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PROFESSOR: OK.

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00:00:23,630 --> 00:00:28,710
Now, today we get to move
on from integral formulas

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00:00:28,710 --> 00:00:34,620
and methods of integration
back to some geometry.

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And this is more or
less going to lead

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00:00:36,870 --> 00:00:39,270
into the kinds of
tools you'll be using

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00:00:39,270 --> 00:00:43,120
in multivariable calculus.

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The first thing that
we're going to do today

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00:00:45,420 --> 00:00:58,220
is discuss arc length.

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00:00:58,220 --> 00:01:02,940
Like all of the cumulative
sums that we've worked on,

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00:01:02,940 --> 00:01:06,710
this one has a storyline and
a picture associated to it,

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00:01:06,710 --> 00:01:09,360
which involves
dividing things up.

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00:01:09,360 --> 00:01:11,690
If you have a
roadway, if you like,

20
00:01:11,690 --> 00:01:18,290
and you have mileage markers
along the road, like this,

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00:01:18,290 --> 00:01:22,490
all the way up
to, say, s_n here,

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00:01:22,490 --> 00:01:28,740
then the length along the road
is described by this parameter,

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00:01:28,740 --> 00:01:30,560
s.

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00:01:30,560 --> 00:01:32,930
Which is arc length.

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00:01:32,930 --> 00:01:37,680
And if we look at a graph
of this sort of thing,

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00:01:37,680 --> 00:01:42,160
if this is the last point b,
and this is the first point a,

27
00:01:42,160 --> 00:01:47,120
then you can think in terms of
having points above x_1, x_2,

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00:01:47,120 --> 00:01:50,460
x_3, etc.

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00:01:50,460 --> 00:01:53,940
The same as we did
with Riemann sums.

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00:01:53,940 --> 00:01:56,050
And then the way
that we're going

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00:01:56,050 --> 00:02:07,540
to approximate this is by taking
the straight lines between each

32
00:02:07,540 --> 00:02:09,470
of these points.

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00:02:09,470 --> 00:02:11,360
As things get
smaller and smaller,

34
00:02:11,360 --> 00:02:15,210
the straight line is going to
be fairly close to the curve.

35
00:02:15,210 --> 00:02:16,980
And that's the main idea.

36
00:02:16,980 --> 00:02:20,050
So let me just depict
one little chunk of this.

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00:02:20,050 --> 00:02:20,900
Which is like this.

38
00:02:20,900 --> 00:02:24,530
One straight line, and here's
the curved surface there.

39
00:02:24,530 --> 00:02:26,830
And the distance along
the curved surface is what

40
00:02:26,830 --> 00:02:31,690
I'm calling delta s, the
change in the length between--

41
00:02:31,690 --> 00:02:34,920
so this would be s_2 - s_1
if I depicted that one.

42
00:02:34,920 --> 00:02:42,490
So this would be delta s
is, say s. s_i - s_(i-1),

43
00:02:42,490 --> 00:02:45,240
some increment there.

44
00:02:45,240 --> 00:02:49,670
And then I can figure
out what the length

45
00:02:49,670 --> 00:02:51,750
of the orange segment is.

46
00:02:51,750 --> 00:02:54,630
Because the horizontal
distance is delta x.

47
00:02:54,630 --> 00:02:57,290
And the vertical
distance is delta y.

48
00:02:57,290 --> 00:03:02,950
And so the formula is that the
hypotenuse is delta (delta x)^2

49
00:03:02,950 --> 00:03:06,470
+ (delta y)^2.

50
00:03:06,470 --> 00:03:09,420
Square root.

51
00:03:09,420 --> 00:03:11,870
And delta s is
approximately that.

52
00:03:11,870 --> 00:03:15,871
So what we're saying is that
(delta s)^2 is approximately

53
00:03:15,871 --> 00:03:16,370
this.

54
00:03:16,370 --> 00:03:21,870
So this is the hypotenuse.

55
00:03:21,870 --> 00:03:23,500
Squared.

56
00:03:23,500 --> 00:03:29,520
And it's very close to
the length of the curve.

57
00:03:29,520 --> 00:03:34,940
And the whole idea of calculus
is that in the infinitesimal,

58
00:03:34,940 --> 00:03:44,150
this is exactly correct.

59
00:03:44,150 --> 00:03:48,010
So that's what's going
to happen in the limit.

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00:03:48,010 --> 00:03:52,000
And that is the basis for
calculating arc length.

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00:03:52,000 --> 00:03:56,060
I'm going to rewrite that
formula on the next board.

62
00:03:56,060 --> 00:03:59,460
But I'm going to write it in
the more customary fashion.

63
00:03:59,460 --> 00:04:02,200
We've done this before,
a certain amount.

64
00:04:02,200 --> 00:04:03,970
But I just want to
emphasize it here

65
00:04:03,970 --> 00:04:09,390
because this handwriting
is a little bit peculiar.

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00:04:09,390 --> 00:04:11,710
This ds is really all one thing.

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00:04:11,710 --> 00:04:15,690
What I really mean is to put
the parenthesis around it.

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00:04:15,690 --> 00:04:16,660
It's one thing.

69
00:04:16,660 --> 00:04:19,070
It's not d times s, it's ds.

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00:04:19,070 --> 00:04:20,020
It's one thing.

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00:04:20,020 --> 00:04:20,940
And we square it.

72
00:04:20,940 --> 00:04:22,750
But for whatever
reason people have

73
00:04:22,750 --> 00:04:26,112
gotten into the habit of
omitting the parentheses.

74
00:04:26,112 --> 00:04:28,070
So you're just going to
have to live with that.

75
00:04:28,070 --> 00:04:31,050
And realize that this is not d
of s^2 or anything like that.

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00:04:31,050 --> 00:04:32,755
And similarly, this
is a single number,

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00:04:32,755 --> 00:04:34,130
and this is a single number.

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00:04:34,130 --> 00:04:36,060
Infinitesimal.

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00:04:36,060 --> 00:04:39,400
So that's just the way
that this idea here

80
00:04:39,400 --> 00:04:42,194
gets written in our notation.

81
00:04:42,194 --> 00:04:43,860
And this is the first
time we're dealing

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00:04:43,860 --> 00:04:46,070
with squares of infinitesimals.

83
00:04:46,070 --> 00:04:47,600
So it's just a little different.

