 00:39:31,780 --> 00:39:33,990 So area inside of this section. 624 00:39:33,990 --> 00:39:37,950 And then the question that we're going to answer right now is, 625 00:39:37,950 --> 00:39:47,040 what is the chance that - or, it's usually called probability 626 00:39:47,040 --> 00:39:56,980 - that x > 1/2. 627 00:39:56,980 --> 00:40:03,620 Let me show you what's going on here. 628 00:40:03,620 --> 00:40:08,540 And this is always the case with things in probability. 629 00:40:08,540 --> 00:40:10,290 So, first of all, we have a name for this. 630 00:40:10,290 --> 00:40:12,510 This is called the probability that x > 1/2. 631 00:40:16,220 --> 00:40:21,060 And so that's what it's called in our notation here. 632 00:40:21,060 --> 00:40:26,920 And what it is, is the probability 633 00:40:26,920 --> 00:40:32,760 is always equal to the part divided by the whole. 634 00:40:32,760 --> 00:40:36,846 It's a ratio just like the one over there. 635 00:40:36,846 --> 00:40:38,720 And which is the part and which is the whole? 636 00:40:38,720 --> 00:40:43,470 Well, in this picture, the whole is the whole parabola. 637 00:40:43,470 --> 00:40:48,910 And the part is the section x > 1/2. 638 00:40:48,910 --> 00:41:00,460 And it's just the ratio of those two areas. 639 00:41:00,460 --> 00:41:01,970 Let's write that down. 640 00:41:01,970 --> 00:41:08,680 That's the integral from 1/2 to 1 of (1 - x^2) dx, 641 00:41:08,680 --> 00:41:16,000 divided by the integral from -1 to 1, (1 - x^2) dx. 642 00:41:16,000 --> 00:41:23,350 And again, the weighting factor here is 1 - x^2. 643 00:41:23,350 --> 00:41:25,990 And to be a little bit more specific here, 644 00:41:25,990 --> 00:41:33,390 the starting point a = -1 and the endpoint is +1. 645 00:41:33,390 --> 00:41:37,540 So this is P(x < 1/2). 646 00:41:37,540 --> 00:41:46,970 And if you work it out, it turns out to be 5/18, we won't do it. 647 00:41:46,970 --> 00:41:47,470 Yeah. 648 00:41:47,470 --> 00:42:21,004 STUDENT: [INAUDIBLE] 649 00:42:21,004 --> 00:42:23,170 PROFESSOR: What we're trying to do with probability. 650 00:42:23,170 --> 00:42:26,760 So I can't repeat your question. 651 00:42:26,760 --> 00:42:29,670 But I can try to say-- because it was 652 00:42:29,670 --> 00:42:31,580 a little bit too complicated. 653 00:42:31,580 --> 00:42:35,020 But it was not correct, OK. 654 00:42:35,020 --> 00:42:38,720 What we're taking is, we have two possible things 655 00:42:38,720 --> 00:42:39,940 that could happen. 656 00:42:39,940 --> 00:42:42,640 Either, let's put it this way. 657 00:42:42,640 --> 00:42:43,910 Let's make it a gamble. 658 00:42:43,910 --> 00:42:47,150 Somebody picks a point in here at random. 659 00:42:47,150 --> 00:42:53,040 And we're trying to figure out what 660 00:42:53,040 --> 00:42:54,527 your chances are of winning. 661 00:42:54,527 --> 00:42:57,110 In other words, the chances the person picks something in here 662 00:42:57,110 --> 00:42:59,646 versus something in there. 663 00:42:59,646 --> 00:43:02,020 And the interesting thing is, so what percent of the time 664 00:43:02,020 --> 00:43:04,220 do you win. 665 00:43:04,220 --> 00:43:06,384 The answer is it's some fraction of 1. 666 00:43:06,384 --> 00:43:07,800 And in order to figure that out, I 667 00:43:07,800 --> 00:43:11,900 have to figure out the total area here. 668 00:43:11,900 --> 00:43:16,020 Versus the total of the entire, all the way from -1 to 1, 669 00:43:16,020 --> 00:43:18,190 the beginning to the end. 670 00:43:18,190 --> 00:43:22,770 So in the numerator, I put success, and in the denominator 671 00:43:22,770 --> 00:43:25,250 I put all possibilities. 