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PROFESSOR: Today we're going
to hold off just a little
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bit on boiling water.
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And talk about another
application of integrals, and
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we'll get to the witches'
cauldron in the middle.
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00:00:36 --> 00:00:45
The that I'd like to start
with today is average value.
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00:00:45 --> 00:00:48
This is something that I
mentioned a little bit earlier,
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and there was a misprint on the
board, so I want to make
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sure that we have the
definitions straight.
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And also the
reasoning straight.
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00:00:56 --> 00:00:59
This is one of the most
important applications of
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00:00:59 --> 00:01:03
integrals, one of the
most important examples.
20
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If you take the average
of a bunch of numbers,
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that looks like this.
22
00:01:10 --> 00:01:15
And we can view this as
sampling a function.
23
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As we would with Riemann's sum.
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And what I said last week was
that this tends to this
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expression here, which is
called the continuous average.
26
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So this guy is the
continuous average.
27
00:01:43 --> 00:01:49
Or just the average of f.
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00:01:49 --> 00:01:53
And I want to explain that,
just to make sure that we're
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all on the same page.
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00:01:56 --> 00:01:59
In general, if you have a
function and you want to
31
00:01:59 --> 00:02:03
interpret the integral, our
first interpretation was that
32
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it's something like the
area under the curve.
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00:02:09 --> 00:02:15
But average value is another
reasonable interpretation.
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00:02:15 --> 00:02:20
Namely, if you take equally
spaced points here, starting
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00:02:20 --> 00:02:28
with x0, x1, x2, all the way up
to xn, which is the left point
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00:02:28 --> 00:02:35
b, and then we have values y1,
which = f ( x1), y2, which = f
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00:02:35 --> 00:02:43
( x2), all the way up to
yn, which = f ( xn).
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00:02:43 --> 00:02:46
And again, the spacing
here that we're talking
39
00:02:46 --> 00:02:50
about is b - a / n.
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00:02:50 --> 00:02:52
So remember that spacing,
that's going to be the
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connection that we'll draw.
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00:02:57 --> 00:03:07
Then the Riemann sum is y1
through yn, the sum of
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00:03:07 --> 00:03:12
(y1 ... yn) delta x.
44
00:03:12 --> 00:03:23
And that's what tends, as delta
x goes to 0, to the integral. .
45
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The only change in point of
view if I want to write this
46
00:03:27 --> 00:03:32
limiting property, which is
right above here, the only
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change between here and here is
that I want to divide by the
48
00:03:36 --> 00:03:38
length of the interval. b - a.
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00:03:38 --> 00:03:42
So I will divide by b - a here.
50
00:03:42 --> 00:03:48
And divide by b - a over here.
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00:03:48 --> 00:03:53
And then I'll just check what
this thing actually is.
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Delta x / b - a, what
is that factor?
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Well, if we look over here
to what delta x is, if you
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00:04:01 --> 00:04:07
divide by b - a, it's 1 / n.
55
00:04:07 --> 00:04:10
So the factor delta
x / b - a = 1 / n.
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00:04:10 --> 00:04:15
That's what I put over here,
the sum of y1 through yn / n.
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And as this tends to
0, it's the same as
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n going to infinity.
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Those are the same things.
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The average value and
the integral are very
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closely related.
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There's only this difference
that we're dividing by the
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length of the interval.
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I want to give an example which
is an incredibly simpleminded
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one, but it'll come
into play later on.
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So let's take the
example of a constant.
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And this is, I hope, will make
you slightly less confused
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about what I just wrote.
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As well as making you think
that this is as simpleminded
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and reasonable as it should be.
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If I check what the average
value of this constant is, it's
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given by this relatively
complicated formula here.
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That is, I have to
integrate the function c.
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Well, it's just the constant c.
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And however you do this, as an
antiderivative I was thinking
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of it as a rectangle, the
answer that you're going
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to get is c here.
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So work that out.
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The answer is c.
80
00:05:20 --> 00:05:26
And so the fact that the
average of c = c, which had
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00:05:26 --> 00:05:30
better be the case for
averages, explains
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00:05:30 --> 00:05:31
the denominator.
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Explains the 1 / b - a there.
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That's cooked up exactly so
that the average of a constant
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is what it's supposed to be.
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Otherwise we have the
wrong normalizing factor.
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We've clearly got a piece
of nonsense on our hands.
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And incidentally, it also
explains the 1 / n in the
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very first formula
that I wrote down.
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The reason why this is called
the average, or one reason why
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it's the right thing, is that
if you took the same constant
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00:05:59 --> 00:06:02
c, for y all the way across
there n times, if you divide
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00:06:02 --> 00:06:03
it by n, you get back c.
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00:06:03 --> 00:06:05
That's what we mean by
average value and that's
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why the n is there.
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00:06:11 --> 00:06:14
So that was an easy example.
97
00:06:14 --> 00:06:17
Now none of the examples that
we are going to give are going
98
00:06:17 --> 00:06:20
to be all that complicated.
99
00:06:20 --> 00:06:23
But they will get sort of
steadily more sophisticated.
100
00:06:23 --> 00:06:35
The second example is going
to be the average height of
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00:06:35 --> 00:06:47
a point on a semicircle.
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00:06:47 --> 00:06:52
And maybe I'll draw a picture
of the semicircle first here.
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And we'll just make it
the standard circle,
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00:06:56 --> 00:06:58
the unit circle.
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So maybe I should have called
it a unit semicircle.
106
00:07:04 --> 00:07:07
This is the point negative
1, this is the point 1.
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00:07:07 --> 00:07:11
And we're picking a point
over here and we're going
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00:07:11 --> 00:07:17
to take the typical, or
the average, height here.
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Integrating with respect to dx.
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00:07:18 --> 00:07:21
So sort of continuously
with respect to dx.
111
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Well, what is that?
112
00:07:25 --> 00:07:29
Well, according to the rule,
it's 1 / b - a times - sorry,
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it's up here in the box.
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1 / b - a, the interval
from a to b, f ( x) dx.
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00:07:34 --> 00:07:39
That's 1 / + 1 - (- 1).
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00:07:39 --> 00:07:45
The integral from - 1 to 1,
square root of 1 - x ^2 dx.
117
00:07:45 --> 00:07:52
Right, because the height
is y = , this is y = the
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00:07:52 --> 00:07:57
square root of 1 - x^2.
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00:07:57 --> 00:08:02
And to evaluate this is not
as difficult as it seems.
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00:08:02 --> 00:08:06
This is 1/2 times this
quantity here, which we
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00:08:06 --> 00:08:08
can interpret as an area.
122
00:08:08 --> 00:08:12
It's the area of
the semicircle.
123
00:08:12 --> 00:08:21
So this is the area of the
semicircle, which we know to be
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half the area of the circle.
125
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So it's pi / 2.
126
00:08:26 --> 00:08:35
And so the answer, here the
average height, is pi/ 4.