84
00:04:47,600 --> 00:04:49,641
But immediately the first
thing we're going to do

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00:04:49,641 --> 00:04:51,750
is take the square root.

86
00:04:51,750 --> 00:04:56,660
If I take the square root,
that's the square root of dx^2

87
00:04:56,660 --> 00:04:58,640
+ dy^2.

88
00:04:58,640 --> 00:05:02,400
And this is the form in which
I always remember this formula.

89
00:05:02,400 --> 00:05:07,030
Let's put it in some
brightly decorated form.

90
00:05:07,030 --> 00:05:12,380
But there are about four,
five, six other forms

91
00:05:12,380 --> 00:05:16,110
that you'll derive from this,
which all mean the same thing.

92
00:05:16,110 --> 00:05:18,479
So this is, as I say,
the way I remember it.

93
00:05:18,479 --> 00:05:20,270
But there are other
ways of thinking of it.

94
00:05:20,270 --> 00:05:23,070
And let's just write
a couple of them down.

95
00:05:23,070 --> 00:05:27,380
The first one is that you
can factor out the dx.

96
00:05:27,380 --> 00:05:29,480
So that looks like this.

97
00:05:29,480 --> 00:05:34,680
1 + (dy / dx)^2.

98
00:05:34,680 --> 00:05:37,630
And then I factored out the dx.

99
00:05:37,630 --> 00:05:39,820
So this is a variant.

100
00:05:39,820 --> 00:05:43,190
And this is the one
which actually we'll

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00:05:43,190 --> 00:05:47,490
be using in practice
right now on our examples.

102
00:05:47,490 --> 00:05:55,600
So the conclusion is
that the arc length,

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00:05:55,600 --> 00:06:05,320
which if you like is this
total s_n - s_0, if you like,

104
00:06:05,320 --> 00:06:09,390
is going to be equal to
the integral from a to b

105
00:06:09,390 --> 00:06:16,430
of the square root
of 1 + (dy/dx)^2, dx.

106
00:06:20,490 --> 00:06:27,500
In practice, it's also very
often written informally

107
00:06:27,500 --> 00:06:29,090
as this.

108
00:06:29,090 --> 00:06:30,750
The integral ds.

109
00:06:30,750 --> 00:06:32,830
So the change in
this little variable

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00:06:32,830 --> 00:06:40,009
s, and this is what you'll see
notationally in many textbooks.

111
00:06:40,009 --> 00:06:42,550
So that's one way of writing
it, and of course the second way

112
00:06:42,550 --> 00:06:45,950
of writing it which is
practically the same thing is

113
00:06:45,950 --> 00:06:50,040
square root of 1 + f'(x)^2, dx.

114
00:06:50,040 --> 00:06:52,330
Mixing in a little bit
of Newton's notation.

115
00:06:52,330 --> 00:06:57,250
And this is with y = f(x).

116
00:06:57,250 --> 00:07:03,310
So this is the formula
for arc length.

117
00:07:03,310 --> 00:07:05,770
And as I say, I
remember it this way.

118
00:07:05,770 --> 00:07:09,060
But you're going to have to
derive various variants of it.

119
00:07:09,060 --> 00:07:11,010
And you'll have to
use some arithmetic

120
00:07:11,010 --> 00:07:12,650
to get to various formulas.

121
00:07:12,650 --> 00:07:15,190
And there will be more later.

122
00:07:15,190 --> 00:07:16,030
Yeah, question.

123
00:07:16,030 --> 00:07:20,820
STUDENT: [INAUDIBLE]

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00:07:20,820 --> 00:07:27,350
PROFESSOR: OK, the question
is, is f'(x)^2 equal to f''(x).

125
00:07:27,350 --> 00:07:31,020
And the answer is no.

126
00:07:31,020 --> 00:07:33,090
And let's just see what it is.

127
00:07:33,090 --> 00:07:39,490
So, for example, if f(x) = x^2,
which is an example which will

128
00:07:39,490 --> 00:07:47,910
come up in a few minutes,
then f'(x) = 2x and f'(x)^2 =

129
00:07:47,910 --> 00:07:52,380
(2x)^2, which is 4x^2.

130
00:07:52,380 --> 00:07:56,490
Whereas f''(x) is equal to, if
I differentiate this another

131
00:07:56,490 --> 00:07:58,750
time, it's equal to 2.

132
00:07:58,750 --> 00:08:03,440
So they don't mean
the same thing.

133
00:08:03,440 --> 00:08:04,700
The same thing over here.

134
00:08:04,700 --> 00:08:07,200
You can see this dy / dx,
this is the rate of change

135
00:08:07,200 --> 00:08:08,190
of y with respect to x.

136
00:08:08,190 --> 00:08:09,632
The quantity squared.

137
00:08:09,632 --> 00:08:11,340
So in other words,
this thing is supposed

138
00:08:11,340 --> 00:08:13,300
to mean the same as that.

139
00:08:13,300 --> 00:08:13,800
Yeah.

140
00:08:13,800 --> 00:08:19,580
Another question.

141
00:08:19,580 --> 00:08:25,660
STUDENT: [INAUDIBLE]

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00:08:25,660 --> 00:08:27,970
PROFESSOR: So the
question is, you

143
00:08:27,970 --> 00:08:30,430
got a little nervous because
I left out these limits.

144
00:08:30,430 --> 00:08:32,120
And indeed, I did
that on purpose

145
00:08:32,120 --> 00:08:34,530
because I didn't want to
specify what was going on.

146
00:08:34,530 --> 00:08:36,590
Really, if you wrote
it in terms of ds,

147
00:08:36,590 --> 00:08:38,710
you'd have to write
it as starting at s_0

148
00:08:38,710 --> 00:08:42,180
and ending at s_n to be
consistent with the variable s.

149
00:08:42,180 --> 00:08:45,730
But of course, if you write it
in terms of another variable,

150
00:08:45,730 --> 00:08:46,940
you put that variable in.

151
00:08:46,940 --> 00:08:49,190
So this is what we do when
we change variables, right?

152
00:08:49,190 --> 00:08:51,570
We have many different
choices for these limits.

153
00:08:51,570 --> 00:08:54,530
And this is the clue as
to which variable we use.

154
00:08:54,530 --> 00:08:59,060
STUDENT: [INAUDIBLE]

155
00:08:59,060 --> 00:09:01,720
PROFESSOR: Correct.
s_0 and s_n are not

156
00:09:01,720 --> 00:09:03,310
the same thing as a and b.

