1
00:00:00 --> 00:00:00
 


2
00:00:00 --> 00:00:02
The following content is
provided under a Creative

3
00:00:02 --> 00:00:02
Commons license.
 

4
00:00:02 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to

5
00:00:06 --> 00:00:09
offer high quality educational
resources for free.

6
00:00:09 --> 00:00:13
To make a donation, or to view
additional materials from

7
00:00:13 --> 00:00:16
hundreds of MIT courses, visit
MIT OpenCourseWare

8
00:00:16 --> 00:00:21
at ocw.mit.edu.
 

9
00:00:21 --> 00:00:24
PROFESSOR: Today we're going
to hold off just a little

10
00:00:24 --> 00:00:27
bit on boiling water.
 

11
00:00:27 --> 00:00:32
And talk about another
application of integrals, and

12
00:00:32 --> 00:00:36
we'll get to the witches'
cauldron in the middle.

13
00:00:36 --> 00:00:45
The that I'd like to start
with today is average value.

14
00:00:45 --> 00:00:48
This is something that I
mentioned a little bit earlier,

15
00:00:48 --> 00:00:50
and there was a misprint on the
board, so I want to make

16
00:00:50 --> 00:00:54
sure that we have the
definitions straight.

17
00:00:54 --> 00:00:56
And also the
reasoning straight.

18
00:00:56 --> 00:00:59
This is one of the most
important applications of

19
00:00:59 --> 00:01:03
integrals, one of the
most important examples.

20
00:01:03 --> 00:01:08
If you take the average
of a bunch of numbers,

21
00:01:08 --> 00:01:10
that looks like this.
 

22
00:01:10 --> 00:01:15
And we can view this as
sampling a function.

23
00:01:15 --> 00:01:17
As we would with Riemann's sum.
 

24
00:01:17 --> 00:01:25
And what I said last week was
that this tends to this

25
00:01:25 --> 00:01:32
expression here, which is
called the continuous average.

26
00:01:32 --> 00:01:43
So this guy is the
continuous average.

27
00:01:43 --> 00:01:49
Or just the average of f.
 

28
00:01:49 --> 00:01:53
And I want to explain that,
just to make sure that we're

29
00:01:53 --> 00:01:56
all on the same page.
 

30
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
00:02:03 --> 00:02:09
it's something like the
area under the curve.

33
00:02:09 --> 00:02:15
But average value is another
reasonable interpretation.

34
00:02:15 --> 00:02:20
Namely, if you take equally
spaced points here, starting

35
00:02:20 --> 00:02:28
with x0, x1, x2, all the way up
to xn, which is the left point

36
00:02:28 --> 00:02:35
b, and then we have values y1,
which = f ( x1), y2, which = f

37
00:02:35 --> 00:02:43
( x2), all the way up to
yn, which = f ( xn).

38
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.
 

40
00:02:50 --> 00:02:52
So remember that spacing,
that's going to be the

41
00:02:52 --> 00:02:57
connection that we'll draw.
 

42
00:02:57 --> 00:03:07
Then the Riemann sum is y1
through yn, the sum of

43
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
00:03:23 --> 00:03:27
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

47
00:03:32 --> 00:03:36
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.
 

49
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.
 

51
00:03:48 --> 00:03:53
And then I'll just check what
this thing actually is.

52
00:03:53 --> 00:03:57
Delta x / b - a, what
is that factor?

53
00:03:57 --> 00:04:01
Well, if we look over here
to what delta x is, if you

54
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.

56
00:04:10 --> 00:04:15
That's what I put over here,
the sum of y1 through yn / n.

57
00:04:15 --> 00:04:20
And as this tends to
0, it's the same as

58
00:04:20 --> 00:04:22
n going to infinity.
 

59
00:04:22 --> 00:04:25
Those are the same things.
 

60
00:04:25 --> 00:04:27
The average value and
the integral are very

61
00:04:27 --> 00:04:28
closely related.
 

62
00:04:28 --> 00:04:32
There's only this difference
that we're dividing by the

63
00:04:32 --> 00:04:36
length of the interval.
 

64
00:04:36 --> 00:04:39
I want to give an example which
is an incredibly simpleminded

65
00:04:39 --> 00:04:41
one, but it'll come
into play later on.

66
00:04:41 --> 00:04:47
So let's take the
example of a constant.

