WEBVTT 00:00:00.000 --> 00:00:02.230 The following content is provided under a Creative 00:00:02.230 --> 00:00:02.896 Commons license. 00:00:02.896 --> 00:00:06.110 Your support will help MIT OpenCourseWare 00:00:06.110 --> 00:00:09.950 continue to offer high quality educational resources for free. 00:00:09.950 --> 00:00:12.650 To make a donation, or to view additional materials 00:00:12.650 --> 00:00:16.160 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.160 --> 00:00:21.670 at ocw.mit.edu. 00:00:21.670 --> 00:00:24.040 PROFESSOR: Today we're going to hold off just 00:00:24.040 --> 00:00:27.250 a little bit on boiling water. 00:00:27.250 --> 00:00:31.920 And talk about another application of integrals, 00:00:31.920 --> 00:00:36.030 and we'll get to the witches' cauldron in the middle. 00:00:36.030 --> 00:00:45.230 The thing that I'd like to start with today is average value. 00:00:45.230 --> 00:00:47.877 This is something that I mentioned a little bit earlier, 00:00:47.877 --> 00:00:49.460 and there was a misprint on the board, 00:00:49.460 --> 00:00:54.980 so I want to make sure that we have the definitions straight. 00:00:54.980 --> 00:00:56.770 And also the reasoning straight. 00:00:56.770 --> 00:00:58.970 This is one of the most important applications 00:00:58.970 --> 00:01:03.840 of integrals, one of the most important examples. 00:01:03.840 --> 00:01:08.300 If you take the average of a bunch of numbers, 00:01:08.300 --> 00:01:10.420 that looks like this. 00:01:10.420 --> 00:01:15.500 And we can view this as sampling a function, 00:01:15.500 --> 00:01:17.570 as we would with the Riemann sum. 00:01:17.570 --> 00:01:23.180 And what I said last week was that this 00:01:23.180 --> 00:01:28.020 tends to this expression here, which is 00:01:28.020 --> 00:01:32.960 called the continuous average. 00:01:32.960 --> 00:01:43.550 So this guy is the continuous average. 00:01:43.550 --> 00:01:49.270 Or just the average of f. 00:01:49.270 --> 00:01:53.710 And I want to explain that, just to make sure that we're all 00:01:53.710 --> 00:01:56.380 on the same page. 00:01:56.380 --> 00:01:58.930 In general, if you have a function 00:01:58.930 --> 00:02:01.220 and you want to interpret the integral, 00:02:01.220 --> 00:02:04.270 our first interpretation was that it's something 00:02:04.270 --> 00:02:09.860 like the area under the curve. 00:02:09.860 --> 00:02:15.440 But average value is another reasonable interpretation. 00:02:15.440 --> 00:02:19.230 Namely, if you take equally spaced 00:02:19.230 --> 00:02:24.590 points here, starting with x_0, x_1, x_2, 00:02:24.590 --> 00:02:29.940 all the way up to x_n, which is the left point b, 00:02:29.940 --> 00:02:32.320 and then we have values y_1, which 00:02:32.320 --> 00:02:38.750 is f(x_1); y_2, which is f(x_2); all the way up 00:02:38.750 --> 00:02:40.130 to y_n, which is f(x_n). 00:02:43.040 --> 00:02:47.260 And again, the spacing here that we're talking about is (b-a) / 00:02:47.260 --> 00:02:47.760 n. 00:02:50.690 --> 00:02:52.540 So remember that spacing, that's going to be 00:02:52.540 --> 00:02:57.100 the connection that we'll draw. 00:02:57.100 --> 00:03:08.070 Then the Riemann sum is y_1 through y_n, the sum of y_1 00:03:08.070 --> 00:03:12.080 through y_n, multiplied by delta x. 00:03:12.080 --> 00:03:16.370 And that's what tends, as delta x goes to 0, to the integral. 00:03:23.400 --> 00:03:26.060 The only change in point of view if I 00:03:26.060 --> 00:03:28.450 want to write this limiting property, which 00:03:28.450 --> 00:03:33.780 is right above here, the only change between here 00:03:33.780 --> 00:03:36.810 and here is that I want to divide by the length 00:03:36.810 --> 00:03:38.850 of the interval. b - a. 00:03:38.850 --> 00:03:42.000 So I will divide by b - a here. 00:03:42.000 --> 00:03:48.230 And divide by b - a over here. 00:03:48.230 --> 00:03:53.510 And then I'll just check what this thing actually is. 00:03:53.510 --> 00:03:57.560 Delta x / (b-a), what is that factor? 00:03:57.560 --> 00:04:01.650 Well, if we look over here to what delta x is, if you 00:04:01.650 --> 00:04:07.040 divide by b - a, it's 1 / n. 00:04:07.040 --> 00:04:10.210 So the factor delta x / (b-a) is 1 / n. 00:04:10.210 --> 00:04:12.660 That's what I put over here, the sum of y_1 00:04:12.660 --> 00:04:15.210 through y_n divided by n. 00:04:15.210 --> 00:04:22.760 And as this tends to 0, it's the same as n going to infinity. 00:04:22.760 --> 00:04:25.610 Those are the same things. 00:04:25.610 --> 00:04:27.170 The average value and the integral 00:04:27.170 --> 00:04:28.460 are very closely related. 00:04:28.460 --> 00:04:30.790 There's only this difference that we're dividing 00:04:30.790 --> 00:04:36.440 by the length of the interval. 00:04:36.440 --> 00:04:38.120 I want to give an example which is 00:04:38.120 --> 00:04:40.330 an incredibly simpleminded one, but it'll 00:04:40.330 --> 00:04:41.800 come into play later on. 00:04:41.800 --> 00:04:47.040 So let's take the example of a constant. 00:04:47.040 --> 00:04:50.730 And this is, I hope-- will make you slightly less confused 00:04:50.730 --> 00:04:52.540 about what I just wrote. 00:04:52.540 --> 00:04:54.220 As well as making you think that this 00:04:54.220 --> 00:04:57.370 is as simpleminded and reasonable as it should be. 00:04:57.370 --> 00:05:00.310 If I check what the average value of this constant is, 00:05:00.310 --> 00:05:04.680 it's given by this relatively complicated formula here. 00:05:04.680 --> 00:05:07.170 That is, I have to integrate the function c. 00:05:07.170 --> 00:05:09.570 Well, it's just the constant c. 00:05:09.570 --> 00:05:12.320 And however you do this, as an antiderivative or as thinking 00:05:12.320 --> 00:05:14.730 of it as a rectangle, the answer that you're going to get 00:05:14.730 --> 00:05:18.310 is c here. 00:05:18.310 --> 00:05:19.490 So work that out. 00:05:19.490 --> 00:05:20.730 The answer is c. 00:05:20.730 --> 00:05:23.780 And so the fact that the average of c 00:05:23.780 --> 00:05:28.480 is equal to c, which had better be the case for averages, 00:05:28.480 --> 00:05:31.760 explains the denominator. 00:05:31.760 --> 00:05:36.690 Explains the 1 / (b-a) there. 00:05:36.690 --> 00:05:39.107 That's cooked up exactly so that the average of a constant 00:05:39.107 --> 00:05:40.273 is what it's supposed to be. 00:05:40.273 --> 00:05:42.250 Otherwise we have the wrong normalizing factor. 00:05:42.250 --> 00:05:45.450 We've clearly got a piece of nonsense on our hands. 