157
00:09:03,310 --> 00:09:05,290
In fact, this is x_n.

158
00:09:05,290 --> 00:09:07,740
And this x_0, over here.

159
00:09:07,740 --> 00:09:08,990
That's what a and b are.

160
00:09:08,990 --> 00:09:13,242
But s_0 and s_n are mileage
markers on the road.

161
00:09:13,242 --> 00:09:15,450
They're not the same thing
as keeping track of what's

162
00:09:15,450 --> 00:09:16,950
happening on the x axis.

163
00:09:16,950 --> 00:09:18,710
So when we measure
arc length, remember

164
00:09:18,710 --> 00:09:27,980
it's mileage along
the curved path.

165
00:09:27,980 --> 00:09:32,410
So now, I need to give
you some examples.

166
00:09:32,410 --> 00:09:40,820
And my first example is
going to be really basic.

167
00:09:40,820 --> 00:09:45,630
But I hope that it helps to
give some perspective here.

168
00:09:45,630 --> 00:09:48,830
So I'm going to take
the example y = m

169
00:09:48,830 --> 00:09:52,550
x, which is a linear
function, a straight line.

170
00:09:52,550 --> 00:09:58,630
And then y' would be m, and so
ds is going to be the square

171
00:09:58,630 --> 00:10:02,650
root of 1 + (y')^2, dx.

172
00:10:02,650 --> 00:10:10,840
Which is the square
root of 1 + m^2, dx.

173
00:10:10,840 --> 00:10:17,700
And now, the length, say,
if we go from, I don't know,

174
00:10:17,700 --> 00:10:27,300
let's say 0 to 10, let's say,
of the graph is going to be

175
00:10:27,300 --> 00:10:33,990
the integral from 0 to 10 of
the square root of 1 + m^2, dx.

176
00:10:33,990 --> 00:10:39,910
Which of course is just
10 square root of 1 + m^2.

177
00:10:39,910 --> 00:10:41,810
Not very surprising.

178
00:10:41,810 --> 00:10:43,360
This is a constant.

179
00:10:43,360 --> 00:10:46,830
It just factors out and the
integral from 0 to 10 of dx

180
00:10:46,830 --> 00:10:51,210
is 10.

181
00:10:51,210 --> 00:10:54,340
Let's just draw a
picture of this.

182
00:10:54,340 --> 00:10:57,530
This is something
which has slope m here.

183
00:10:57,530 --> 00:10:59,020
And it's going to 10.

184
00:10:59,020 --> 00:11:02,340
So this horizontal is 10.

185
00:11:02,340 --> 00:11:05,930
And the vertical is 10m.

186
00:11:05,930 --> 00:11:08,030
Those are the
dimensions of this.

187
00:11:08,030 --> 00:11:11,810
And the Pythagorean theorem
says that the hypotenuse,

188
00:11:11,810 --> 00:11:15,457
not surprisingly, let's draw
it in here in orange to remind

189
00:11:15,457 --> 00:11:18,040
ourselves that it was the same
type of orange that we had over

190
00:11:18,040 --> 00:11:27,160
there, this length here is the
square root of 10^2 + (10m)^2.

191
00:11:27,160 --> 00:11:31,400
That's the formula
for the hypotenuse.

192
00:11:31,400 --> 00:11:38,520
And that's exactly
the same as this.

193
00:11:38,520 --> 00:11:40,540
Maybe you're saying
duh, this is obvious.

194
00:11:40,540 --> 00:11:44,160
But the point that I'm
trying to make is this.

195
00:11:44,160 --> 00:11:48,220
If you can figure out these
formulas for linear functions,

196
00:11:48,220 --> 00:11:52,510
calculus tells you how to
do it for every function.

197
00:11:52,510 --> 00:11:56,370
The idea of calculus is that
this easy calculation here,

198
00:11:56,370 --> 00:11:58,530
which you can do
without any calculus

199
00:11:58,530 --> 00:12:04,360
at all, all of the tools, the
notations of differentials

200
00:12:04,360 --> 00:12:06,115
and limits and
integrals, is going

201
00:12:06,115 --> 00:12:10,040
to make you be able to
do it for any curve.

202
00:12:10,040 --> 00:12:12,760
Because we can break things up
into these little infinitesimal

203
00:12:12,760 --> 00:12:13,260
bits.

204
00:12:13,260 --> 00:12:15,820
This is the whole idea
of all of the methods

205
00:12:15,820 --> 00:12:18,620
that we had to set
up integrals here.

206
00:12:18,620 --> 00:12:25,770
This is the main point
of these integrals.

207
00:12:25,770 --> 00:12:32,480
Now, so let's do something
slightly more interesting.

208
00:12:32,480 --> 00:12:39,970
Our next example is
going to be the circle,

209
00:12:39,970 --> 00:12:41,540
so y = square root of 1-x^2.

210
00:12:48,590 --> 00:12:51,820
If you like, that's the
graph of a semicircle.

211
00:12:51,820 --> 00:12:57,040
And maybe we'll set
it up here this way.

212
00:12:57,040 --> 00:13:00,030
So that the semicircle
goes around like this.

213
00:13:00,030 --> 00:13:02,310
And we'll start
it here at x = 0.

214
00:13:02,310 --> 00:13:04,250
And we'll go over to a.

215
00:13:04,250 --> 00:13:06,560
And we'll take this little
piece of the circle.

216
00:13:06,560 --> 00:13:08,150
So down to here.

217
00:13:08,150 --> 00:13:11,850
If you like.

218
00:13:11,850 --> 00:13:14,970
So here's the
portion of the circle

219
00:13:14,970 --> 00:13:17,660
that I'm going to
measure the length of.

220
00:13:17,660 --> 00:13:19,110
Now, we know that length.

221
00:13:19,110 --> 00:13:20,160
It's called arc length.

222
00:13:20,160 --> 00:13:21,659
And I'm going to
give it a name, I'm

223
00:13:21,659 --> 00:13:23,400
going to call it alpha here.

224
00:13:23,400 --> 00:13:39,640
So alpha's the arc
length along the circle.

225
00:13:39,640 --> 00:13:42,660
Now, let's figure
out what it is.

226
00:13:42,660 --> 00:13:45,970
First, in order to do this, I
have to figure out what y' is.

227
00:13:45,970 --> 00:13:47,960
Or, if you like, dy/dx.