67
00:04:47 --> 00:04:50
And this is, I hope, will make
you slightly less confused

68
00:04:50 --> 00:04:52
about what I just wrote.
 

69
00:04:52 --> 00:04:55
As well as making you think
that this is as simpleminded

70
00:04:55 --> 00:04:57
and reasonable as it should be.
 

71
00:04:57 --> 00:05:00
If I check what the average
value of this constant is, it's

72
00:05:00 --> 00:05:04
given by this relatively
complicated formula here.

73
00:05:04 --> 00:05:07
That is, I have to
integrate the function c.

74
00:05:07 --> 00:05:09
Well, it's just the constant c.
 

75
00:05:09 --> 00:05:12
And however you do this, as an
antiderivative I was thinking

76
00:05:12 --> 00:05:14
of it as a rectangle, the
answer that you're going

77
00:05:14 --> 00:05:18
to get is c here.
 

78
00:05:18 --> 00:05:19
So work that out.
 

79
00:05:19 --> 00:05:20
The answer is c.
 

80
00:05:20 --> 00:05:26
And so the fact that the
average of c = c, which had

81
00:05:26 --> 00:05:30
better be the case for
averages, explains

82
00:05:30 --> 00:05:31
the denominator.
 

83
00:05:31 --> 00:05:36
Explains the 1 / b - a there.
 

84
00:05:36 --> 00:05:39
That's cooked up exactly so
that the average of a constant

85
00:05:39 --> 00:05:40
is what it's supposed to be.
 

86
00:05:40 --> 00:05:42
Otherwise we have the
wrong normalizing factor.

87
00:05:42 --> 00:05:45
We've clearly got a piece
of nonsense on our hands.

88
00:05:45 --> 00:05:51
And incidentally, it also
explains the 1 / n in the

89
00:05:51 --> 00:05:53
very first formula
that I wrote down.

90
00:05:53 --> 00:05:56
The reason why this is called
the average, or one reason why

91
00:05:56 --> 00:05:59
it's the right thing, is that
if you took the same constant

92
00:05:59 --> 00:06:02
c, for y all the way across
there n times, if you divide

93
00:06:02 --> 00:06:03
it by n, you get back c.
 

94
00:06:03 --> 00:06:05
That's what we mean by
average value and that's

95
00:06:05 --> 00:06:11
why the n is there.
 

96
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

101
00:06:35 --> 00:06:47
a point on a semicircle.
 

102
00:06:47 --> 00:06:52
And maybe I'll draw a picture
of the semicircle first here.

103
00:06:52 --> 00:06:56
And we'll just make it
the standard circle,

104
00:06:56 --> 00:06:58
the unit circle.
 

105
00:06:58 --> 00:07:04
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.

107
00:07:07 --> 00:07:11
And we're picking a point
over here and we're going

108
00:07:11 --> 00:07:17
to take the typical, or
the average, height here.

109
00:07:17 --> 00:07:18
Integrating with respect to dx.
 

110
00:07:18 --> 00:07:21
So sort of continuously
with respect to dx.

111
00:07:21 --> 00:07:25
Well, what is that?
 

112
00:07:25 --> 00:07:29
Well, according to the rule,
it's 1 / b - a times - sorry,

113
00:07:29 --> 00:07:30
it's up here in the box.
 

114
00:07:30 --> 00:07:34
1 / b - a, the interval
from a to b, f ( x) dx.

115
00:07:34 --> 00:07:39
That's 1 / + 1 - (- 1).
 

116
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

118
00:07:52 --> 00:07:57
square root of 1 - x^2.
 

119
00:07:57 --> 00:08:02
And to evaluate this is not
as difficult as it seems.

120
00:08:02 --> 00:08:06
This is 1/2 times this
quantity here, which we

121
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

124
00:08:21 --> 00:08:22
half the area of the circle.
 

125
00:08:22 --> 00:08:26
So it's pi / 2.
 

126
00:08:26 --> 00:08:35
And so the answer, here the
average height, is pi/ 4.

127
00:08:35 --> 00:08:38
Now, later in the class and
actually not in this unit,

128
00:08:38 --> 00:08:40
we'll actually be able
to calculate the

129
00:08:40 --> 00:08:42
antiderivative of this.
 

130
00:08:42 --> 00:08:44
So in other words, we'll
be able to calculate

131
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

133
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.

152
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.
 

170
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

183
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