00:05:45.450 --> 00:05:50.470 And incidentally, it also explains the 1/n 00:05:50.470 --> 00:05:53.460 in the very first formula that I wrote down. 00:05:53.460 --> 00:05:55.360 The reason why this is called the average, 00:05:55.360 --> 00:05:57.580 or one reason why it's the right thing, 00:05:57.580 --> 00:06:00.177 is that if you took the same constant c, for y 00:06:00.177 --> 00:06:02.510 all the way across there n times, if you divide it by n, 00:06:02.510 --> 00:06:03.930 you get back c. 00:06:03.930 --> 00:06:05.490 That's what we mean by average value 00:06:05.490 --> 00:06:11.370 and that's why the n is there. 00:06:11.370 --> 00:06:14.960 So that was an easy example. 00:06:14.960 --> 00:06:17.610 Now none of the examples that we are going to give 00:06:17.610 --> 00:06:20.000 are going to be all that complicated. 00:06:20.000 --> 00:06:23.460 But they will get sort of steadily more sophisticated. 00:06:23.460 --> 00:06:32.610 The second example is going to be the average height 00:06:32.610 --> 00:06:47.860 of a point on a semicircle. 00:06:47.860 --> 00:06:52.800 And maybe I'll draw a picture of the semicircle first here. 00:06:52.800 --> 00:06:57.250 And we'll just make it the standard circle, the unit 00:06:57.250 --> 00:06:58.840 circle. 00:06:58.840 --> 00:07:04.890 So maybe I should have called it a unit semicircle. 00:07:04.890 --> 00:07:07.680 This is the point negative 1, this is the point 1. 00:07:07.680 --> 00:07:09.890 And we're picking a point over here 00:07:09.890 --> 00:07:15.540 and we're going to take the typical, or the average, height 00:07:15.540 --> 00:07:17.210 here. 00:07:17.210 --> 00:07:18.550 Integrating with respect to dx. 00:07:18.550 --> 00:07:21.770 So sort of continuously with respect to dx. 00:07:21.770 --> 00:07:25.030 Well, what is that? 00:07:25.030 --> 00:07:29.380 Well, according to the rule, it's 1/(b-a) times - sorry, 00:07:29.380 --> 00:07:30.820 it's up here in the box. 00:07:30.820 --> 00:07:34.080 1/(b-a), the integral from a to b, f(x) dx. 00:07:34.080 --> 00:07:39.000 That's 1 / (+1 - (-1)). 00:07:39.000 --> 00:07:45.240 The integral from - 1 to 1, square root of 1 - x^2, dx. 00:07:45.240 --> 00:07:52.026 Right, because the height is y is equal to-- this is y is 00:07:52.026 --> 00:07:57.260 equal to the square root of 1 - x^2. 00:07:57.260 --> 00:08:02.650 And to evaluate this is not as difficult as it seems. 00:08:02.650 --> 00:08:06.490 This is 1/2 times this quantity here, 00:08:06.490 --> 00:08:08.940 which we can interpret as an area. 00:08:08.940 --> 00:08:12.520 It's the area of the semicircle. 00:08:12.520 --> 00:08:19.410 So this is the area of the semicircle, 00:08:19.410 --> 00:08:22.730 which we know to be half the area of the circle. 00:08:22.730 --> 00:08:26.700 So it's pi/2. 00:08:26.700 --> 00:08:29.200 And so the answer, here the average height, is pi/4. 00:08:35.500 --> 00:08:38.670 Now, later in the class and actually not in this unit, 00:08:38.670 --> 00:08:40.330 we'll actually be able to calculate 00:08:40.330 --> 00:08:42.424 the antiderivative of this. 00:08:42.424 --> 00:08:45.090 So in other words, we'll be able to calculate this analytically. 00:08:45.090 --> 00:08:47.090 For right now we just have the geometric reason 00:08:47.090 --> 00:08:51.190 why the value of this is pi/2. 00:08:51.190 --> 00:08:53.070 And we'll do that in the fourth unit 00:08:53.070 --> 00:08:57.140 when we do a lot of techniques of integration. 00:08:57.140 --> 00:08:58.290 So here's an example. 00:08:58.290 --> 00:09:00.260 Turns out, the average height of this is pi/4. 00:09:03.770 --> 00:09:06.700 Now, the next example that I want to give 00:09:06.700 --> 00:09:09.440 introduces a little bit of confusion. 00:09:09.440 --> 00:09:13.370 And I'm not going to resolve this confusion completely, 00:09:13.370 --> 00:09:15.870 but I'm going to try to get you used to it. 00:09:15.870 --> 00:09:22.770 I'm going to take the average height again. 00:09:22.770 --> 00:09:33.690 But now, with respect to arc length. 00:09:33.690 --> 00:09:37.250 Which is usually denoted theta. 00:09:37.250 --> 00:09:43.080 Now, this brings up an extremely important feature of averages. 00:09:43.080 --> 00:09:45.560 Which is that you have to specify 00:09:45.560 --> 00:09:48.740 the variable with respect to which the average is 00:09:48.740 --> 00:09:51.290 taking place. 00:09:51.290 --> 00:09:54.620 And the answer will be different depending on the variable. 00:09:54.620 --> 00:09:56.060 So it's not going to be the same. 00:09:56.060 --> 00:09:58.130 Wow, can't spell the word length here. 00:09:58.130 --> 00:10:02.230 Just like the plural of witches the last time. 00:10:02.230 --> 00:10:03.150 We'll work on that. 00:10:03.150 --> 00:10:11.410 We'll fix all of our, that's an ancient Gaelic word, I think. 00:10:11.410 --> 00:10:16.560 Lengh. 00:10:16.560 --> 00:10:22.190 So now, let me show you that it's not quite the same here. 00:10:22.190 --> 00:10:24.660 It's especially exaggerated if maybe I 00:10:24.660 --> 00:10:30.440 shift this little interval dx over to the right-hand end. 00:10:30.440 --> 00:10:33.230 And you can see that the little portion that corresponds 00:10:33.230 --> 00:10:35.630 to it, which is the d theta piece, 00:10:35.630 --> 00:10:39.130 has a different length from the dx piece. 00:10:39.130 --> 00:10:40.690 And indeed, as you come down here, 00:10:40.690 --> 00:10:43.760 these very short portions of dx length 00:10:43.760 --> 00:10:47.770 have much longer portions of theta length. 00:10:47.770 --> 00:10:51.570 So that the average that we're taking when we do it 00:10:51.570 --> 00:10:54.670 with respect to theta is going to emphasize 00:10:54.670 --> 00:10:56.060 the low values more. 00:10:56.060 --> 00:10:58.090 They're going to be more exaggerated. 00:10:58.090 --> 00:11:02.322 And the average should be lower than the average 00:11:02.322 --> 00:11:03.030 that we got here. 00:11:03.030 --> 00:11:04.880 So we should expect a different number. 00:11:04.880 --> 00:11:06.726 And it's not going to be / 4, it's 00:11:06.726 --> 00:11:07.850 going to be something else. 00:11:07.850 --> 00:11:12.170 Whatever it is, it should be smaller than pi / 4. 00:11:12.170 --> 00:11:14.200 Now, let's set up the integral. 00:11:14.200 --> 00:11:17.100 The integral follows the same rule. 00:11:17.100 --> 00:11:19.760 It's just 1 over the length of the interval times 00:11:19.760 --> 00:11:24.390 the integral over the interval of the function. 00:11:24.390 --> 00:11:27.480 That's the integral, but now where does theta range? 