228
00:13:47,960 --> 00:13:50,680
Now, that's a calculation that
we've done a number of times.

229
00:13:50,680 --> 00:13:52,830
And I'm going to do
it slightly faster.

230
00:13:52,830 --> 00:13:57,160
But you remember it gives you a
square root in the denominator.

231
00:13:57,160 --> 00:13:59,500
And then you have the
derivative of what's

232
00:13:59,500 --> 00:14:01,180
inside the square root.

233
00:14:01,180 --> 00:14:02,260
Which is -2x.

234
00:14:02,260 --> 00:14:05,590
But then there's also 1/2,
because in disguise it's really

235
00:14:05,590 --> 00:14:07,840
(1 - x^2)^(1/2).

236
00:14:07,840 --> 00:14:10,584
So we've done this
calculation enough times

237
00:14:10,584 --> 00:14:12,500
that I'm not going to
carry it out completely.

238
00:14:12,500 --> 00:14:14,290
I want you to think
about what it is.

239
00:14:14,290 --> 00:14:17,690
It turns out to -x up here,
because the 1/2 and the 2

240
00:14:17,690 --> 00:14:22,890
cancel.

241
00:14:22,890 --> 00:14:25,870
And now the thing that
we have to integrate

242
00:14:25,870 --> 00:14:30,190
is this arc length element,
as it's called, ds.

243
00:14:30,190 --> 00:14:38,350
And that's going to be the
square root of 1 + (y')^2, dx.

244
00:14:38,350 --> 00:14:41,060
And so I'm going to have to
carry out the calculation,

245
00:14:41,060 --> 00:14:42,970
some messy calculation here.

246
00:14:42,970 --> 00:14:47,020
Which is that this is 1 plus
the quantity -x over square root

247
00:14:47,020 --> 00:14:49,274
of 1 - x^2, squared.

248
00:14:49,274 --> 00:14:51,440
So I have to figure out
what's under the square root

249
00:14:51,440 --> 00:14:55,740
sign over here in order to
carry out this calculation.

250
00:14:55,740 --> 00:14:58,570
Now let's do that.

251
00:14:58,570 --> 00:15:03,310
This is 1 + x^2 / (1 - x^2).

252
00:15:03,310 --> 00:15:06,140
That's what this simplifies to.

253
00:15:06,140 --> 00:15:11,190
And then that's equal to, over
a common denominator, 1 - x^2.

254
00:15:11,190 --> 00:15:13,580
1 - x^2 + x^2.

255
00:15:13,580 --> 00:15:16,500
And there is a little bit
of simplification now.

256
00:15:16,500 --> 00:15:19,190
Because the two x^2's cancel.

257
00:15:19,190 --> 00:15:20,390
And we get 1/(1-x^2).

258
00:15:28,640 --> 00:15:35,750
So now I get to finish
off the calculation

259
00:15:35,750 --> 00:15:40,580
by actually figuring out
what the arc length is.

260
00:15:40,580 --> 00:15:42,460
And what is it?

261
00:15:42,460 --> 00:15:51,834
Well, this alpha is equal to
the integral from 0 to a of ds.

262
00:15:51,834 --> 00:15:53,750
Well, it's going to be
the square root of what

263
00:15:53,750 --> 00:15:55,110
I have here.

264
00:15:55,110 --> 00:15:57,210
This was a square,
this is just what

265
00:15:57,210 --> 00:15:58,710
was underneath the
square root sign.

266
00:15:58,710 --> 00:16:01,500
This is 1 + (y')^2.

267
00:16:01,500 --> 00:16:03,270
Have to take the
square root of that.

268
00:16:03,270 --> 00:16:08,380
So what I get here is dx over
the square root of 1 - x^2.

269
00:16:13,350 --> 00:16:18,590
And now, we recognize this.

270
00:16:18,590 --> 00:16:21,100
The antiderivative of this
is something that we know.

271
00:16:21,100 --> 00:16:23,790
This is the inverse sine.

272
00:16:23,790 --> 00:16:25,830
Evaluated at 0 and a.

273
00:16:25,830 --> 00:16:29,490
Which is just giving
us the inverse sine

274
00:16:29,490 --> 00:16:35,860
of a, because the inverse
sine of 0 is equal to 0.

275
00:16:35,860 --> 00:16:43,980
So alpha is equal to
the inverse sine of a.

276
00:16:43,980 --> 00:16:51,320
That's a very fancy way of
saying that sin(alpha) = a.

277
00:16:51,320 --> 00:16:54,870
That's the equivalent
statement here.

278
00:16:54,870 --> 00:16:59,830
And what's going on here
is something that's just

279
00:16:59,830 --> 00:17:01,950
a little deeper than it looks.

280
00:17:01,950 --> 00:17:03,120
Which is this.

281
00:17:03,120 --> 00:17:08,030
We've just figured out a
geometric interpretation

282
00:17:08,030 --> 00:17:09,350
of what's going on here.

283
00:17:09,350 --> 00:17:13,680
That is, that we went a
distance alpha along this arc.

284
00:17:13,680 --> 00:17:28,570
And now remember that
the radius here is 1.

285
00:17:28,570 --> 00:17:34,430
And this horizontal
distance here is a.

286
00:17:34,430 --> 00:17:37,450
This distance here is a.

287
00:17:37,450 --> 00:17:40,600
And so the geometric
interpretation of this

288
00:17:40,600 --> 00:17:51,000
is that this angle
is alpha radians.

289
00:17:51,000 --> 00:17:54,950
And sin(alpha) = a.

290
00:17:54,950 --> 00:17:57,570
So this is consistent
with our definition

291
00:17:57,570 --> 00:18:00,430
previously, our previous
geometric definition

292
00:18:00,430 --> 00:18:02,530
of radians.

293
00:18:02,530 --> 00:18:07,190
But this is really your first
true definition of radians.

294
00:18:07,190 --> 00:18:09,850
We never actually--
People told you

295
00:18:09,850 --> 00:18:12,570
that radians were the arc
length along this curve.

296
00:18:12,570 --> 00:18:14,770
This is the first time
you're deriving it.

297
00:18:14,770 --> 00:18:18,420
This is the first time you're
seeing it correctly done.

298
00:18:18,420 --> 00:18:20,134
And furthermore, this
is the first time

299
00:18:20,134 --> 00:18:21,550
you're seeing a
correct definition

300
00:18:21,550 --> 00:18:24,100
of the sine function.

