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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.