00:11:27.480 --> 00:11:32.430 This time, theta goes from 0 to pi. 00:11:32.430 --> 00:11:34.990 So the integral is from 0 to pi. 00:11:34.990 --> 00:11:40.140 And the thing we divide by is pi. 00:11:40.140 --> 00:11:42.590 And the integration requires us to know 00:11:42.590 --> 00:11:44.160 the formula for the height. 00:11:44.160 --> 00:11:47.410 Which is sin theta. 00:11:47.410 --> 00:11:48.680 In terms of theta, of course. 00:11:48.680 --> 00:11:50.960 It's the same as square root of 1 - x^2, 00:11:50.960 --> 00:11:53.360 but it's expressed in terms of theta. 00:11:53.360 --> 00:11:55.200 So it's this. 00:11:55.200 --> 00:11:58.340 And here's our average. 00:11:58.340 --> 00:12:01.210 I'll put this up here. 00:12:01.210 --> 00:12:09.010 So that's the formula for the height. 00:12:09.010 --> 00:12:10.810 So let's work it out. 00:12:10.810 --> 00:12:12.724 This one, we have the advantage of being 00:12:12.724 --> 00:12:15.140 able to work out because we know the antiderivative of sin 00:12:15.140 --> 00:12:16.920 theta. 00:12:16.920 --> 00:12:25.530 It happens with this factor of pi, it's -cos theta. 00:12:25.530 --> 00:12:35.730 And so, that's -1/pi cos pi-- sorry. (cos pi - cos 0). 00:12:35.730 --> 00:12:42.820 Which is -1/pi (-2), which is 2 / pi. 00:12:47.340 --> 00:12:51.020 And sure enough, if you check it, 00:12:51.020 --> 00:13:02.250 you'll see that 2 / pi < pi / 4, because pi^2 > 8. 00:13:02.250 --> 00:13:02.980 Yeah, question. 00:13:02.980 --> 00:13:06.610 STUDENT: [INAUDIBLE] 00:13:06.610 --> 00:13:09.700 PROFESSOR: The question is how do I get sin theta. 00:13:09.700 --> 00:13:16.080 And the answer is, on this diagram, if theta is over here, 00:13:16.080 --> 00:13:19.950 then this height is this, and this is the angle theta, 00:13:19.950 --> 00:13:22.790 then the height is the sine. 00:13:22.790 --> 00:13:26.600 OK. 00:13:26.600 --> 00:13:27.320 Another question. 00:13:27.320 --> 00:13:32.160 STUDENT: [INAUDIBLE] 00:13:32.160 --> 00:13:41.240 PROFESSOR: The question is, what is the first one, 00:13:41.240 --> 00:13:43.230 the first one is an average of height, 00:13:43.230 --> 00:13:51.310 of a point on a semicircle and this one is with respect to x. 00:13:51.310 --> 00:13:52.890 So what this reveals is that it's 00:13:52.890 --> 00:13:55.490 ambiguous to say what the average value of something is, 00:13:55.490 --> 00:13:59.240 unless you've explained what the underlying averaging 00:13:59.240 --> 00:14:00.200 variable is. 00:14:00.200 --> 00:14:08.140 STUDENT: [INAUDIBLE] 00:14:08.140 --> 00:14:10.800 PROFESSOR: The next question is how should you 00:14:10.800 --> 00:14:13.860 interpret this value. 00:14:13.860 --> 00:14:17.250 That is, what came out of this calculation? 00:14:17.250 --> 00:14:22.460 And the answer is only sort of embedded 00:14:22.460 --> 00:14:26.050 in this calculation itself. 00:14:26.050 --> 00:14:27.690 So here's a way of thinking of it 00:14:27.690 --> 00:14:29.730 which is anticipating our next subject. 00:14:29.730 --> 00:14:31.520 Which is probability. 00:14:31.520 --> 00:14:35.510 Which is, suppose you picked a number at random 00:14:35.510 --> 00:14:36.660 in this interval. 00:14:36.660 --> 00:14:39.440 With equal likelihood, one place and another. 00:14:39.440 --> 00:14:42.080 And then you saw what height was above that. 00:14:42.080 --> 00:14:45.272 That would be the interpretation of this first average value. 00:14:45.272 --> 00:14:46.980 And the second one is, I picked something 00:14:46.980 --> 00:14:50.110 at random on this circle. 00:14:50.110 --> 00:14:53.240 And equally likely, any possible point on this circle 00:14:53.240 --> 00:14:54.970 according to its length. 00:14:54.970 --> 00:14:58.180 And then I ask what the height of that point is. 00:14:58.180 --> 00:15:04.670 And those are just different things. 00:15:04.670 --> 00:15:05.460 Another question. 00:15:05.460 --> 00:15:09.510 STUDENT: [INAUDIBLE] 00:15:09.510 --> 00:15:12.340 PROFESSOR: cos pi, shouldn't it be 0? 00:15:12.340 --> 00:15:19.440 No. cos of-- it's -1. cos pi is -1. 00:15:19.440 --> 00:15:25.110 Cosine, sorry. 00:15:25.110 --> 00:15:29.240 No, cos 0 = 1. cos pi = -1. 00:15:29.240 --> 00:15:30.476 And so they cancel. 00:15:30.476 --> 00:15:31.600 That is, they don't cancel. 00:15:31.600 --> 00:15:35.880 It's -1 - 1, which is -2. 00:15:35.880 --> 00:15:37.830 Key point. 00:15:37.830 --> 00:15:38.330 Yeah. 00:15:38.330 --> 00:15:49.680 STUDENT: [INAUDIBLE] 00:15:49.680 --> 00:15:51.260 PROFESSOR: All right, let me repeat. 00:15:51.260 --> 00:15:55.150 So the question was to repeat the reasoning 00:15:55.150 --> 00:16:00.580 by which I guessed in advance that probably this 00:16:00.580 --> 00:16:04.220 was going to be the relationship between the average value 00:16:04.220 --> 00:16:06.460 with respect to arc length versus the average value 00:16:06.460 --> 00:16:10.790 with respect to this horizontal distance. 00:16:10.790 --> 00:16:15.405 And it had to do with the previous way this diagram was 00:16:15.405 --> 00:16:16.980 drawn. 00:16:16.980 --> 00:16:21.690 Which is comparing an interval in dx 00:16:21.690 --> 00:16:25.800 with an interval in theta. 00:16:25.800 --> 00:16:27.780 A little section in theta. 00:16:27.780 --> 00:16:32.530 And when you're near the top, they're nearly this same. 00:16:32.530 --> 00:16:34.424 That is, it's more or less balanced. 00:16:34.424 --> 00:16:36.340 It's a little curved here, a little different. 00:16:36.340 --> 00:16:39.010 But here it becomes very exaggerated. 00:16:39.010 --> 00:16:42.390 The d theta lengths are much longer than the dx lengths. 00:16:42.390 --> 00:16:45.990 Which means that importance given 00:16:45.990 --> 00:16:49.750 by the theta variable to these parts of the circle 00:16:49.750 --> 00:16:52.930 is larger, relative to these parts. 00:16:52.930 --> 00:16:55.830 Whereas if you look at this section versus this section 00:16:55.830 --> 00:16:57.660 for the dx, they give equal weights 00:16:57.660 --> 00:16:59.090 to these two equal lengths. 00:16:59.090 --> 00:17:01.660 But here, with respect to theta, this is relatively short 00:17:01.660 --> 00:17:03.090 and this is much larger. 00:17:03.090 --> 00:17:05.150 So, as I say, the theta variable's 00:17:05.150 --> 00:17:09.140 emphasizing the lower parts of the semicircle more. 00:17:09.140 --> 00:17:10.930 That's because this length is shorter 00:17:10.930 --> 00:17:12.340 and this length is longer. 00:17:12.340 --> 00:17:16.130 Whereas these two are the same. 00:17:16.130 --> 00:17:20.110 It's a balancing act of the relative weights. 00:17:20.110 --> 00:17:23.190 I'm going to say that again in a different way, and maybe 00:17:23.190 --> 00:17:31.357 this will-- The lower part is more important for theta. 