301
00:18:24,100 --> 00:18:26,156
Remember we had
this crazy way, we

302
00:18:26,156 --> 00:18:27,530
defined the
exponential function,

303
00:18:27,530 --> 00:18:29,661
then we had another
way of defining the log

304
00:18:29,661 --> 00:18:30,660
function as an integral.

305
00:18:30,660 --> 00:18:32,900
Then we defined the
exponential in terms of it.

306
00:18:32,900 --> 00:18:34,690
Well, this is the
same sort of thing.

307
00:18:34,690 --> 00:18:36,700
What's really happening
here is that if you

308
00:18:36,700 --> 00:18:38,190
want to know what
radians are, you

309
00:18:38,190 --> 00:18:40,847
have to calculate this number.

310
00:18:40,847 --> 00:18:42,430
If you've calculated
this number, then

311
00:18:42,430 --> 00:18:47,930
by definition if sine is
the thing whose alpha radian

312
00:18:47,930 --> 00:18:49,890
amount gives you
a, then it must be

313
00:18:49,890 --> 00:18:52,520
that this is sine inverse of a.

314
00:18:52,520 --> 00:18:55,430
And so the first thing that
gets defined is the arcsine.

315
00:18:55,430 --> 00:18:56,930
And the next thing
that gets defined

316
00:18:56,930 --> 00:19:00,250
is the sine, afterwards.

317
00:19:00,250 --> 00:19:04,500
This is the way the
foundational approach actually

318
00:19:04,500 --> 00:19:06,730
works when you start
from first principles.

319
00:19:06,730 --> 00:19:10,200
This arc length being one
of the first principles.

320
00:19:10,200 --> 00:19:13,720
So now we have a solid
foundation for trig functions.

321
00:19:13,720 --> 00:19:15,760
And this is giving
that connection.

322
00:19:15,760 --> 00:19:18,260
Of course, it's consistent with
everything you already knew,

323
00:19:18,260 --> 00:19:22,079
so you don't have to make any
transitional thinking here.

324
00:19:22,079 --> 00:19:23,620
It's just that this
is the first time

325
00:19:23,620 --> 00:19:25,570
it's being done rigorously.

326
00:19:25,570 --> 00:19:36,380
Because you only
now have arc length.

327
00:19:36,380 --> 00:19:41,220
So these are examples, as I say,
that maybe you already know.

328
00:19:41,220 --> 00:19:44,820
And maybe we'll do one that
we don't know quite as well.

329
00:19:44,820 --> 00:19:49,300
Let's find the
length of a parabola.

330
00:19:49,300 --> 00:19:59,180
This is Example 3.

331
00:19:59,180 --> 00:20:00,660
Now, that was what
I was suggesting

332
00:20:00,660 --> 00:20:03,140
we were going to do earlier.

333
00:20:03,140 --> 00:20:06,730
So this is the function y x^2.

334
00:20:06,730 --> 00:20:09,800
y' = 2x.

335
00:20:09,800 --> 00:20:20,120
And so ds is equal to the
square root of 1 + (2x)^2, dx.

336
00:20:20,120 --> 00:20:24,600
And now I can figure out what
a piece of a parabola is.

337
00:20:24,600 --> 00:20:28,220
So I'll draw the piece
of parabola up to a,

338
00:20:28,220 --> 00:20:30,840
let's say, starting from 0.

339
00:20:30,840 --> 00:20:32,680
So that's the chunk.

340
00:20:32,680 --> 00:20:45,450
And then its arc length,
between 0 and a of this curve,

341
00:20:45,450 --> 00:21:02,400
is the integral from 0 to a of
square root of 1 + 4x^2, dx.

342
00:21:02,400 --> 00:21:08,490
OK, now if you like, this is
the answer to the question.

343
00:21:08,490 --> 00:21:11,040
But people hate
looking at answers

344
00:21:11,040 --> 00:21:13,800
when they're integrals
if they can be evaluated.

345
00:21:13,800 --> 00:21:16,510
So one of the reasons why we
went through all this rigmarole

346
00:21:16,510 --> 00:21:18,800
of calculating these
things is to show you

347
00:21:18,800 --> 00:21:22,120
that we can actually evaluate
a bunch of these functions

348
00:21:22,120 --> 00:21:23,340
here more explicitly.

349
00:21:23,340 --> 00:21:28,180
It doesn't help a lot, but
there is an explicit calculation

350
00:21:28,180 --> 00:21:28,680
of this.

351
00:21:28,680 --> 00:21:30,560
So remember how
you would do this.

352
00:21:30,560 --> 00:21:33,040
So this is just a
little bit of review.

353
00:21:33,040 --> 00:21:35,420
What we did in techniques
of integration.

354
00:21:35,420 --> 00:21:39,000
The first step is what?

355
00:21:39,000 --> 00:21:40,700
A substitution.

356
00:21:40,700 --> 00:21:43,950
It's a trig substitution.

357
00:21:43,950 --> 00:21:45,030
And what is it?

358
00:21:45,030 --> 00:21:47,250
STUDENT: [INAUDIBLE]

359
00:21:47,250 --> 00:21:50,270
PROFESSOR: So x equals
something tan(theta).

360
00:21:50,270 --> 00:21:54,587
I claim that it's 1/2 tan,
and I'm going to call it u.

361
00:21:54,587 --> 00:21:56,420
Because I'm going to
use theta for something

362
00:21:56,420 --> 00:21:58,161
else in a couple of days.

363
00:21:58,161 --> 00:21:58,660
OK?

364
00:21:58,660 --> 00:22:01,420
So this is the substitution.

365
00:22:01,420 --> 00:22:10,620
And then of course dx
= 1/2 sec^2 u du, etc.

366
00:22:10,620 --> 00:22:12,750
So what happens if you do this?

367
00:22:12,750 --> 00:22:15,160
I'll write down the
answer, but I'm not

368
00:22:15,160 --> 00:22:16,330
going to carry this out.

369
00:22:16,330 --> 00:22:19,090
Because every one of
these is horrendous.

370
00:22:19,090 --> 00:22:22,190
But I think I worked it out.

371
00:22:22,190 --> 00:22:23,370
Let's see if I'm lucky.

372
00:22:23,370 --> 00:22:24,300
Oh yeah.

373
00:22:24,300 --> 00:22:26,380
I think this is what it is.