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Now, later in the class and
actually not in this unit,
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we'll actually be able
to calculate the
129
00:08:40 --> 00:08:42
antiderivative of this.
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00:08:42 --> 00:08:44
So in other words, we'll
be able to calculate
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00:08:44 --> 00:08:45
this analytically.
132
00:08:45 --> 00:08:48
For right now we just have
the geometric reason why the
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00:08:48 --> 00:08:51
value of this is pi / 2.
134
00:08:51 --> 00:08:53
And we'll do that in the fourth
unit when we do a lot of
135
00:08:53 --> 00:08:57
techniques of integration.
136
00:08:57 --> 00:08:58
So here's an example.
137
00:08:58 --> 00:09:03
Turns out, the average
height of this is pi / 4.
138
00:09:03 --> 00:09:07
Now, the next example that I
want to give introduces a
139
00:09:07 --> 00:09:09
little bit of confusion.
140
00:09:09 --> 00:09:13
And I'm not going to resolve
this confusion completely,
141
00:09:13 --> 00:09:15
but I'm going to try
to get you used to it.
142
00:09:15 --> 00:09:22
I'm going to take the
average height again.
143
00:09:22 --> 00:09:33
But now, with respect
to arc length.
144
00:09:33 --> 00:09:37
Which is usually denoted theta.
145
00:09:37 --> 00:09:40
Now, this brings up an
extremely important
146
00:09:40 --> 00:09:43
feature of averages.
147
00:09:43 --> 00:09:47
Which is that you have to
specify the variable with
148
00:09:47 --> 00:09:51
respect to which the
average is taking place.
149
00:09:51 --> 00:09:53
And the answer will be
different depending
150
00:09:53 --> 00:09:54
on the variable.
151
00:09:54 --> 00:09:56
So it's not going
to be the same.
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00:09:56 --> 00:09:58
Wow, can't spell the
word length here.
153
00:09:58 --> 00:10:02
Just like the plural of
witches the last time.
154
00:10:02 --> 00:10:03
We'll work on that.
155
00:10:03 --> 00:10:11
We'll fix all of our, that's an
ancient Gaelic word, I think.
156
00:10:11 --> 00:10:16
Lengh.
157
00:10:16 --> 00:10:22
So now, let me show you that
it's not quite the same here.
158
00:10:22 --> 00:10:25
It's especially exaggerated
if maybe I shift this
159
00:10:25 --> 00:10:30
little interval dx over
to the right-hand end.
160
00:10:30 --> 00:10:33
And you can see that the little
portion that corresponds to it,
161
00:10:33 --> 00:10:37
which is the d theta piece, has
a different length
162
00:10:37 --> 00:10:39
from the dx piece.
163
00:10:39 --> 00:10:42
And indeed, as you come down
here, these very short portions
164
00:10:42 --> 00:10:47
of dx length have much longer
portions of theta length.
165
00:10:47 --> 00:10:51
So that the average that we're
taking when we do it with
166
00:10:51 --> 00:10:56
respect to theta, is going to
emphasize the low values more.
167
00:10:56 --> 00:10:58
They're going to be
more exaggerated.
168
00:10:58 --> 00:11:02
And the average should be
lower than the average
169
00:11:02 --> 00:11:03
that we got here.
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00:11:03 --> 00:11:04
So we should expect
a different number.
171
00:11:04 --> 00:11:06
And it's not going to be
pi / 4, it's going to
172
00:11:06 --> 00:11:07
be something else.
173
00:11:07 --> 00:11:12
Whatever it is, it should
be smaller than pi / 4.
174
00:11:12 --> 00:11:14
Now, let's set up the integral.
175
00:11:14 --> 00:11:17
The integral follows
the same rule.
176
00:11:17 --> 00:11:20
It's just 1 over the length of
the interval times the integral
177
00:11:20 --> 00:11:24
over the interval
of the function.
178
00:11:24 --> 00:11:27
That's the integral, but now
where does theta range?
179
00:11:27 --> 00:11:32
This time, theta
goes from 0 to pi.
180
00:11:32 --> 00:11:34
So the integral
is from 0 to pi.
181
00:11:34 --> 00:11:40
And the thing we
divide by is pi.
182
00:11:40 --> 00:11:43
And the integration requires
us to know the formula
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00:11:43 --> 00:11:44
for the height.
184
00:11:44 --> 00:11:47
Which is sin theta.
185
00:11:47 --> 00:11:48
In terms of theta, of course.
186
00:11:48 --> 00:11:51
It's the same as square root
of - x^2, but it's expressed
187
00:11:51 --> 00:11:53
in terms of theta.
188
00:11:53 --> 00:11:55
So it's this.
189
00:11:55 --> 00:11:58
And here's our average.
190
00:11:58 --> 00:12:01
I'll put this up here.
191
00:12:01 --> 00:12:09
So that's the formula
for the height.
192
00:12:09 --> 00:12:10
So let's work it out.
193
00:12:10 --> 00:12:13
This one, we have the advantage
of being able to work
194
00:12:13 --> 00:12:16
out because we know the
antiderivative of sin theta.
195
00:12:16 --> 00:12:25
It happens with this factor
of pi, it's - cos theta.
196
00:12:25 --> 00:12:35
And so, that's - 1 / pi ( cos
pi), sorry. (cos pi - cos) 0.
197
00:12:35 --> 00:12:47
Which is - 1 / pi ( -
2), which is 2 / pi.
198
00:12:47 --> 00:12:53
And sure enough, if you check
it, you'll see that 2 / pi
199
00:12:53 --> 00:13:02
8.
200
00:13:02 --> 00:13:02
Yeah, question.
201
00:13:02 --> 00:13:06
STUDENT: [INAUDIBLE]
202
00:13:06 --> 00:13:09
PROFESSOR: The question is
how do I get sin theta.
203
00:13:09 --> 00:13:16
And the answer is, on this
diagram, if theta is over here
204
00:13:16 --> 00:13:20
then this height is this, and
this is the angle theta, then
205
00:13:20 --> 00:13:22
the height is the sine.
206
00:13:22 --> 00:13:26
OK.
207
00:13:26 --> 00:13:27
Another question.
208
00:13:27 --> 00:13:32
STUDENT: [INAUDIBLE]
209
00:13:32 --> 00:13:41
PROFESSOR: The question is,
what is the first one, the
210
00:13:41 --> 00:13:44
first one is an average of
height, of a point on a
211
00:13:44 --> 00:13:51
semicircle and this one
is with respect to x.
212
00:13:51 --> 00:13:54
So what this reveals is that
it's ambiguous to say what the
213
00:13:54 --> 00:13:57
average value of something is,
unless you've explained
214
00:13:57 --> 00:14:00
what the underlying
averaging variable is.
215
00:14:00 --> 00:14:08
STUDENT: [INAUDIBLE]
216
00:14:08 --> 00:14:10
PROFESSOR: The next
question is how should
217
00:14:10 --> 00:14:13
you interpret this value.