00:17:31.357 --> 00:17:32.190 STUDENT: [INAUDIBLE] 00:17:32.190 --> 00:17:33.595 PROFESSOR: So the question is, but shouldn't it 00:17:33.595 --> 00:17:35.920 have a bigger value because it's a longer length. 00:17:35.920 --> 00:17:37.620 Never with averages. 00:17:37.620 --> 00:17:39.540 Whatever the length is, we're always dividing. 00:17:39.540 --> 00:17:42.760 We're always compensating by the total. 00:17:42.760 --> 00:17:45.410 We have the integral from 0 to pi, but we're dividing by pi. 00:17:45.410 --> 00:17:49.760 Here we had the integral from -1 to 1, but we're dividing by 2. 00:17:49.760 --> 00:17:52.220 So we divide by something different each time. 00:17:52.220 --> 00:17:53.850 And this is very, very important. 00:17:53.850 --> 00:17:55.410 It's that the average of a constant 00:17:55.410 --> 00:17:58.070 is that same constant regardless of which one we did. 00:17:58.070 --> 00:18:00.070 So if it were a constant, we would always 00:18:00.070 --> 00:18:01.270 compensate for the length. 00:18:01.270 --> 00:18:03.780 So the length never matters. 00:18:03.780 --> 00:18:09.160 If it's the integral from 0 to 1,000,000, or 100, let's say, 00:18:09.160 --> 00:18:12.036 1/100 c dx, it's just the same. 00:18:12.036 --> 00:18:14.160 It's always that, it doesn't matter how long it is. 00:18:14.160 --> 00:18:17.405 Because we compensate. 00:18:17.405 --> 00:18:19.405 That's really the difference between an integral 00:18:19.405 --> 00:18:24.750 and an average, is that we're dividing by the total. 00:18:24.750 --> 00:18:28.320 Now I want to introduce another notion, which is actually 00:18:28.320 --> 00:18:31.561 what's underlying these two examples that I just wrote 00:18:31.561 --> 00:18:32.060 down. 00:18:32.060 --> 00:18:35.210 And this is by far the one which you should emphasize 00:18:35.210 --> 00:18:39.470 the most in your thoughts. 00:18:39.470 --> 00:18:42.480 Because it is much more flexible, 00:18:42.480 --> 00:18:49.470 and is much more typical of real life problems. 00:18:49.470 --> 00:18:53.920 So the idea of a weighted average is the following. 00:18:53.920 --> 00:18:57.580 You take the integral, say from a to b, of some function. 00:18:57.580 --> 00:19:02.550 But now you multiply by a weight. 00:19:02.550 --> 00:19:05.560 And you have to divide by the total. 00:19:05.560 --> 00:19:07.410 And what's the total going to be? 00:19:07.410 --> 00:19:11.690 It's the integral from a to b of this total weighting 00:19:11.690 --> 00:19:14.020 that we have. 00:19:14.020 --> 00:19:17.190 Now, why is this the correct notion? 00:19:17.190 --> 00:19:20.420 I'm going to explain it to you in two ways. 00:19:20.420 --> 00:19:24.800 The first is this very simpleminded thing 00:19:24.800 --> 00:19:30.480 that I wrote on the board there, with the constants. 00:19:30.480 --> 00:19:38.480 What we want is the average value of c to be c. 00:19:38.480 --> 00:19:40.820 Otherwise this makes no sense as an average. 00:19:40.820 --> 00:19:43.070 Now, let's just look at this definition here. 00:19:43.070 --> 00:19:44.660 And see that that's correct. 00:19:44.660 --> 00:19:49.920 If you integrate c, from a to b, w(x) dx, 00:19:49.920 --> 00:19:54.710 and you divide by the integral from a to b, w(x) dx, 00:19:54.710 --> 00:19:57.020 not surprisingly, the c factors out. 00:19:57.020 --> 00:19:59.010 It's a constant. 00:19:59.010 --> 00:20:03.840 So this is c times the integral a to b, w(x) dx, divided 00:20:03.840 --> 00:20:05.380 by the same thing. 00:20:05.380 --> 00:20:08.212 And that's why we picked it. 00:20:08.212 --> 00:20:10.170 We picked it so that these things would cancel. 00:20:10.170 --> 00:20:14.660 And this would give c. 00:20:14.660 --> 00:20:17.260 So in the previous case, this property 00:20:17.260 --> 00:20:20.140 explains the denominator. 00:20:20.140 --> 00:20:29.760 And again over here, it explains the denominator. 00:20:29.760 --> 00:20:32.510 And let me just give you one more explanation. 00:20:32.510 --> 00:20:38.340 Which is maybe a real-life-- pretend real-life example. 00:20:38.340 --> 00:20:43.060 You have a stock which you bought for $10 one year. 00:20:43.060 --> 00:20:46.740 And then six months later you brought some more for $20. 00:20:46.740 --> 00:20:50.360 And then you bought some more for $30. 00:20:50.360 --> 00:20:53.810 Now, what's the average price of your stock? 00:20:53.810 --> 00:20:58.370 Well, it depends on how many shares you bought. 00:20:58.370 --> 00:21:00.700 If you bought this many shares the first time, 00:21:00.700 --> 00:21:02.440 and this many shares the second time, 00:21:02.440 --> 00:21:04.310 and this many shares the third time, 00:21:04.310 --> 00:21:07.230 this is the total amount that you spent. 00:21:07.230 --> 00:21:13.870 And the average price is the total price divided 00:21:13.870 --> 00:21:17.120 by the total number of shares. 00:21:17.120 --> 00:21:23.150 And this is the discrete analog of this continuous averaging 00:21:23.150 --> 00:21:24.070 process here. 00:21:24.070 --> 00:21:28.260 The function f now, so I use w for weight, the function f now 00:21:28.260 --> 00:21:31.150 is the function whose values are 10, 20 and 30. 00:21:31.150 --> 00:21:34.830 And the weightings are the relative importance 00:21:34.830 --> 00:21:42.030 of the different purchases. 00:21:42.030 --> 00:21:51.586 So again, these w_i's are weights. 00:21:51.586 --> 00:21:52.710 There was another question. 00:21:52.710 --> 00:21:54.400 Out in the audience, at some point. 00:21:54.400 --> 00:21:55.300 Over here, yes. 00:21:55.300 --> 00:22:04.520 STUDENT: [INAUDIBLE] 00:22:04.520 --> 00:22:06.920 PROFESSOR: Very, very good point. 00:22:06.920 --> 00:22:15.710 So in this numerator here, the statement is-- in this example, 00:22:15.710 --> 00:22:17.020 we factored out c. 00:22:17.020 --> 00:22:20.710 But here we cannot factor out f(x). 00:22:20.710 --> 00:22:23.450 That's extremely important and that is the whole point. 00:22:23.450 --> 00:22:26.930 So, in other words, the weighted average is very interesting 00:22:26.930 --> 00:22:31.680 - you have to do two different integrals to figure it out 00:22:31.680 --> 00:22:32.380 in general. 00:22:32.380 --> 00:22:34.050 When it happens that this is c, it's 00:22:34.050 --> 00:22:35.430 an extremely boring integral. 00:22:35.430 --> 00:22:37.100 Which in fact because, it's an average, 00:22:37.100 --> 00:22:38.799 you don't even have to calculate at all. 00:22:38.799 --> 00:22:40.340 Factor it out and cancel these things 00:22:40.340 --> 00:22:43.530 and never bother to calculate these two numbers. 00:22:43.530 --> 00:22:46.580 So these massive numbers just cancel. 