374
00:22:26,380 --> 00:22:40,979
It's a 1/4 ln(2x + square root
of (1+4x^2) + 1/2 x square root

375
00:22:40,979 --> 00:22:41,520
of (1+4x^2)).

376
00:22:46,540 --> 00:22:52,190
Evaluated at a and 0.

377
00:22:52,190 --> 00:22:53,090
So yick.

378
00:22:53,090 --> 00:22:53,810
I mean, you know.

379
00:22:53,810 --> 00:22:55,720
STUDENT: [INAUDIBLE]

380
00:22:55,720 --> 00:22:58,850
PROFESSOR: Why I
did I make it 1/2?

381
00:22:58,850 --> 00:23:00,989
Because it turns out that
when you differentiate.

382
00:23:00,989 --> 00:23:02,780
So the question is,
why is there 1/2 there?

383
00:23:02,780 --> 00:23:05,734
If you differentiate it without
the 1/2, you get this term

384
00:23:05,734 --> 00:23:07,650
and it looks like it's
going to be just right.

385
00:23:07,650 --> 00:23:10,191
But then if you differentiate
this one you get another thing.

386
00:23:10,191 --> 00:23:12,170
And it all mixes together.

387
00:23:12,170 --> 00:23:13,660
And it turns out
that there's more.

388
00:23:13,660 --> 00:23:15,280
So it turns out that it's 1/2.

389
00:23:15,280 --> 00:23:18,750
Differentiate it and check.

390
00:23:18,750 --> 00:23:21,530
So this just an incredibly
long calculation.

391
00:23:21,530 --> 00:23:24,220
It would take fifteen minutes
or something like that.

392
00:23:24,220 --> 00:23:26,220
But the point is, you
do know in principle

393
00:23:26,220 --> 00:23:27,840
how to do these things.

394
00:23:27,840 --> 00:23:43,455
STUDENT: [INAUDIBLE]

395
00:23:43,455 --> 00:23:45,330
PROFESSOR: Oh, he was
talking about this 1/2,

396
00:23:45,330 --> 00:23:47,100
not this crazy 1/2 here.

397
00:23:47,100 --> 00:23:47,600
Sorry.

398
00:23:47,600 --> 00:23:48,637
STUDENT: [INAUDIBLE]

399
00:23:48,637 --> 00:23:49,470
PROFESSOR: Yeah, OK.

400
00:23:49,470 --> 00:23:50,740
So sorry about that.

401
00:23:50,740 --> 00:23:53,080
Thank you for helping.

402
00:23:53,080 --> 00:23:56,270
This factor of 1/2 here comes
about because when you square

403
00:23:56,270 --> 00:23:58,690
x, you don't get tan^2.

404
00:23:58,690 --> 00:24:02,510
When you square 2x, you
get (4x)^2 and that matches

405
00:24:02,510 --> 00:24:03,930
perfectly with this thing.

406
00:24:03,930 --> 00:24:07,300
And that's why you
need this factor here.

407
00:24:07,300 --> 00:24:07,800
Yeah.

408
00:24:07,800 --> 00:24:09,216
Another question,
way in the back.

409
00:24:09,216 --> 00:24:18,190
STUDENT: [INAUDIBLE]

410
00:24:18,190 --> 00:24:20,780
PROFESSOR: The question is,
when you do this substitution,

411
00:24:20,780 --> 00:24:25,040
doesn't the limit
from 0 to a change.

412
00:24:25,040 --> 00:24:27,020
And the answer is,
absolutely yes.

413
00:24:27,020 --> 00:24:30,230
The limits in terms
of u are not the same

414
00:24:30,230 --> 00:24:31,800
as the limits in terms of a.

415
00:24:31,800 --> 00:24:34,870
But if I then translate back
to the x variables, which

416
00:24:34,870 --> 00:24:40,820
I've done here in this bottom
formula, of x = 0 and x = a,

417
00:24:40,820 --> 00:24:44,740
it goes back to those in
the original variables.

418
00:24:44,740 --> 00:24:46,840
So if I write things in
the original variables,

419
00:24:46,840 --> 00:24:48,980
I have the original limits.

420
00:24:48,980 --> 00:24:51,962
If I use the u variables, I
would have to change limits.

421
00:24:51,962 --> 00:24:53,670
But I'm not carrying
out the integration,

422
00:24:53,670 --> 00:24:55,060
because I don't want to.

423
00:24:55,060 --> 00:25:00,630
So I brought it back
to the x formula.

424
00:25:00,630 --> 00:25:07,080
Other questions.

425
00:25:07,080 --> 00:25:11,580
OK, so now we're ready to launch
into three-space a little bit

426
00:25:11,580 --> 00:25:14,250
here.

427
00:25:14,250 --> 00:25:41,310
We're going to talk
about surface area.

428
00:25:41,310 --> 00:25:43,750
You're going to be
doing a lot with surface

429
00:25:43,750 --> 00:25:48,550
area in multivariable calculus.

430
00:25:48,550 --> 00:25:50,990
It's one of the
really fun things.

431
00:25:50,990 --> 00:25:55,430
And just remember, when
it gets complicated,

432
00:25:55,430 --> 00:25:57,926
that the simplest things
are the most important.

433
00:25:57,926 --> 00:25:59,300
And the simple
things are, if you

434
00:25:59,300 --> 00:26:01,820
can handle things
for linear functions,

435
00:26:01,820 --> 00:26:02,920
you know all the rest.

436
00:26:02,920 --> 00:26:04,544
So there's going to
be some complicated

437
00:26:04,544 --> 00:26:06,390
stuff but it'll
really only involve

438
00:26:06,390 --> 00:26:09,480
what's happening on planes.

439
00:26:09,480 --> 00:26:11,680
So let's start
with surface area.

440
00:26:11,680 --> 00:26:15,615
And the example that
I'd like to give

441
00:26:15,615 --> 00:26:20,110
- this is the only type of
example that we'll have -

442
00:26:20,110 --> 00:26:28,560
is the surface of rotation.

443
00:26:28,560 --> 00:26:31,990
And as long as we have
our parabola there,

444
00:26:31,990 --> 00:26:33,300
we'll use that one.

445
00:26:33,300 --> 00:26:51,920
So we have y = x^2,
rotated around the x-axis.

446
00:26:51,920 --> 00:26:54,480
So let's take a look at
what this looks like.