218
00:14:13 --> 00:14:17
That is, what came out
of this calculation?
219
00:14:17 --> 00:14:22
And the answer is only
sort of embedded in this
220
00:14:22 --> 00:14:26
calculation itself.
221
00:14:26 --> 00:14:28
So here's a way of thinking
of it which is anticipating
222
00:14:28 --> 00:14:29
our next subject.
223
00:14:29 --> 00:14:31
Which is probability.
224
00:14:31 --> 00:14:34
Which is, suppose you
picked a number at
225
00:14:34 --> 00:14:36
random in this interval.
226
00:14:36 --> 00:14:39
With equal likelihood,
one place and another.
227
00:14:39 --> 00:14:42
And then you saw what
height was above that.
228
00:14:42 --> 00:14:43
That would be the
interpretation of this
229
00:14:43 --> 00:14:45
first average value.
230
00:14:45 --> 00:14:47
And the second one is,
I picked something at
231
00:14:47 --> 00:14:50
random on this circle.
232
00:14:50 --> 00:14:53
And equally likely, any
possible point on this circle
233
00:14:53 --> 00:14:54
according to its length.
234
00:14:54 --> 00:14:58
And then I ask what the
height of that point is.
235
00:14:58 --> 00:15:04
And those are just
different things.
236
00:15:04 --> 00:15:05
Another question.
237
00:15:05 --> 00:15:09
STUDENT: [INAUDIBLE]
238
00:15:09 --> 00:15:12
PROFESSOR: Cos pi,
shouldn't it be 0?
239
00:15:12 --> 00:15:16
No. cos of, it's - 1.
240
00:15:16 --> 00:15:19
Cos pi = - 1.
241
00:15:19 --> 00:15:25
Cosine, sorry.
242
00:15:25 --> 00:15:29
No, cos 0 = 1. cos pi = - 1.
243
00:15:29 --> 00:15:30
And so they cancel.
244
00:15:30 --> 00:15:31
That is, they don't cancel.
245
00:15:31 --> 00:15:35
It's - 1 - 1, which = - 2.
246
00:15:35 --> 00:15:38
Key point.
247
00:15:38 --> 00:15:38
Yeah.
248
00:15:38 --> 00:15:49
STUDENT: [INAUDIBLE]
249
00:15:49 --> 00:15:51
PROFESSOR: All right,
let me repeat.
250
00:15:51 --> 00:15:57
So the question was to repeat
the reasoning by which I
251
00:15:57 --> 00:16:01
guessed in advance that
probably this was going to be
252
00:16:01 --> 00:16:04
the relationship between the
average value with respect to
253
00:16:04 --> 00:16:07
arc length versus the average
value with respect to this
254
00:16:07 --> 00:16:10
horizontal distance.
255
00:16:10 --> 00:16:13
And it had to do with
the previous way this
256
00:16:13 --> 00:16:16
diagram was drawn.
257
00:16:16 --> 00:16:21
Which is comparing an
interval in dx with
258
00:16:21 --> 00:16:25
an interval in theta.
259
00:16:25 --> 00:16:27
A little section in theta.
260
00:16:27 --> 00:16:32
And when you're near the top,
they're nearly this same.
261
00:16:32 --> 00:16:34
That is, it's more
or less balanced.
262
00:16:34 --> 00:16:36
It's a little curved here,
a little different.
263
00:16:36 --> 00:16:39
But here it becomes
very exaggerated.
264
00:16:39 --> 00:16:42
The d theta lengths are much
longer than the dx lengths.
265
00:16:42 --> 00:16:47
Which means that importance
given by the theta of variable
266
00:16:47 --> 00:16:51
to these parts of the circle is
larger, relative
267
00:16:51 --> 00:16:52
to these parts.
268
00:16:52 --> 00:16:55
Whereas if you look at this
section versus this section for
269
00:16:55 --> 00:16:59
the dx, they give equal weights
to these two equal lengths.
270
00:16:59 --> 00:17:01
But here, with respect to
theta, this is relatively short
271
00:17:01 --> 00:17:03
and this is much larger.
272
00:17:03 --> 00:17:05
So, as I say, the theta
variable's emphasizing
273
00:17:05 --> 00:17:09
the lower parts of
the semicircle more.
274
00:17:09 --> 00:17:11
That's because this length
is shorter and this
275
00:17:11 --> 00:17:12
length is longer.
276
00:17:12 --> 00:17:16
Whereas these two are the same.
277
00:17:16 --> 00:17:20
It's a balancing act of
the relative weights.
278
00:17:20 --> 00:17:22
I'm going to say that
again in a different way,
279
00:17:22 --> 00:17:25
and maybe this will.
280
00:17:25 --> 00:17:31
The lower part is more
important for theta.
281
00:17:31 --> 00:17:32
STUDENT: [INAUDIBLE]
282
00:17:32 --> 00:17:33
PROFESSOR: So the question is,
but shouldn't it have a bigger
283
00:17:33 --> 00:17:35
value because it's
a longer length.
284
00:17:35 --> 00:17:37
Never with averages.
285
00:17:37 --> 00:17:39
Whatever the length is,
we're always dividing.
286
00:17:39 --> 00:17:42
We're always compensating
by the total.
287
00:17:42 --> 00:17:45
We have the integral from 0 to
pi, but we're dividing by pi.
288
00:17:45 --> 00:17:48
Here we had the integral
from - 1 to 1, but
289
00:17:48 --> 00:17:49
we're dividing by 2.
290
00:17:49 --> 00:17:52
So we divide by something
different each time.
291
00:17:52 --> 00:17:53
And this is very,
very important.
292
00:17:53 --> 00:17:56
It's that the average of a
constant is that same constant
293
00:17:56 --> 00:17:58
regardless of which one we did.
294
00:17:58 --> 00:18:00
So if it were a constant,
we would always compensate
295
00:18:00 --> 00:18:01
for the length.
296
00:18:01 --> 00:18:03
So the length never matters.
297
00:18:03 --> 00:18:09
If it's the integral from 0 to
1,000,000, or 100, let's say,
298
00:18:09 --> 00:18:12
1/100 cdx, it's just the same.
299
00:18:12 --> 00:18:14
It's always that, it doesn't
matter how long it is.
300
00:18:14 --> 00:18:18
Because we compensate.
301
00:18:18 --> 00:18:19
That's really the difference
between an integral and an
302
00:18:19 --> 00:18:24
average, is that we're
dividing by the total.
303
00:18:24 --> 00:18:28
Now I want to introduce another
notion, which is actually
304
00:18:28 --> 00:18:30
what's underlying these
two examples that
305
00:18:30 --> 00:18:32
I just wrote down.
306
00:18:32 --> 00:18:35
And this is by far the one
which you should emphasize
307
00:18:35 --> 00:18:39
the most in your thoughts.