00:22:46.580 --> 00:22:48.880 So it's a very special property of a constant, 00:22:48.880 --> 00:22:55.650 that it factors out. 00:22:55.650 --> 00:22:59.960 That was our first discussion, and now with this example 00:22:59.960 --> 00:23:02.900 I'm going to go back to the heating up of the witches' 00:23:02.900 --> 00:23:06.190 cauldron and we'll use average value 00:23:06.190 --> 00:23:19.700 to illustrate the integral that we get in that context as well. 00:23:19.700 --> 00:23:20.910 I remind you, let's see. 00:23:20.910 --> 00:23:25.480 The situation with the witches' cauldron was this. 00:23:25.480 --> 00:23:40.130 The first important thing is that there were-- so this 00:23:40.130 --> 00:23:42.770 is the big cauldron here. 00:23:42.770 --> 00:23:47.480 This is the one whose height is 1 meter and whose 00:23:47.480 --> 00:23:53.190 width is 2 meters. 00:23:53.190 --> 00:23:56.540 And it's a parabola of revolution here. 00:23:56.540 --> 00:24:06.500 And it had about approximately 1600 liters in it. 00:24:06.500 --> 00:24:14.640 And this curve was y = x^2. 00:24:14.640 --> 00:24:17.580 And the situation that I described 00:24:17.580 --> 00:24:24.900 at the end of last time was that the initial temperature 00:24:24.900 --> 00:24:28.560 was T = 0 degrees Celsius. 00:24:28.560 --> 00:24:34.250 And the final temperature, instead 00:24:34.250 --> 00:24:37.590 of being a constant temperature, we 00:24:37.590 --> 00:24:41.290 were heating this guy up from the bottom. 00:24:41.290 --> 00:24:48.340 And it was hotter on the bottom than on the top. 00:24:48.340 --> 00:24:50.370 And the final temperature was given 00:24:50.370 --> 00:24:57.880 by the formula T is equal to 100 minus 30 times the height y. 00:24:57.880 --> 00:25:04.090 So at y = 0, at the bottom, it's 100. 00:25:04.090 --> 00:25:10.730 And at the top, T = 70 degrees. 00:25:10.730 --> 00:25:14.610 OK, so this is the final configuration 00:25:14.610 --> 00:25:16.020 for the temperature. 00:25:16.020 --> 00:25:33.100 And the question was how much energy do we need. 00:25:33.100 --> 00:25:35.210 So, the first observation here, and this 00:25:35.210 --> 00:25:39.500 is the reason for giving this example, 00:25:39.500 --> 00:25:42.630 is that it's important to realize 00:25:42.630 --> 00:25:54.700 that you want to use the method of disks in this case. 00:25:54.700 --> 00:25:57.630 The reason-- So it doesn't have to do with, 00:25:57.630 --> 00:26:00.070 you shouldn't think of the disks first. 00:26:00.070 --> 00:26:05.010 But what you should think of is the horizontal. 00:26:05.010 --> 00:26:10.840 We must use horizontals because T is constant on horizontals. 00:26:10.840 --> 00:26:12.740 It's not constant on verticals. 00:26:12.740 --> 00:26:16.130 If we set things up with shells, as we did last time, 00:26:16.130 --> 00:26:18.790 to compute the volume of this, then T 00:26:18.790 --> 00:26:21.760 will vary along the shell. 00:26:21.760 --> 00:26:24.130 And we will still have an averaging problem, 00:26:24.130 --> 00:26:26.730 an integral problem along the vertical portion. 00:26:26.730 --> 00:26:30.340 But if we do it this way, T is constant on this whole level 00:26:30.340 --> 00:26:31.770 here. 00:26:31.770 --> 00:26:34.300 And so there's no more calculus involved 00:26:34.300 --> 00:26:36.760 in calculating what the contribution is of any given 00:26:36.760 --> 00:26:39.840 level. 00:26:39.840 --> 00:26:49.620 So T is constant on horizontals. 00:26:49.620 --> 00:26:52.620 Actually, in disguise, this is that same trick 00:26:52.620 --> 00:26:53.370 that we have here. 00:26:53.370 --> 00:26:55.596 We can factor constants out of integrals. 00:26:55.596 --> 00:26:56.970 You could view it as an integral, 00:26:56.970 --> 00:27:03.480 but the point is that it's more elementary than that. 00:27:03.480 --> 00:27:06.430 Now I have to set it up for you. 00:27:06.430 --> 00:27:08.920 And in order to do that, I need to remember 00:27:08.920 --> 00:27:10.470 what the equation is. 00:27:10.470 --> 00:27:12.960 Which is y = x^2. 00:27:12.960 --> 00:27:18.500 And the formula for the total amount of energy 00:27:18.500 --> 00:27:25.000 is going to be volume times the number of degrees. 00:27:25.000 --> 00:27:31.480 That's going to be equal to the energy that we need here. 00:27:31.480 --> 00:27:33.250 And so let's add it up. 00:27:33.250 --> 00:27:38.540 It's the integral from 0 to 1, and this is with respect to y. 00:27:38.540 --> 00:27:41.300 So the y level goes from 0 to 1. 00:27:41.300 --> 00:27:47.310 This top level is y = 1, this bottom level is y = 0. 00:27:47.310 --> 00:27:55.090 And the disk that we get, this is the point (x, y) here, 00:27:55.090 --> 00:27:56.220 is rotated around. 00:27:56.220 --> 00:28:01.230 And its radius is x. 00:28:01.230 --> 00:28:09.480 So the thickness is dy, and the area of the disk is pi x^2. 00:28:09.480 --> 00:28:11.430 And the thing that we're averaging 00:28:11.430 --> 00:28:14.550 is T. Well, we're not yet averaging, 00:28:14.550 --> 00:28:16.010 we're just integrating it. 00:28:16.010 --> 00:28:24.170 We're just adding up the total. 00:28:24.170 --> 00:28:29.080 Now I'm just going to plug in the various values for this. 00:28:29.080 --> 00:28:36.580 And what I'm going to get is T, again, is 100 - 30y. 00:28:36.580 --> 00:28:40.960 And this radius is measured up to this very end. 00:28:40.960 --> 00:28:42.440 So x^2 = y. 00:28:42.440 --> 00:28:45.410 So this is pi y dy. 00:28:45.410 --> 00:28:47.800 And this is the integral that we'll be able to evaluate. 00:28:47.800 --> 00:28:48.660 Yeah, question. 00:28:48.660 --> 00:28:50.250 STUDENT: [INAUDIBLE] 00:28:50.250 --> 00:29:00.070 PROFESSOR: All right. 00:29:00.070 --> 00:29:05.710 Well, let's carry this out. 00:29:05.710 --> 00:29:09.160 Let's finish off the calculation here. 00:29:09.160 --> 00:29:10.230 Let's see. 00:29:10.230 --> 00:29:16.380 This is equal to, what it it equal to? 00:29:16.380 --> 00:29:19.580 Well, I'll put it over here. 00:29:19.580 --> 00:29:26.590 It's equal to 50 pi y ^2 minus-- right, 00:29:26.590 --> 00:29:30.010 because this is 100 pi y, and then there's a 30, 00:29:30.010 --> 00:29:37.480 this is 100 pi y - 30 pi y^2, and I have to take 00:29:37.480 --> 00:29:38.780 the antiderivative of that. 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. 00:29:48.120 --> 00:29:57.930 And that is 40 pi. 00:29:57.930 --> 00:30:03.510 Now, I spent a tremendous amount of time 00:30:03.510 --> 00:30:07.900 last time focusing on units. 