447
00:26:54,480 --> 00:26:57,910
It's the parabola, which
is going like that.

448
00:26:57,910 --> 00:27:01,850
And then it's being
spun around the x-axis.

449
00:27:01,850 --> 00:27:08,390
So some kind of shape like
this with little circles.

450
00:27:08,390 --> 00:27:17,880
It's some kind of
trumpet shape, right?

451
00:27:17,880 --> 00:27:20,170
And that's the shape
that we're-- Now, again,

452
00:27:20,170 --> 00:27:20,980
it's the surface.

453
00:27:20,980 --> 00:27:27,770
It's just the metal of the
trumpet, not the insides.

454
00:27:27,770 --> 00:27:33,280
Now, the principle for figuring
out what the formula for area

455
00:27:33,280 --> 00:27:36,470
is, is not that
different from what we

456
00:27:36,470 --> 00:27:38,530
did for surfaces of revolution.

457
00:27:38,530 --> 00:27:42,610
But it just requires a little
bit of thought and imagination.

458
00:27:42,610 --> 00:27:50,360
We have a little chunk
of arc length along here.

459
00:27:50,360 --> 00:27:55,580
And we're going to spin
that around this axis.

460
00:27:55,580 --> 00:28:01,610
Now, if this were a horizontal
piece of arc length,

461
00:28:01,610 --> 00:28:04,500
then it would spin
around just like a shell.

462
00:28:04,500 --> 00:28:07,550
It would just be a surface.

463
00:28:07,550 --> 00:28:11,670
But if it's tilted,
if it's tilted,

464
00:28:11,670 --> 00:28:15,030
then there's more surface area
proportional to the amount

465
00:28:15,030 --> 00:28:17,040
that it's tilted.

466
00:28:17,040 --> 00:28:19,310
So it's proportional to
the length of the segment

467
00:28:19,310 --> 00:28:22,890
that you spin around.

468
00:28:22,890 --> 00:28:29,430
So the total is going to be ds,
that's one of the factors here.

469
00:28:29,430 --> 00:28:32,220
Maybe I'll write that second.

470
00:28:32,220 --> 00:28:33,560
That's one of the dimensions.

471
00:28:33,560 --> 00:28:36,660
And then the other dimension
is the circumference.

472
00:28:36,660 --> 00:28:43,180
Which is 2 pi, in this case, y.

473
00:28:43,180 --> 00:28:46,300
So that's the end
of the calculation.

474
00:28:46,300 --> 00:28:55,990
This is the area
element of surface area.

475
00:28:55,990 --> 00:28:59,860
Now, when you get to 18.02,
and maybe even before that,

476
00:28:59,860 --> 00:29:03,000
you'll also see some people
referring to this area element

477
00:29:03,000 --> 00:29:09,110
when it's a curvy surface
like this with a notation dS.

478
00:29:09,110 --> 00:29:10,610
That's a little
confusing because we

479
00:29:10,610 --> 00:29:12,070
have a lower case s here.

480
00:29:12,070 --> 00:29:15,450
We're not going to
use it right now.

481
00:29:15,450 --> 00:29:17,620
But the lower case s
is usually arc length.

482
00:29:17,620 --> 00:29:23,950
The upper case S is
usually surface area.

483
00:29:23,950 --> 00:29:25,650
So.

484
00:29:25,650 --> 00:29:32,510
Also used for dA.

485
00:29:32,510 --> 00:29:33,730
The area element.

486
00:29:33,730 --> 00:29:39,780
Because this is a
curved area element.

487
00:29:39,780 --> 00:29:47,630
So let's figure
out this example.

488
00:29:47,630 --> 00:29:54,090
So in the example-- ...is equal
to x ^2 then the situation is,

489
00:29:54,090 --> 00:30:01,050
we have the surface area is
equal to the integral from,

490
00:30:01,050 --> 00:30:03,770
I don't know, 0 to a if those
are the limits that we wanted

491
00:30:03,770 --> 00:30:05,040
to choose.

492
00:30:05,040 --> 00:30:10,470
Of 2 pi x^2, right?

493
00:30:10,470 --> 00:30:11,160
Because y = x^2.

494
00:30:11,160 --> 00:30:17,960
Times the square
root of 1 + 4x^2, dx.

495
00:30:17,960 --> 00:30:20,300
Remember we had this from
our previous example.

496
00:30:20,300 --> 00:30:32,850
This was ds from previous.

497
00:30:32,850 --> 00:30:41,930
And this, of course, is 2 pi y.

498
00:30:41,930 --> 00:30:49,150
Now again, the calculation of
this integral is kind of long.

499
00:30:49,150 --> 00:30:52,060
And I'm going to omit it.

500
00:30:52,060 --> 00:30:54,300
But let me just point
out that it follows

501
00:30:54,300 --> 00:30:56,350
from the same substitution.

502
00:30:56,350 --> 00:31:05,150
Namely, x = 1/2 tan u is going
to work for this integral.

503
00:31:05,150 --> 00:31:06,200
It's kind of a mess.

504
00:31:06,200 --> 00:31:08,590
There's a tan squared here
and the secant squared.

505
00:31:08,590 --> 00:31:10,170
There's another
secant and so on.

506
00:31:10,170 --> 00:31:12,640
So it's one of
these trig integrals

507
00:31:12,640 --> 00:31:19,670
that then takes a while to do.

508
00:31:19,670 --> 00:31:22,940
So that just is going to
trail off into nothing.

509
00:31:22,940 --> 00:31:25,540
And the reason is that
what's important here

510
00:31:25,540 --> 00:31:27,250
is more the method.

511
00:31:27,250 --> 00:31:29,650
And the setup of the integrals.

512
00:31:29,650 --> 00:31:32,902
The actual computation, in
fact, you could go to a program

513
00:31:32,902 --> 00:31:34,610
and you could plug in
something like this

514
00:31:34,610 --> 00:31:37,300
and you would spit out
an answer immediately.

515
00:31:37,300 --> 00:31:41,100
So really what we just want is
for you to have enough control

516
00:31:41,100 --> 00:31:43,640
to see that it's an integral
that's a manageable one.

517
00:31:43,640 --> 00:31:45,630
And also to know that
if you plugged it in,

518
00:31:45,630 --> 00:31:50,970
you would get an answer.