672 00:43:25,250 --> 00:43:26,720 So that-- Right? 673 00:43:26,720 --> 00:43:29,039 STUDENT: [INAUDIBLE] 674 00:43:29,039 --> 00:43:31,080 PROFESSOR: And that's the interpretation of this. 675 00:43:31,080 --> 00:43:33,060 So maybe I didn't understand your question. 676 00:43:33,060 --> 00:43:37,900 STUDENT: [INAUDIBLE] 677 00:43:37,900 --> 00:43:40,130 PROFESSOR: Ah, why is 1 - x^2. the weighting factor. 678 00:43:40,130 --> 00:43:44,550 That has to do with how you compute areas under curves. 679 00:43:44,550 --> 00:43:49,040 The curve here is y = 1 - x^2. 680 00:43:49,040 --> 00:43:51,810 And so, in order to calculate how much area is between 1/2 681 00:43:51,810 --> 00:43:52,940 and 1, I have to integrate. 682 00:43:52,940 --> 00:43:54,440 That's the interpretation of this. 683 00:43:54,440 --> 00:43:56,450 This is the area under that curve. 684 00:43:56,450 --> 00:43:57,220 This integral. 685 00:43:57,220 --> 00:44:01,200 And the denominator's the area under the whole thing. 686 00:44:01,200 --> 00:44:02,052 OK, yeah. 687 00:44:02,052 --> 00:44:02,760 Another question. 688 00:44:02,760 --> 00:44:06,060 STUDENT: [INAUDIBLE] 689 00:44:06,060 --> 00:44:08,100 PROFESSOR: Ah. 690 00:44:08,100 --> 00:44:09,460 Yikes. 691 00:44:09,460 --> 00:44:12,531 It was supposed to be the same question as over here. 692 00:44:12,531 --> 00:44:13,030 Thank you. 693 00:44:13,030 --> 00:44:19,700 STUDENT: [INAUDIBLE] PROFESSOR: This has something 694 00:44:19,700 --> 00:44:21,420 to do with weighting factors. 695 00:44:21,420 --> 00:44:25,410 Here's the weight factor. 696 00:44:25,410 --> 00:44:27,720 Well, it's the relative importance 697 00:44:27,720 --> 00:44:29,680 from the point of view of this probability 698 00:44:29,680 --> 00:44:33,160 of these places versus those. 699 00:44:33,160 --> 00:44:36,940 That is, so this is a weighting factor because it's telling me 700 00:44:36,940 --> 00:44:45,560 that in some sense this number 5/18-- actually that makes me 701 00:44:45,560 --> 00:44:48,420 think that this number is probably wrong. 702 00:44:48,420 --> 00:44:53,170 Well, I'll let you calculate it out. 703 00:44:53,170 --> 00:44:55,380 It looks like it should be less than 1/4 here, 704 00:44:55,380 --> 00:44:57,760 because this is 1/4 of the total distance 705 00:44:57,760 --> 00:44:59,370 and there's a little less in here 706 00:44:59,370 --> 00:45:00,812 than there is in the middle. 707 00:45:00,812 --> 00:45:03,270 So in fact it probably should be less than 1/4, the answer. 708 00:45:03,270 --> 00:45:09,070 STUDENT: [INAUDIBLE] 709 00:45:09,070 --> 00:45:11,070 PROFESSOR: The equation of the curve is 1 - x^2. 710 00:45:13,406 --> 00:45:14,780 The reason why it's the weighting 711 00:45:14,780 --> 00:45:17,490 factor is that we're interpreting-- The question 712 00:45:17,490 --> 00:45:20,120 has to do with the area under that curve. 713 00:45:20,120 --> 00:45:24,020 And so, this is showing us how much is relatively important 714 00:45:24,020 --> 00:45:25,200 versus how much is not. 715 00:45:25,200 --> 00:45:27,369 This is-- These parts are relatively important, 716 00:45:27,369 --> 00:45:28,660 these parts are less important. 717 00:45:28,660 --> 00:45:29,960 According to area. 718 00:45:29,960 --> 00:45:31,730 Because we've said that area is the way 719 00:45:31,730 --> 00:45:35,920 we're making the choice. 720 00:45:35,920 --> 00:45:38,420 So I don't have quite enough time 721 00:45:38,420 --> 00:45:43,132 to tell you about my next example. 