00:30:07.900 --> 00:30:11.250 Because I want to tell you how to get a realistic number out 00:30:11.250 --> 00:30:12.080 of this. 00:30:12.080 --> 00:30:16.400 And there's a subtle point here that I pointed out 00:30:16.400 --> 00:30:19.160 last time that had to do with changing meters to centimeters. 00:30:19.160 --> 00:30:22.570 I claim that I've treated those correctly. 00:30:22.570 --> 00:30:25.970 So, what we have here is that the answer 00:30:25.970 --> 00:30:34.800 is in degrees, that is Celsius, times cubic meters. 00:30:34.800 --> 00:30:36.570 These are the correct units. 00:30:36.570 --> 00:30:43.160 And now, I can translate this into-- Celsius is spelled 00:30:43.160 --> 00:30:44.896 with a C. That's interesting. 00:30:44.896 --> 00:30:46.630 Celsius. 00:30:46.630 --> 00:30:50.850 I can translate this into units that you're more familiar with. 00:30:50.850 --> 00:30:57.620 So let's try 40 pi degrees times m^3, 00:30:57.620 --> 00:30:59.710 and then do the conversion factors. 00:30:59.710 --> 00:31:08.840 First of all there's one calorie per degree times a milliliter. 00:31:08.840 --> 00:31:11.150 That's one conversion. 00:31:11.150 --> 00:31:14.600 And then let's see. 00:31:14.600 --> 00:31:18.580 I'm going to have to translate from centimeters so I have here 00:31:18.580 --> 00:31:22.350 (100 cm / m)^3. 00:31:25.110 --> 00:31:30.550 So these are the two conversion factors that I need. 00:31:30.550 --> 00:31:38.690 And so, I get 40 pi 10^6, that's 100^3. 00:31:38.690 --> 00:31:46.550 And this is in calories. 00:31:46.550 --> 00:31:48.470 So how much is this? 00:31:48.470 --> 00:31:51.180 Well, it's a little better, maybe, 00:31:51.180 --> 00:31:57.777 to do it in 40 pi * 1,000 kilocalories, 00:31:57.777 --> 00:31:59.610 because these are the ones that you actually 00:31:59.610 --> 00:32:05.710 see on your nutrition labels of foods. 00:32:05.710 --> 00:32:12.600 And so this number is around 125 or so. 00:32:12.600 --> 00:32:15.960 Let's see, is that about right? 00:32:15.960 --> 00:32:17.930 Let's make sure I've got these numbers right. 00:32:17.930 --> 00:32:20.550 Yeah, this is about 125. 00:32:20.550 --> 00:32:22.780 40 times pi. 00:32:22.780 --> 00:32:32.220 And so one candy bar-- This is a Halloween example, so. 00:32:32.220 --> 00:32:38.460 One candy bar is about 250 kilocalories. 00:32:38.460 --> 00:32:44.620 So this is half a candy bar. 00:32:44.620 --> 00:32:57.230 So the answer to our question is that it takes 500 candy bars 00:32:57.230 --> 00:33:02.900 to heat up this thing. 00:33:02.900 --> 00:33:07.570 OK, so that's our example. 00:33:07.570 --> 00:33:08.630 Now, yeah. 00:33:08.630 --> 00:33:09.160 Question. 00:33:09.160 --> 00:33:13.710 STUDENT: [INAUDIBLE] 00:33:13.710 --> 00:33:16.980 PROFESSOR: What does the integral give us? 00:33:16.980 --> 00:33:22.971 This integral is-- the integral represents the following 00:33:22.971 --> 00:33:23.470 things. 00:33:23.470 --> 00:33:26.330 So the question is, what does this integral give us. 00:33:26.330 --> 00:33:27.640 So here's the integral. 00:33:27.640 --> 00:33:30.640 Here it is, rewritten so that it can be calculated. 00:33:30.640 --> 00:33:34.170 And what this integral is giving us is the following thing. 00:33:34.170 --> 00:33:36.160 You have to imagine the following idea. 00:33:36.160 --> 00:33:39.220 You've got a little chunk of water in here. 00:33:39.220 --> 00:33:41.690 And you're going to raise is from 0 degrees 00:33:41.690 --> 00:33:46.840 all the way up to whatever the target temperature is. 00:33:46.840 --> 00:33:50.810 And so that little milliliter of water, if you like, 00:33:50.810 --> 00:33:53.700 has to be raised from 0 to some number which 00:33:53.700 --> 00:33:56.730 is a function of the height. 00:33:56.730 --> 00:33:59.950 It's something between 70 and 100 degrees. 00:33:59.950 --> 00:34:02.790 And the one right above it also has 00:34:02.790 --> 00:34:05.240 to be raised to a temperature, although a slightly 00:34:05.240 --> 00:34:06.280 different temperature. 00:34:06.280 --> 00:34:08.840 And what we're doing with the integral is we're adding up 00:34:08.840 --> 00:34:15.647 all of those degrees and the calorie represents how much it 00:34:15.647 --> 00:34:17.230 takes, one calorie represents how much 00:34:17.230 --> 00:34:21.980 it takes to raise by 1 degree 1 milliliter of water. 00:34:21.980 --> 00:34:26.620 One cubic centimeter of water. 00:34:26.620 --> 00:34:31.610 That's the definition of a calorie. 00:34:31.610 --> 00:34:32.990 And we're adding it up. 00:34:32.990 --> 00:34:35.430 So in other words, each of these cubes is one thing. 00:34:35.430 --> 00:34:38.010 And now we have to add it up over this massive thing, which 00:34:38.010 --> 00:34:40.150 is 1600 liters. 00:34:40.150 --> 00:34:42.140 And we have a lot of different little cubes. 00:34:42.140 --> 00:34:43.420 And that's what we did. 00:34:43.420 --> 00:34:45.320 When we glommed them all together. 00:34:45.320 --> 00:34:48.800 That's what the integral is doing for us. 00:34:48.800 --> 00:34:54.200 Other questions. 00:34:54.200 --> 00:34:57.400 Now I want to connect this with weighted averages 00:34:57.400 --> 00:34:58.800 before we go on. 00:34:58.800 --> 00:35:03.640 Because that was the reason why I did weighted averages first. 00:35:03.640 --> 00:35:14.810 I'm going to compute also the average final temperature. 00:35:14.810 --> 00:35:17.502 So, final because this is the interesting one, 00:35:17.502 --> 00:35:19.460 the average starting temperature's very boring, 00:35:19.460 --> 00:35:21.030 it's 0. 00:35:21.030 --> 00:35:26.090 The average final temperature is-- individually 00:35:26.090 --> 00:35:27.840 the temperatures are different. 00:35:27.840 --> 00:35:29.770 And the answer here is it's the integral 00:35:29.770 --> 00:35:37.930 from 0 to 1 of T pi y dy divided by the integral 00:35:37.930 --> 00:35:42.040 from 0 to 1 of pi y dy. 00:35:42.040 --> 00:35:43.970 So this is the total temperature, 00:35:43.970 --> 00:35:46.820 weighted appropriately to the volume of water that's 00:35:46.820 --> 00:35:49.180 involved at that temperature, divided 00:35:49.180 --> 00:35:52.880 by the total volume of water. 00:35:52.880 --> 00:35:55.130 And we computed these two numbers. 00:35:55.130 --> 00:35:58.350 The number in the numerator is what we call 40 pi. 00:35:58.350 --> 00:36:00.420 And the number in the denominator, actually this 00:36:00.420 --> 00:36:02.870 is easier than what we did last time with shells; 00:36:02.870 --> 00:36:04.800 you can just look at this and see that it's 00:36:04.800 --> 00:36:06.160 the area under a triangle. 00:36:06.160 --> 00:36:08.500 It's pi / 2. 00:36:08.500 --> 00:36:11.760 And so the answer here is 80 degrees. 