308
00:18:39 --> 00:18:45
Because it is much more
flexible, and is much more
309
00:18:45 --> 00:18:49
typical of real life problems.
310
00:18:49 --> 00:18:53
So the idea of a weighted
average is the following.
311
00:18:53 --> 00:18:57
You take the integral, say from
a to b, of some function.
312
00:18:57 --> 00:19:02
But now you multiply
by a weight.
313
00:19:02 --> 00:19:05
And you have to
divide by the total.
314
00:19:05 --> 00:19:07
And what's the
total going to be?
315
00:19:07 --> 00:19:10
It's the integral from
a to b of this total
316
00:19:10 --> 00:19:14
weighting that we have.
317
00:19:14 --> 00:19:17
Now, why is this the
correct notion?
318
00:19:17 --> 00:19:20
I'm going to explain it
to you in two ways.
319
00:19:20 --> 00:19:24
The first is this very
simpleminded thing that I
320
00:19:24 --> 00:19:30
wrote on the board there,
with the constants.
321
00:19:30 --> 00:19:38
What we want is the average
value of c to be c.
322
00:19:38 --> 00:19:40
Otherwise this makes no
sense as an average.
323
00:19:40 --> 00:19:43
Now, let's just look at
this definition here.
324
00:19:43 --> 00:19:44
And see that that's correct.
325
00:19:44 --> 00:19:51
If you integrate c, from a to
b, w ( x) dx, and you divide by
326
00:19:51 --> 00:19:55
the integral from a to b, w (
x) dx, not surprisingly,
327
00:19:55 --> 00:19:57
the c factors out.
328
00:19:57 --> 00:19:59
It's a constant.
329
00:19:59 --> 00:20:03
So this is c times the integral
a to b, w (x), dx, divided
330
00:20:03 --> 00:20:05
by the same thing.
331
00:20:05 --> 00:20:08
And that's why we picked it.
332
00:20:08 --> 00:20:10
We picked it so that these
things would cancel.
333
00:20:10 --> 00:20:14
And this would give c.
334
00:20:14 --> 00:20:17
So in the previous case,
this property explains
335
00:20:17 --> 00:20:20
the denominator.
336
00:20:20 --> 00:20:29
And again over here, it
explains the denominator.
337
00:20:29 --> 00:20:32
And let me just give you
one more explanation.
338
00:20:32 --> 00:20:38
Which is maybe a real life
pretend real life example.
339
00:20:38 --> 00:20:43
You have a stock which you
bought for $10 one year.
340
00:20:43 --> 00:20:46
And then six months later you
brought some more for $20.
341
00:20:46 --> 00:20:50
And then you bought
some more for $30.
342
00:20:50 --> 00:20:53
Now, what's the average
price of your stock?
343
00:20:53 --> 00:20:58
Well, it depends on how
many shares you bought.
344
00:20:58 --> 00:21:01
If you bought this many shares
the first time, and this many
345
00:21:01 --> 00:21:03
shares the second time, and
this many shares the third
346
00:21:03 --> 00:21:07
time, this is the total
amount that you spent.
347
00:21:07 --> 00:21:14
And the average price is the
total price divided by the
348
00:21:14 --> 00:21:17
total number of shares.
349
00:21:17 --> 00:21:22
And this is the discrete
analog of this continuous
350
00:21:22 --> 00:21:24
averaging process here.
351
00:21:24 --> 00:21:27
The function f now, so I use
w for weight, the function
352
00:21:27 --> 00:21:31
f now is the function whose
values are 10, 20 and 30.
353
00:21:31 --> 00:21:35
And the weightings are the
relative importance of
354
00:21:35 --> 00:21:42
the different purchases.
355
00:21:42 --> 00:21:51
So again, these
wi's are weights.
356
00:21:51 --> 00:21:52
There was another question.
357
00:21:52 --> 00:21:54
Out in the audience,
at some point.
358
00:21:54 --> 00:21:55
Over here, yes.
359
00:21:55 --> 00:22:04
STUDENT: [INAUDIBLE]
360
00:22:04 --> 00:22:06
PROFESSOR: Very,
very good point.
361
00:22:06 --> 00:22:15
So in this numerator here,
the statement is, in this
362
00:22:15 --> 00:22:17
example, we factored out c.
363
00:22:17 --> 00:22:20
But here we cannot
factor out f ( x).
364
00:22:20 --> 00:22:23
That's extremely important
and that is the whole point.
365
00:22:23 --> 00:22:25
So, in other words, the
weighted average is very
366
00:22:25 --> 00:22:29
interesting you have to do
two different integrals to
367
00:22:29 --> 00:22:32
figure it out in general.
368
00:22:32 --> 00:22:34
When it happens that this
is c, it's an extremely
369
00:22:34 --> 00:22:35
boring integral.
370
00:22:35 --> 00:22:37
Which in fact because, it's
an average, you don't even
371
00:22:37 --> 00:22:38
have to calculate at all.
372
00:22:38 --> 00:22:41
Factor it out and cancel these
things and never bother to
373
00:22:41 --> 00:22:43
calculate these two numbers.
374
00:22:43 --> 00:22:46
So these massive
numbers just cancel.
375
00:22:46 --> 00:22:48
So it's a very special
property of a constant,
376
00:22:48 --> 00:22:55
that it factors out.
377
00:22:55 --> 00:23:00
That was our first discussion,
and now with this example I'm
378
00:23:00 --> 00:23:03
going to go back to the heating
up of the witches' cauldron and
379
00:23:03 --> 00:23:08
we'll use average value to
illustrate the integral that we
380
00:23:08 --> 00:23:19
get in that context as well.
381
00:23:19 --> 00:23:20
I remind you, let's see.
382
00:23:20 --> 00:23:25
The situation with the
witches' cauldron was this.
383
00:23:25 --> 00:23:40
The first important thing is
that there were, so this
384
00:23:40 --> 00:23:42
is the big cauldron here.
385
00:23:42 --> 00:23:47
This is the one whose height
is 1 meter and whose
386
00:23:47 --> 00:23:53
width is 2 meters.
387
00:23:53 --> 00:23:56
And it's a parabola
of revolution here.
388
00:23:56 --> 00:24:06
And it had about approximately
1600 liters in it.
389
00:24:06 --> 00:24:14
And this curve was y = x^2.
390
00:24:14 --> 00:24:18
And the situation that I
described at the end of last
391
00:24:18 --> 00:24:26
time was that the initial
temperature was T =
392
00:24:26 --> 00:24:28
0 degrees Celsius.
393
00:24:28 --> 00:24:35
And the final temperature,
instead of being a constant
394
00:24:35 --> 00:24:41
temperature, we were heating
this guy up from the bottom.
395
00:24:41 --> 00:24:48
And it was hotter on the
bottom than on the top.
396
00:24:48 --> 00:24:53
And the final temperature was
given by the formula T = 100
397
00:24:53 --> 00:24:57
- 30 times the height y.