722 00:45:43,132 --> 00:45:44,590 Instead, I'm just going to tell you 723 00:45:44,590 --> 00:45:46,820 what the general formula is. 724 00:45:46,820 --> 00:45:48,920 And we'll do our example next time. 725 00:45:48,920 --> 00:45:51,790 I'll tell you what it's going to be. 726 00:45:51,790 --> 00:46:04,190 So here's the general formula for probability here. 727 00:46:04,190 --> 00:46:12,020 We're going to imagine that we have a total range which 728 00:46:12,020 --> 00:46:13,940 is maybe going from a to b, and we 729 00:46:13,940 --> 00:46:18,320 have some intermediate values x_1 and x_2, 730 00:46:18,320 --> 00:46:23,020 and then we're going to try to compute the probability 731 00:46:23,020 --> 00:46:28,150 that some variable that we picked at random 732 00:46:28,150 --> 00:46:31,560 occurs between x_1 and x_2. 733 00:46:31,560 --> 00:46:36,490 And by definition, we're saying that it's an integral. 734 00:46:36,490 --> 00:46:41,280 It's the integral from x_1 to x_2 of the weight dx, 735 00:46:41,280 --> 00:46:46,710 divided by the integral all the way from a to b. 736 00:46:46,710 --> 00:46:47,390 Of the weight. 737 00:46:47,390 --> 00:46:55,900 So, again, this is the part divided by the whole. 738 00:46:55,900 --> 00:46:59,890 And the relationship between this and the weighted average 739 00:46:59,890 --> 00:47:02,950 that we had earlier was that the function f f(x) 740 00:47:02,950 --> 00:47:04,450 is kind of a strange function. 741 00:47:04,450 --> 00:47:06,410 It's 0 and 1. 742 00:47:06,410 --> 00:47:09,200 It's just-- The picture, if you like, 743 00:47:09,200 --> 00:47:11,860 is that you have this weighting factor. 744 00:47:11,860 --> 00:47:14,180 And it's going from a to b. 745 00:47:14,180 --> 00:47:16,649 But then in between there, we have the part 746 00:47:16,649 --> 00:47:17,690 that we're interested in. 747 00:47:17,690 --> 00:47:20,130 Which is between x_1 and x_2. 748 00:47:20,130 --> 00:47:23,910 And it's the ratio of this inner part to the whole thing 749 00:47:23,910 --> 00:47:34,650 that we're interested in. 750 00:47:34,650 --> 00:47:39,549 Tomorrow I'm going to try to do a realistic example. 751 00:47:39,549 --> 00:47:41,090 And I'm going to tell you what it is, 752 00:47:41,090 --> 00:47:43,620 but we'll take it up tomorrow. 753 00:47:43,620 --> 00:47:45,460 I told you it was going to be tomorrow, 754 00:47:45,460 --> 00:47:46,835 but we still have a whole minute, 755 00:47:46,835 --> 00:47:49,270 so I'm going to tell you what the problem is. 756 00:47:49,270 --> 00:47:53,660 So this is going to be a target practice problem. 757 00:47:53,660 --> 00:47:55,210 You have a target here and you're 758 00:47:55,210 --> 00:48:00,810 throwing darts at this target. 759 00:48:00,810 --> 00:48:05,960 And so you're throwing darts at this target. 760 00:48:05,960 --> 00:48:13,020 And somebody is standing next to the dartboard. 761 00:48:13,020 --> 00:48:18,160 Your little brother is standing next to the dartboard here. 762 00:48:18,160 --> 00:48:21,830 And the question is, how likely you 763 00:48:21,830 --> 00:48:24,320 are to hit your little brother. 764 00:48:24,320 --> 00:48:26,834 So this will, let's see. 765 00:48:26,834 --> 00:48:28,500 You'll see whether you like that or not. 766 00:48:28,500 --> 00:48:29,960 Actually, I was the little brother. 767 00:48:29,960 --> 00:48:31,710 So, I don't know which way you want to go. 768 00:48:31,710 --> 00:48:32,680 We'll go either way. 769 00:48:32,680 --> 00:48:35,240 We'll find out next time.