00:36:11.760 --> 00:36:14.910 This is the average temperature. 00:36:14.910 --> 00:36:17.670 Note that this is a weighted average. 00:36:17.670 --> 00:36:22.810 The weighting here is different according to the height. 00:36:22.810 --> 00:36:28.190 The weighting factor is pi y. 00:36:28.190 --> 00:36:30.010 That's the weighting factor. 00:36:30.010 --> 00:36:32.270 And that's not surprising. 00:36:32.270 --> 00:36:35.630 When y is small, there's less volume down here. 00:36:35.630 --> 00:36:38.510 Up above, those are more important volumes, 00:36:38.510 --> 00:36:41.720 because there's more water up at the top of the cauldron 00:36:41.720 --> 00:36:43.960 than there is down at the bottom of the cauldron. 00:36:43.960 --> 00:36:46.480 If you compare this to the ordinary average, 00:36:46.480 --> 00:36:48.570 if you take the maximum temperature 00:36:48.570 --> 00:36:52.630 plus the minimum temperature, divided by 2, 00:36:52.630 --> 00:36:56.470 that would be (100 + 70) / 2. 00:36:56.470 --> 00:36:59.830 You would get 85 degrees. 00:36:59.830 --> 00:37:01.610 And that's bigger. 00:37:01.610 --> 00:37:02.440 Why? 00:37:02.440 --> 00:37:05.030 Because the cooler water is on top. 00:37:05.030 --> 00:37:08.530 And the actual average, the correct weighted average, 00:37:08.530 --> 00:37:11.800 is lower than this fake average. 00:37:11.800 --> 00:37:15.520 Which is not the true average in this context. 00:37:15.520 --> 00:37:17.780 All right so the weighting is that the thing 00:37:17.780 --> 00:37:33.160 is getting fatter near the top. 00:37:33.160 --> 00:37:38.980 So now I'm going to do another example of weighted average. 00:37:38.980 --> 00:37:46.020 And this example is also very much worth your while. 00:37:46.020 --> 00:37:48.940 It's the other incredibly important one 00:37:48.940 --> 00:37:51.960 in interpreting integrals. 00:37:51.960 --> 00:37:56.790 And it's a very, very simple example of a function f. 00:37:56.790 --> 00:37:59.240 The weightings will be different, 00:37:59.240 --> 00:38:03.320 but the function f, will be of a very particular kind. 00:38:03.320 --> 00:38:07.450 Namely, the function f will be practically a constant. 00:38:07.450 --> 00:38:08.480 But not quite. 00:38:08.480 --> 00:38:10.090 It's going to be a constant on one 00:38:10.090 --> 00:38:13.880 interval, and then 0 on the rest. 00:38:13.880 --> 00:38:16.950 So we'll do those weighted averages now. 00:38:16.950 --> 00:38:34.900 And this subject is called probability. 00:38:34.900 --> 00:38:39.130 In probability, what we do, so I'm just 00:38:39.130 --> 00:38:43.910 going to give some examples here. 00:38:43.910 --> 00:38:54.620 I'm going to pick a point in quotation marks - at random. 00:38:54.620 --> 00:39:00.900 In the region y < x < 1 - x^2. 00:39:00.900 --> 00:39:05.060 That's this shape here. 00:39:05.060 --> 00:39:08.770 Well, let's draw it right down here. 00:39:08.770 --> 00:39:09.310 For now. 00:39:09.310 --> 00:39:10.380 So, somewhere in here. 00:39:10.380 --> 00:39:13.310 Some point, (x, y). 00:39:13.310 --> 00:39:18.335 And then I need to tell you, according to what 00:39:18.335 --> 00:39:20.330 this random really means. 00:39:20.330 --> 00:39:31.780 This is proportional to area, if you like. 00:39:31.780 --> 00:39:33.990 So area inside of this section. 00:39:33.990 --> 00:39:37.950 And then the question that we're going to answer right now is, 00:39:37.950 --> 00:39:47.040 what is the chance that - or, it's usually called probability 00:39:47.040 --> 00:39:56.980 - that x > 1/2. 00:39:56.980 --> 00:40:03.620 Let me show you what's going on here. 00:40:03.620 --> 00:40:08.540 And this is always the case with things in probability. 00:40:08.540 --> 00:40:10.290 So, first of all, we have a name for this. 00:40:10.290 --> 00:40:12.510 This is called the probability that x > 1/2. 00:40:16.220 --> 00:40:21.060 And so that's what it's called in our notation here. 00:40:21.060 --> 00:40:26.920 And what it is, is the probability 00:40:26.920 --> 00:40:32.760 is always equal to the part divided by the whole. 00:40:32.760 --> 00:40:36.846 It's a ratio just like the one over there. 00:40:36.846 --> 00:40:38.720 And which is the part and which is the whole? 00:40:38.720 --> 00:40:43.470 Well, in this picture, the whole is the whole parabola. 00:40:43.470 --> 00:40:48.910 And the part is the section x > 1/2. 00:40:48.910 --> 00:41:00.460 And it's just the ratio of those two areas. 00:41:00.460 --> 00:41:01.970 Let's write that down. 00:41:01.970 --> 00:41:08.680 That's the integral from 1/2 to 1 of (1 - x^2) dx, 00:41:08.680 --> 00:41:16.000 divided by the integral from -1 to 1, (1 - x^2) dx. 00:41:16.000 --> 00:41:23.350 And again, the weighting factor here is 1 - x^2. 00:41:23.350 --> 00:41:25.990 And to be a little bit more specific here, 00:41:25.990 --> 00:41:33.390 the starting point a = -1 and the endpoint is +1. 00:41:33.390 --> 00:41:37.540 So this is P(x < 1/2). 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. 00:41:46.970 --> 00:41:47.470 Yeah. 00:41:47.470 --> 00:42:21.004 STUDENT: [INAUDIBLE] 00:42:21.004 --> 00:42:23.170 PROFESSOR: What we're trying to do with probability. 00:42:23.170 --> 00:42:26.760 So I can't repeat your question. 00:42:26.760 --> 00:42:29.670 But I can try to say-- because it was 00:42:29.670 --> 00:42:31.580 a little bit too complicated. 00:42:31.580 --> 00:42:35.020 But it was not correct, OK. 00:42:35.020 --> 00:42:38.720 What we're taking is, we have two possible things 00:42:38.720 --> 00:42:39.940 that could happen. 00:42:39.940 --> 00:42:42.640 Either, let's put it this way. 00:42:42.640 --> 00:42:43.910 Let's make it a gamble. 00:42:43.910 --> 00:42:47.150 Somebody picks a point in here at random. 00:42:47.150 --> 00:42:53.040 And we're trying to figure out what 00:42:53.040 --> 00:42:54.527 your chances are of winning. 00:42:54.527 --> 00:42:57.110 In other words, the chances the person picks something in here 00:42:57.110 --> 00:42:59.646 versus something in there. 00:42:59.646 --> 00:43:02.020 And the interesting thing is, so what percent of the time 00:43:02.020 --> 00:43:04.220 do you win. 00:43:04.220 --> 00:43:06.384 The answer is it's some fraction of 1. 00:43:06.384 --> 00:43:07.800 And in order to figure that out, I 00:43:07.800 --> 00:43:11.900 have to figure out the total area here. 00:43:11.900 --> 00:43:16.020 Versus the total of the entire, all the way from -1 to 1, 00:43:16.020 --> 00:43:18.190 the beginning to the end. 00:43:18.190 --> 00:43:22.770 So in the numerator, I put success, and in the denominator 00:43:22.770 --> 00:43:25.250 I put all possibilities. 