398
00:24:57 --> 00:25:04
So at y = 0, at the
bottom, it's 100.
399
00:25:04 --> 00:25:10
And at the top, T = 70 degrees.
400
00:25:10 --> 00:25:14
OK, so this is the final
configuration for
401
00:25:14 --> 00:25:16
the temperature.
402
00:25:16 --> 00:25:33
And the question was how
much energy do we need.
403
00:25:33 --> 00:25:37
So, the first observation here,
and this is the reason for
404
00:25:37 --> 00:25:42
giving this example, is that
it's important to realize that
405
00:25:42 --> 00:25:54
you want to use the method
of disks in this case.
406
00:25:54 --> 00:25:58
The reason, so it doesn't have
to do with, you shouldn't
407
00:25:58 --> 00:26:00
think of the disks first.
408
00:26:00 --> 00:26:05
But what you should think
of is the horizontal.
409
00:26:05 --> 00:26:10
We must use horizontals because
T is constant on horizontals.
410
00:26:10 --> 00:26:12
It's not constant on verticals.
411
00:26:12 --> 00:26:16
If we set things up with
shells, as we did last time, to
412
00:26:16 --> 00:26:19
compute the volume of this,
then T will vary
413
00:26:19 --> 00:26:21
along the shell.
414
00:26:21 --> 00:26:24
And we will still have an
averaging problem, an
415
00:26:24 --> 00:26:26
integral problem along
the vertical portion.
416
00:26:26 --> 00:26:29
But if we do it this way,
T is constant on this
417
00:26:29 --> 00:26:31
whole level here.
418
00:26:31 --> 00:26:35
And so there's no more calculus
involved in calculating what
419
00:26:35 --> 00:26:39
the contribution is
of any given level.
420
00:26:39 --> 00:26:49
So t is constant
on horizontals.
421
00:26:49 --> 00:26:52
Actually, in disguise,
this is that same trick
422
00:26:52 --> 00:26:53
that we have here.
423
00:26:53 --> 00:26:55
We can factor constants
out of integrals.
424
00:26:55 --> 00:26:57
You could view it as an
integral, but the point is that
425
00:26:57 --> 00:27:03
it's more elementary than that.
426
00:27:03 --> 00:27:06
Now I have to set
it up for you.
427
00:27:06 --> 00:27:08
And in order to do that,
I need to remember
428
00:27:08 --> 00:27:10
what the equation is.
429
00:27:10 --> 00:27:12
Which is y = x ^2.
430
00:27:12 --> 00:27:19
And the formula for the total
amount of energy is going to
431
00:27:19 --> 00:27:25
be volume times the
number of degrees.
432
00:27:25 --> 00:27:31
That's going to be equal to
the energy that we need here.
433
00:27:31 --> 00:27:33
And so let's add it up.
434
00:27:33 --> 00:27:38
It's the integral from 0 to 1,
and this is with respect to y.
435
00:27:38 --> 00:27:41
So the y level
goes from 0 to 1.
436
00:27:41 --> 00:27:47
This top level's y = 1,
this bottom level's y = 0.
437
00:27:47 --> 00:27:54
And the disk that we get,
this is the point (x, y)
438
00:27:54 --> 00:27:56
here, is rotated around.
439
00:27:56 --> 00:28:01
And its radius is x.
440
00:28:01 --> 00:28:07
So the thickness is dy, and the
area of the disk is pi x^2.
441
00:28:09 --> 00:28:13
And the thing that
we're averaging is T.
442
00:28:13 --> 00:28:16
Well, we're not yet averaging,
we're just integrating it.
443
00:28:16 --> 00:28:24
We're just adding up the total.
444
00:28:24 --> 00:28:29
Now I'm just going to plug in
the various values for this.
445
00:28:29 --> 00:28:36
And what I'm going to get
is T, again, = 100 - 30y.
446
00:28:36 --> 00:28:40
And this radius is measured
up to this very end.
447
00:28:40 --> 00:28:42
So x^2 = y.
448
00:28:42 --> 00:28:45
So this is pi y dy.
449
00:28:45 --> 00:28:47
And this is the integral that
we'll be able to evaluate.
450
00:28:47 --> 00:28:48
Yeah, question.
451
00:28:48 --> 00:28:50
STUDENT: [INAUDIBLE]
452
00:28:50 --> 00:29:00
PROFESSOR: All right.
453
00:29:00 --> 00:29:05
Well, let's carry this out.
454
00:29:05 --> 00:29:09
Let's finish off the
calculation here.
455
00:29:09 --> 00:29:10
Let's see.
456
00:29:10 --> 00:29:16
This is equal to, what
does it equal to.
457
00:29:16 --> 00:29:19
Well, I'll put it over here.
458
00:29:19 --> 00:29:27
It's equal to 50 pi y ^2 -
right, because this is 100 pi
459
00:29:27 --> 00:29:36
y, and then there's a 30, this
is 100 pi y - 30 pi y ^2,
460
00:29:36 --> 00:29:38
and I have to take the
antiderivative of that.
461
00:29:38 --> 00:29:45
So I get 50 pi y ^2,
and I get 10 pi y ^3.
462
00:29:45 --> 00:29:48
Evaluate it at 0 and 1.
463
00:29:48 --> 00:29:57
And that is 40 pi.
464
00:29:57 --> 00:30:03
Now, I spent a tremendous
amount of time last
465
00:30:03 --> 00:30:07
time focusing on units.
466
00:30:07 --> 00:30:10
Because I want to tell you
how to get a realistic
467
00:30:10 --> 00:30:12
number out of this.
468
00:30:12 --> 00:30:16
And there's a subtle point here
that I pointed out last time
469
00:30:16 --> 00:30:19
that had to do with changing
meters to centimeters.
470
00:30:19 --> 00:30:22
I claim that I've treated
those correctly.
471
00:30:22 --> 00:30:28
So, what we have here is that
the answer is in degrees, that
472
00:30:28 --> 00:30:34
is Celsius, times cubic meters.
473
00:30:34 --> 00:30:36
These are the correct units.
474
00:30:36 --> 00:30:43
And now, I can translate
this into Celsius is
475
00:30:43 --> 00:30:44
spelled with a C.
476
00:30:44 --> 00:30:44
That's interesting.
477
00:30:44 --> 00:30:46
Celsius.
478
00:30:46 --> 00:30:50
I can translate this into units
that you're more familiar with.
479
00:30:50 --> 00:30:58
So let's try 40 pi deg *
m ^3, and then do the
480
00:30:58 --> 00:30:59
conversion factors.
481
00:30:59 --> 00:31:05
First of all there's
one calorie per degree
482
00:31:05 --> 00:31:08
times a milliliter.
483
00:31:08 --> 00:31:11
That's one conversion.
484
00:31:11 --> 00:31:14
And then let's see.
485
00:31:14 --> 00:31:17
I'm going to have to translate
from centimeters so I have
486
00:31:17 --> 00:31:25
here (100 cm / m)^ 3.
487
00:31:25 --> 00:31:30
So these are the two conversion
factors that I need.
488
00:31:30 --> 00:31:37
And so, I get 40 pi (
10 ^ 6), that's 100^3.
489
00:31:38 --> 00:31:46
And this is in calories.
490
00:31:46 --> 00:31:48
So how much is this?
491
00:31:48 --> 00:31:53
Well, it's a little better,
maybe, to do it in 40 pi *
492
00:31:53 --> 00:31:59
1,000 kilocalories, because
these are the ones that
493
00:31:59 --> 00:32:05
you actually see on your
nutrition labels of foods.
494
00:32:05 --> 00:32:12
And so this number is
around 125 or so.
495
00:32:12 --> 00:32:15
Let's see, is that about right?
496
00:32:15 --> 00:32:17
Let's make sure I've got
these numbers right.
497
00:32:17 --> 00:32:20
Yeah, this is about 125.
498
00:32:20 --> 00:32:22
40 pi.
499
00:32:22 --> 00:32:32
And so one candy bar, this
is a Halloween example, so.
500
00:32:32 --> 00:32:38
One candy bar is about
250 kilocalories.
501
00:32:38 --> 00:32:44
So this is half a candy bar.
502
00:32:44 --> 00:32:53
So the answer to our question
is that it takes 500 candy
503
00:32:53 --> 00:33:02
bars to heat up this thing.
504
00:33:02 --> 00:33:07
OK, so that's our example.
505
00:33:07 --> 00:33:08
Now, yeah.
506
00:33:08 --> 00:33:09
Question.
507
00:33:09 --> 00:33:13
STUDENT: [INAUDIBLE]
508
00:33:13 --> 00:33:16
PROFESSOR: What does
the integral give us?
509
00:33:16 --> 00:33:22
This integral is, the
integral represents
510
00:33:22 --> 00:33:23
the following things.
511
00:33:23 --> 00:33:26
So the question is, what
does this integral give us.
512
00:33:26 --> 00:33:27
So here's the integral.
513
00:33:27 --> 00:33:30
Here it is, rewritten so
that it can be calculated.
514
00:33:30 --> 00:33:33
And what this integral
is giving us is the
515
00:33:33 --> 00:33:34
following thing.
516
00:33:34 --> 00:33:36
You have to imagine
the following idea.
517
00:33:36 --> 00:33:39
You've got a little
chunk of water in here.
518
00:33:39 --> 00:33:42
And you're going to raise is
from 0 degrees all the way up
519
00:33:42 --> 00:33:46
to whatever the target
temperature is.
520
00:33:46 --> 00:33:51
And so that little milliliter
of water, if you like, has
521
00:33:51 --> 00:33:53
to be raised from 0 to
some number which is a
522
00:33:53 --> 00:33:56
function of the height.
523
00:33:56 --> 00:33:59
It's something between
70 and 100 degrees.
524
00:33:59 --> 00:34:04
And the one right above it
also has to be raised to a
525
00:34:04 --> 00:34:06
temperature, although a
slightly different temperature.
526
00:34:06 --> 00:34:08
And what we're doing with the
integral is we're adding up
527
00:34:08 --> 00:34:15
all of those degrees and the
calorie represents how much it
528
00:34:15 --> 00:34:18
takes, one calorie represents
how much it takes to raise by 1
529
00:34:18 --> 00:34:21
degree 1 milliliter of water.
530
00:34:21 --> 00:34:26
One cubic centimeter of water.
531
00:34:26 --> 00:34:31
That's the definition
of a calorie.
532
00:34:31 --> 00:34:32
And we're adding it up.
533
00:34:32 --> 00:34:35
So in other words, each of
these cubes is one thing.
534
00:34:35 --> 00:34:37
And now we have to add it up
over this massive thing,
535
00:34:37 --> 00:34:40
which is 1600 liters.
536
00:34:40 --> 00:34:42
And we have a lot of
different little cubes.
537
00:34:42 --> 00:34:43
And that's what we did.
538
00:34:43 --> 00:34:45
When we glommed
them all together.
539
00:34:45 --> 00:34:48
That's what the integral
is doing for us.
540
00:34:48 --> 00:34:54
Other questions.
541
00:34:54 --> 00:34:57
Now I want to connect this
with weighted averages
542
00:34:57 --> 00:34:58
before we go on.
543
00:34:58 --> 00:35:03
Because that was the reason why
I did weighted averages first.
544
00:35:03 --> 00:35:14
I'm going to compute also the
average final temperature.
545
00:35:14 --> 00:35:18
So, final because this is the
interesting one, the average
546
00:35:18 --> 00:35:21
starting temperature's
very boring, it's 0.
547
00:35:21 --> 00:35:26
The average final temperature
is, individually the
548
00:35:26 --> 00:35:27
temperatures are different.
549
00:35:27 --> 00:35:34
And the answer here is it's the
integral from 0 to 1 of T pi y
550
00:35:34 --> 00:35:42
dy divided by the integral
from 0 to 1 of pi y dy.
551
00:35:42 --> 00:35:44
So this is the total
temperature, weighted
552
00:35:44 --> 00:35:48
appropriately to the volume of
water that's involved at that
553
00:35:48 --> 00:35:52
temperature, divided by the
total volume of water.
554
00:35:52 --> 00:35:55
And we computed
these two numbers.
555
00:35:55 --> 00:35:58
The number in the numerator
is what we call 40 pi.
556
00:35:58 --> 00:36:00
And the number in the
denominator, actually this is
557
00:36:00 --> 00:36:03
easier than what we did last
time with shells you can just
558
00:36:03 --> 00:36:06
look at this and see that it's
the area under a triangle.
559
00:36:06 --> 00:36:08
It's pi / 2.
560
00:36:08 --> 00:36:11
And so the answer
here is 80 degrees.
561
00:36:11 --> 00:36:14
This is the average
temperature.
562
00:36:14 --> 00:36:17
Note that this is a
weighted average.
563
00:36:17 --> 00:36:22
The weighting here is different
according to the height.
564
00:36:22 --> 00:36:28
The weighting factor is pi y.
565
00:36:28 --> 00:36:30
That's the weighting factor.
566
00:36:30 --> 00:36:32
And that's not surprising.
567
00:36:32 --> 00:36:35
When y is small, there's
less volume down here.
568
00:36:35 --> 00:36:38
Up above, those are more
important volumes, because
569
00:36:38 --> 00:36:41
there's more water up at the
top of the cauldron than
570
00:36:41 --> 00:36:43
there is down at the
bottom of the cauldron.
571
00:36:43 --> 00:36:46
If you compare this to the
ordinary average, if you take
572
00:36:46 --> 00:36:51
the maximum temperature plus
the minimum temperature,
573
00:36:51 --> 00:36:56
divided by 2, that
would be 100 + 70 / 2.