00:43:25.250 --> 00:43:26.720 So that-- Right? 00:43:26.720 --> 00:43:29.039 STUDENT: [INAUDIBLE] 00:43:29.039 --> 00:43:31.080 PROFESSOR: And that's the interpretation of this. 00:43:31.080 --> 00:43:33.060 So maybe I didn't understand your question. 00:43:33.060 --> 00:43:37.900 STUDENT: [INAUDIBLE] 00:43:37.900 --> 00:43:40.130 PROFESSOR: Ah, why is 1 - x^2. the weighting factor. 00:43:40.130 --> 00:43:44.550 That has to do with how you compute areas under curves. 00:43:44.550 --> 00:43:49.040 The curve here is y = 1 - x^2. 00:43:49.040 --> 00:43:51.810 And so, in order to calculate how much area is between 1/2 00:43:51.810 --> 00:43:52.940 and 1, I have to integrate. 00:43:52.940 --> 00:43:54.440 That's the interpretation of this. 00:43:54.440 --> 00:43:56.450 This is the area under that curve. 00:43:56.450 --> 00:43:57.220 This integral. 00:43:57.220 --> 00:44:01.200 And the denominator's the area under the whole thing. 00:44:01.200 --> 00:44:02.052 OK, yeah. 00:44:02.052 --> 00:44:02.760 Another question. 00:44:02.760 --> 00:44:06.060 STUDENT: [INAUDIBLE] 00:44:06.060 --> 00:44:08.100 PROFESSOR: Ah. 00:44:08.100 --> 00:44:09.460 Yikes. 00:44:09.460 --> 00:44:12.531 It was supposed to be the same question as over here. 00:44:12.531 --> 00:44:13.030 Thank you. 00:44:13.030 --> 00:44:19.700 STUDENT: [INAUDIBLE] PROFESSOR: This has something 00:44:19.700 --> 00:44:21.420 to do with weighting factors. 00:44:21.420 --> 00:44:25.410 Here's the weight factor. 00:44:25.410 --> 00:44:27.720 Well, it's the relative importance 00:44:27.720 --> 00:44:29.680 from the point of view of this probability 00:44:29.680 --> 00:44:33.160 of these places versus those. 00:44:33.160 --> 00:44:36.940 That is, so this is a weighting factor because it's telling me 00:44:36.940 --> 00:44:45.560 that in some sense this number 5/18-- actually that makes me 00:44:45.560 --> 00:44:48.420 think that this number is probably wrong. 00:44:48.420 --> 00:44:53.170 Well, I'll let you calculate it out. 00:44:53.170 --> 00:44:55.380 It looks like it should be less than 1/4 here, 00:44:55.380 --> 00:44:57.760 because this is 1/4 of the total distance 00:44:57.760 --> 00:44:59.370 and there's a little less in here 00:44:59.370 --> 00:45:00.812 than there is in the middle. 00:45:00.812 --> 00:45:03.270 So in fact it probably should be less than 1/4, the answer. 00:45:03.270 --> 00:45:09.070 STUDENT: [INAUDIBLE] 00:45:09.070 --> 00:45:11.070 PROFESSOR: The equation of the curve is 1 - x^2. 00:45:13.406 --> 00:45:14.780 The reason why it's the weighting 00:45:14.780 --> 00:45:17.490 factor is that we're interpreting-- The question 00:45:17.490 --> 00:45:20.120 has to do with the area under that curve. 00:45:20.120 --> 00:45:24.020 And so, this is showing us how much is relatively important 00:45:24.020 --> 00:45:25.200 versus how much is not. 00:45:25.200 --> 00:45:27.369 This is-- These parts are relatively important, 00:45:27.369 --> 00:45:28.660 these parts are less important. 00:45:28.660 --> 00:45:29.960 According to area. 00:45:29.960 --> 00:45:31.730 Because we've said that area is the way 00:45:31.730 --> 00:45:35.920 we're making the choice. 00:45:35.920 --> 00:45:38.420 So I don't have quite enough time 00:45:38.420 --> 00:45:43.132 to tell you about my next example. 00:45:43.132 --> 00:45:44.590 Instead, I'm just going to tell you 00:45:44.590 --> 00:45:46.820 what the general formula is. 00:45:46.820 --> 00:45:48.920 And we'll do our example next time. 00:45:48.920 --> 00:45:51.790 I'll tell you what it's going to be. 00:45:51.790 --> 00:46:04.190 So here's the general formula for probability here. 00:46:04.190 --> 00:46:12.020 We're going to imagine that we have a total range which 00:46:12.020 --> 00:46:13.940 is maybe going from a to b, and we 00:46:13.940 --> 00:46:18.320 have some intermediate values x_1 and x_2, 00:46:18.320 --> 00:46:23.020 and then we're going to try to compute the probability 00:46:23.020 --> 00:46:28.150 that some variable that we picked at random 00:46:28.150 --> 00:46:31.560 occurs between x_1 and x_2. 00:46:31.560 --> 00:46:36.490 And by definition, we're saying that it's an integral. 00:46:36.490 --> 00:46:41.280 It's the integral from x_1 to x_2 of the weight dx, 00:46:41.280 --> 00:46:46.710 divided by the integral all the way from a to b. 00:46:46.710 --> 00:46:47.390 Of the weight. 00:46:47.390 --> 00:46:55.900 So, again, this is the part divided by the whole. 00:46:55.900 --> 00:46:59.890 And the relationship between this and the weighted average 00:46:59.890 --> 00:47:02.950 that we had earlier was that the function f f(x) 00:47:02.950 --> 00:47:04.450 is kind of a strange function. 00:47:04.450 --> 00:47:06.410 It's 0 and 1. 00:47:06.410 --> 00:47:09.200 It's just-- The picture, if you like, 00:47:09.200 --> 00:47:11.860 is that you have this weighting factor. 00:47:11.860 --> 00:47:14.180 And it's going from a to b. 00:47:14.180 --> 00:47:16.649 But then in between there, we have the part 00:47:16.649 --> 00:47:17.690 that we're interested in. 00:47:17.690 --> 00:47:20.130 Which is between x_1 and x_2. 00:47:20.130 --> 00:47:23.910 And it's the ratio of this inner part to the whole thing 00:47:23.910 --> 00:47:34.650 that we're interested in. 00:47:34.650 --> 00:47:39.549 Tomorrow I'm going to try to do a realistic example. 00:47:39.549 --> 00:47:41.090 And I'm going to tell you what it is, 00:47:41.090 --> 00:47:43.620 but we'll take it up tomorrow. 00:47:43.620 --> 00:47:45.460 I told you it was going to be tomorrow, 00:47:45.460 --> 00:47:46.835 but we still have a whole minute, 00:47:46.835 --> 00:47:49.270 so I'm going to tell you what the problem is. 00:47:49.270 --> 00:47:53.660 So this is going to be a target practice problem. 00:47:53.660 --> 00:47:55.210 You have a target here and you're 00:47:55.210 --> 00:48:00.810 throwing darts at this target. 00:48:00.810 --> 00:48:05.960 And so you're throwing darts at this target. 00:48:05.960 --> 00:48:13.020 And somebody is standing next to the dartboard. 00:48:13.020 --> 00:48:18.160 Your little brother is standing next to the dartboard here. 00:48:18.160 --> 00:48:21.830 And the question is, how likely you 00:48:21.830 --> 00:48:24.320 are to hit your little brother. 00:48:24.320 --> 00:48:26.834 So this will, let's see. 00:48:26.834 --> 00:48:28.500 You'll see whether you like that or not. 00:48:28.500 --> 00:48:29.960 Actually, I was the little brother. 00:48:29.960 --> 00:48:31.710 So, I don't know which way you want to go. 00:48:31.710 --> 00:48:32.680 We'll go either way. 00:48:32.680 --> 00:48:35.240 We'll find out next time.