574
00:36:56 --> 00:36:59
You would get 85 degrees.
575
00:36:59 --> 00:37:01
And that's bigger.
576
00:37:01 --> 00:37:02
Why?
577
00:37:02 --> 00:37:05
Because the cooler
water is on top.
578
00:37:05 --> 00:37:08
And the actual average, the
correct weighted average, is
579
00:37:08 --> 00:37:11
lower than this fake average.
580
00:37:11 --> 00:37:15
Which is not the true
average in this context.
581
00:37:15 --> 00:37:18
All right so the weighting is
that the thing is getting
582
00:37:18 --> 00:37:33
fatter near the top.
583
00:37:33 --> 00:37:38
So now I'm going to do another
example of weighted average.
584
00:37:38 --> 00:37:46
And this example is also
very much worth your while.
585
00:37:46 --> 00:37:49
It's the other incredibly
important one in
586
00:37:49 --> 00:37:51
interpreting integrals.
587
00:37:51 --> 00:37:56
And it's a very, very simple
example of a function, f.
588
00:37:56 --> 00:38:00
The weightings will be
different, but the functions,
589
00:38:00 --> 00:38:03
f, will be of a very
particular kind.
590
00:38:03 --> 00:38:07
Namely, the function f will
be practically a constant.
591
00:38:07 --> 00:38:08
But not quite.
592
00:38:08 --> 00:38:10
It's going to be a constant
on one interval, and
593
00:38:10 --> 00:38:13
then 0 on the rest.
594
00:38:13 --> 00:38:16
So we'll do those
weighted averages now.
595
00:38:16 --> 00:38:34
And this subject is
called probability.
596
00:38:34 --> 00:38:40
In probability, what we
do, so I'm just going to
597
00:38:40 --> 00:38:43
give some examples here.
598
00:38:43 --> 00:38:54
I'm going to pick a point to
in quotation marks at random.
599
00:38:54 --> 00:39:00
In the region y < x < 1 - x ^2.
600
00:39:00 --> 00:39:05
That's this shape here.
601
00:39:05 --> 00:39:08
Well, let's draw it
right down here.
602
00:39:08 --> 00:39:09
For now.
603
00:39:09 --> 00:39:10
So, somewhere in here.
604
00:39:10 --> 00:39:13
Some point, (x, y).
605
00:39:13 --> 00:39:19
And then I need to tell you,
according to what this
606
00:39:19 --> 00:39:20
random really means.
607
00:39:20 --> 00:39:31
This is proportional
to area, if you like.
608
00:39:31 --> 00:39:33
So area inside of this section.
609
00:39:33 --> 00:39:36
And then the question that
we're going to answer right
610
00:39:36 --> 00:39:46
now is, what is the chance
that, it's usually called
611
00:39:46 --> 00:39:56
probability, that x > 1/2.
612
00:39:56 --> 00:40:03
Let me show you what's
going on here.
613
00:40:03 --> 00:40:08
And this is always the case
with things in probability.
614
00:40:08 --> 00:40:10
So, first of all, we
have a name for this.
615
00:40:10 --> 00:40:16
This is called P ( x > 1/2).
616
00:40:16 --> 00:40:21
And so that's what it's
called in our notation here.
617
00:40:21 --> 00:40:27
And what it is, is the
probability is always equal
618
00:40:27 --> 00:40:32
to the part / the whole.
619
00:40:32 --> 00:40:36
It's a ratio just like
the one over there.
620
00:40:36 --> 00:40:38
And which is the part
and which is the whole.
621
00:40:38 --> 00:40:43
Well, in this picture, the
whole is the whole parabola.
622
00:40:43 --> 00:40:48
And the part is the
section x > 1/2.
623
00:40:48 --> 00:41:00
And it's just the ratio
of those two areas.
624
00:41:00 --> 00:41:01
Let's write that down.
625
00:41:01 --> 00:41:09
That's the integral from 1/2 to
1 of (1 - x ^2) dx, divided by
626
00:41:09 --> 00:41:16
the integral from - 1
to 1, (1 - x ^2 ) dx.
627
00:41:16 --> 00:41:23
And again, the weighting
factor here is 1 - x^2.
628
00:41:23 --> 00:41:27
And to be a little bit more
specific here, the starting
629
00:41:27 --> 00:41:33
point a = - 1 and the
endpoint b = + 1.
630
00:41:33 --> 00:41:37
So this is P(x < 1/2).
631
00:41:37 --> 00:41:42
And if you work it out,
it turns out to be
632
00:41:42 --> 00:41:47
5/18, we won't do it.
633
00:41:47 --> 00:41:47
Yeah.
634
00:41:47 --> 00:42:21
STUDENT: [INAUDIBLE]
635
00:42:21 --> 00:42:23
PROFESSOR: What we're trying
to do with probability.
636
00:42:23 --> 00:42:26
So I can't repeat
your question.
637
00:42:26 --> 00:42:30
But I can try say, because
it was a little bit
638
00:42:30 --> 00:42:31
too complicated.
639
00:42:31 --> 00:42:35
But it was not correct, OK.
640
00:42:35 --> 00:42:38
What we're taking is, we
have two possible things
641
00:42:38 --> 00:42:39
that could happen.
642
00:42:39 --> 00:42:42
Either, let's put it this way.
643
00:42:42 --> 00:42:43
Let's make it a gamble.
644
00:42:43 --> 00:42:47
Somebody picks a point
in here at random.
645
00:42:47 --> 00:42:53
And we're trying to figure
out what your chances
646
00:42:53 --> 00:42:54
are of winning.
647
00:42:54 --> 00:42:57
In other words, the chances the
person picks something in here
648
00:42:57 --> 00:42:59
versus something in there.
649
00:42:59 --> 00:43:01
And the interesting thing
is, so what percent of
650
00:43:01 --> 00:43:04
the time do you win.
651
00:43:04 --> 00:43:06
The answer is it's
some fraction of 1.
652
00:43:06 --> 00:43:08
And in order to figure that
out, I have to figure
653
00:43:08 --> 00:43:11
out the total area here.
654
00:43:11 --> 00:43:16
Versus the total of the entire,
all the way from - 1 to 1,
655
00:43:16 --> 00:43:18
the beginning to the end.
656
00:43:18 --> 00:43:22
So in the numerator, I put
success, and in the denominator
657
00:43:22 --> 00:43:25
I put all possibilities.
658
00:43:25 --> 00:43:26
So that, right?
659
00:43:26 --> 00:43:29
STUDENT: [INAUDIBLE]
660
00:43:29 --> 00:43:31
PROFESSOR: And that's the
interpretation of this.
661
00:43:31 --> 00:43:33
So maybe I didn't
understand your question.
662
00:43:33 --> 00:43:37
STUDENT: [INAUDIBLE]
663
00:43:37 --> 00:43:40
PROFESSOR: Ah, why is 1 - x
^2 the weighting factor.
664
00:43:40 --> 00:43:44
That has to do with how you
compute areas under curves.
665
00:43:44 --> 00:43:49
The curve here is y = 1 - x ^2.
666
00:43:49 --> 00:43:51
And so, in order to calculate
how much area is between 1/2
667
00:43:51 --> 00:43:52
and 1, I have to integrate.
668
00:43:52 --> 00:43:54
That's the
interpretation of this.
669
00:43:54 --> 00:43:56
This is the area
under that curve.
670
00:43:56 --> 00:43:57
This integral.
671
00:43:57 --> 00:44:01
And the denominator's the
area under the whole thing.
672
00:44:01 --> 00:44:02
OK, yeah.
673
00:44:02 --> 00:44:02
Another question.
674
00:44:02 --> 00:44:06
STUDENT: [INAUDIBLE]
675
00:44:06 --> 00:44:08
PROFESSOR: Ah.
676
00:44:08 --> 00:44:09
Yikes.
677
00:44:09 --> 00:44:12
It was supposed to be the
same question as over here.
678
00:44:12 --> 00:44:13
Thank you.
679
00:44:13 --> 00:44:18
STUDENT: [INAUDIBLE]
680
00:44:18 --> 00:44:21
PROFESSOR: This has something
to do with weighting factors.
681
00:44:21 --> 00:44:25
Here's the weight factor.
682
00:44:25 --> 00:44:28
Well, it's the relative
importance from the point of
683
00:44:28 --> 00:44:33
view of this probability of
these places versus those.
684
00:44:33 --> 00:44:36
That is, so this is a weighting
factor because it's telling me
685
00:44:36 --> 00:44:45
that in some sense this number
5/18, actually that makes me
686
00:44:45 --> 00:44:48
think that this number
is probably wrong.
687
00:44:48 --> 00:44:53
Well, I'll let you
calculate it out.
688
00:44:53 --> 00:44:56
It looks like it should be less
than 1/4 here, because this is
689
00:44:56 --> 00:44:59
1/4 of the total distance and
there's a little less in here
690
00:44:59 --> 00:45:00
than there is in the middle.
691
00:45:00 --> 00:45:03
So in fact it probably should
be less than 1/4, the answer.
692
00:45:03 --> 00:45:09
STUDENT: [INAUDIBLE]
693
00:45:09 --> 00:45:11
PROFESSOR: The equation
of the curve is 1 - x^2.
694
00:45:11 --> 00:45:13
695
00:45:13 --> 00:45:15
The reason why it's the
weighting factor is that we're
696
00:45:15 --> 00:45:18
interpreting, the question has
to do with the area
697
00:45:18 --> 00:45:20
under that curve.
698
00:45:20 --> 00:45:24
And so, this is showing us how
much is relatively important
699
00:45:24 --> 00:45:25
versus how much is not.
700
00:45:25 --> 00:45:27
This is, these parts are
relatively important, these
701
00:45:27 --> 00:45:28
parts are less important.
702
00:45:28 --> 00:45:29
According to area.
703
00:45:29 --> 00:45:31
Because we've said that
area is the way we're
704
00:45:31 --> 00:45:35
making the choice.
705
00:45:35 --> 00:45:41
So I don't have quite enough
time to tell you about
706
00:45:41 --> 00:45:43
my next example.
707
00:45:43 --> 00:45:44
Instead, I'm just going
to tell you what the
708
00:45:44 --> 00:45:46
general formula is.
709
00:45:46 --> 00:45:48
And we'll do our
example next time.
710
00:45:48 --> 00:45:51
I'll tell you what
it's going to be.
711
00:45:51 --> 00:46:04
So here's the general formula
for probability here.
712
00:46:04 --> 00:46:12
We're going to imagine that we
have a total range which is
713
00:46:12 --> 00:46:16
maybe going from a to b, and we
have some intermediate values
714
00:46:16 --> 00:46:20
x1 and x2, and then we're
going to try to compute the
715
00:46:20 --> 00:46:28
probability that some variable
that we picked at random
716
00:46:28 --> 00:46:31
occurs between x1 and x2.
717
00:46:31 --> 00:46:36
And by definition, we're
saying that it's an integral.
718
00:46:36 --> 00:46:41
It's the integral from x1 to x2
of the weight dx, divided by
719
00:46:41 --> 00:46:46
the integral all the
way from a to b.
720
00:46:46 --> 00:46:47
Of the weight.
721
00:46:47 --> 00:46:55
So, again, this is the part
divided by the whole.
722
00:46:55 --> 00:46:59
And the relationship between
this and the weighted average
723
00:46:59 --> 00:47:03
that we had earlier was that
the function f ( x) is kind
724
00:47:03 --> 00:47:04
of a strange function.
725
00:47:04 --> 00:47:06
It's 0 and 1.
726
00:47:06 --> 00:47:10
It's just the picture, if
you like, is that you have
727
00:47:10 --> 00:47:11
this weighting factor.
728
00:47:11 --> 00:47:14
And it's going from a to b.
729
00:47:14 --> 00:47:16
But then in between there,
we have the part that
730
00:47:16 --> 00:47:17
we're interested in.
731
00:47:17 --> 00:47:20
Which is between x1 and x2.
732
00:47:20 --> 00:47:23
And it's the ratio of this
inner part to the whole thing
733
00:47:23 --> 00:47:34
that we're interested in.
734
00:47:34 --> 00:47:39
Tomorrow I'm going to try
to do a realistic example.
735
00:47:39 --> 00:47:41
And I'm going to tell you
what it is, but we'll
736
00:47:41 --> 00:47:43
take it up tomorrow.
737
00:47:43 --> 00:47:46
I told you it was going to be
tomorrow, but we still have a
738
00:47:46 --> 00:47:49
whole minute, so I'm going to
tell you what the problem is.
739
00:47:49 --> 00:47:53
So this is going to be a
target practice problem.
740
00:47:53 --> 00:47:56
You have a target here and
you're throwing darts
741
00:47:56 --> 00:48:00
at this target.
742
00:48:00 --> 00:48:05
And so you're throwing
darts at this target.
743
00:48:05 --> 00:48:13
And somebody is standing
next to the dartboard.
744
00:48:13 --> 00:48:18
Your little brother is standing
next to the dartboard here.
745
00:48:18 --> 00:48:23
And the question is, how
likely you are to hit
746
00:48:23 --> 00:48:24
your little brother.
747
00:48:24 --> 00:48:27
So this will, let's see.
748
00:48:27 --> 00:48:28
You'll see whether you
like that or not.
749
00:48:28 --> 00:48:30
Actually, I was the
little brother.
750
00:48:30 --> 00:48:31
So, I don't know which
way you want to go.
751
00:48:31 --> 00:48:32
We'll go either way.
752
00:48:32 --> 00:48:35
We'll find out next time.
753
00:48:35 --> 00:48:35