519
00:31:50,970 --> 00:31:53,320
When I actually do carry
out a calculation, though,

520
00:31:53,320 --> 00:31:57,740
what I want to do
is to do something

521
00:31:57,740 --> 00:32:00,190
that has an answer
that you can remember.

522
00:32:00,190 --> 00:32:02,090
And that's a nice answer.

523
00:32:02,090 --> 00:32:05,940
So that turns out to be
the example of the surface

524
00:32:05,940 --> 00:32:07,580
area of a sphere.

525
00:32:07,580 --> 00:32:10,110
So it's analogous
to this 2 here.

526
00:32:10,110 --> 00:32:15,170
And maybe I should
remember this result here.

527
00:32:15,170 --> 00:32:24,390
Which was that the arc length
element was given by this.

528
00:32:24,390 --> 00:32:38,700
So we'll save that for a second.

529
00:32:38,700 --> 00:32:41,710
So we're going to do
this surface area now.

530
00:32:41,710 --> 00:32:43,930
So if you like, this
is another example.

531
00:32:43,930 --> 00:32:54,820
The surface area of a sphere.

532
00:32:54,820 --> 00:32:59,670
This is a good example,
and one, as I say,

533
00:32:59,670 --> 00:33:01,160
that has a really nice answer.

534
00:33:01,160 --> 00:33:07,130
So it's worth doing.

535
00:33:07,130 --> 00:33:09,770
So first of all, I'm not going
to set it up quite the way

536
00:33:09,770 --> 00:33:11,794
I did in Example 2.

537
00:33:11,794 --> 00:33:13,710
Instead, I'm going to
take the general sphere,

538
00:33:13,710 --> 00:33:18,790
because I'd like to watch
the dependence on the radius.

539
00:33:18,790 --> 00:33:22,160
So here this is going
to be the radius.

540
00:33:22,160 --> 00:33:27,250
It's going to be radius a.

541
00:33:27,250 --> 00:33:30,440
And now, if I carry out
the same calculations

542
00:33:30,440 --> 00:33:33,840
as before, if you think
about it for a second,

543
00:33:33,840 --> 00:33:42,322
you're going to get this
result. And then, the rest

544
00:33:42,322 --> 00:33:43,780
of the arithmetic,
which is sitting

545
00:33:43,780 --> 00:33:47,750
up there in the case,
a = 1, will give us

546
00:33:47,750 --> 00:33:53,120
that ds is equal to what?

547
00:33:53,120 --> 00:33:56,670
Well, maybe I'll
just carry it out.

548
00:33:56,670 --> 00:33:58,510
Because that's always nice.

549
00:33:58,510 --> 00:34:03,630
So we have 1 +
x^2 / (a^2 - x^2).

550
00:34:03,630 --> 00:34:07,060
That's 1 + (y')^2.

551
00:34:07,060 --> 00:34:09,630
And now I put this over
a common denominator.

552
00:34:09,630 --> 00:34:11,800
And I get a^2 - x^2.

553
00:34:11,800 --> 00:34:15,030
And I have in the
numerator a^2 - x^2 + x^2.

554
00:34:15,030 --> 00:34:17,590
So the same cancellation occurs.

555
00:34:17,590 --> 00:34:25,390
But now we get an
a^2 in the numerator.

556
00:34:25,390 --> 00:34:28,070
So now I can set up the ds.

557
00:34:28,070 --> 00:34:30,200
And so here's what happens.

558
00:34:30,200 --> 00:34:35,370
The area of a section of
the sphere, so let's see.

559
00:34:35,370 --> 00:34:38,920
We're going to start at
some starting place x_1,

560
00:34:38,920 --> 00:34:40,760
and end at some place x_2.

561
00:34:40,760 --> 00:34:43,110
So what does that look like?

562
00:34:43,110 --> 00:34:45,140
Here's the sphere.

563
00:34:45,140 --> 00:34:47,490
And we're starting
at a place x_1.

564
00:34:47,490 --> 00:34:49,810
And we're ending at a place x_2.

565
00:34:49,810 --> 00:34:53,830
And we're taking more or less
the slice here, if you like.

566
00:34:53,830 --> 00:34:59,480
The section of this sphere.

567
00:34:59,480 --> 00:35:02,150
So the area's going
to equal this.

568
00:35:02,150 --> 00:35:06,730
And what is it going to be?

569
00:35:06,730 --> 00:35:12,820
Well, so I have here 2 pi y.

570
00:35:12,820 --> 00:35:15,310
I'll write it out, just
leave it as y for now.

571
00:35:15,310 --> 00:35:19,190
And then I have ds.

572
00:35:19,190 --> 00:35:22,110
So that's always what the
formula is when you're

573
00:35:22,110 --> 00:35:25,880
revolving around the x-axis.

574
00:35:25,880 --> 00:35:29,920
And then I'll plug
in for those things.

575
00:35:29,920 --> 00:35:35,990
So 2 pi, the formula for y
is square root a^2 - x^2.

576
00:35:38,700 --> 00:35:42,220
And the formula
for ds, well, it's

577
00:35:42,220 --> 00:35:44,970
the square root
of this times dx.

578
00:35:44,970 --> 00:35:48,920
So it's the square root
of a^2 / (a^2 - x^2), dx.

579
00:35:51,760 --> 00:35:54,880
So this part is ds.

580
00:35:54,880 --> 00:36:02,170
And this part is y.

581
00:36:02,170 --> 00:36:07,590
And now, I claim we have a nice
cancellation that takes place.

582
00:36:07,590 --> 00:36:09,770
Square root of a^2 is a.

583
00:36:09,770 --> 00:36:12,740
And then there's another
good cancellation.

584
00:36:12,740 --> 00:36:13,990
As you can see.

585
00:36:13,990 --> 00:36:17,140
Now, what we get here is the
integral from x_1 to x_2,

586
00:36:17,140 --> 00:36:21,510
of 2 pi a dx, which is
about the easiest integral

587
00:36:21,510 --> 00:36:23,340
you can imagine.

588
00:36:23,340 --> 00:36:24,940
It's just the integral
of a constant.

589
00:36:24,940 --> 00:36:27,980
So it's 2 pi a (x_2 - x_1).

590
00:36:36,810 --> 00:36:40,780
Let's check this in
a couple of examples.

591
00:36:40,780 --> 00:36:48,390
And then see what it's saying
geometrically, a little bit.

592
00:36:48,390 --> 00:36:53,220
So what this is saying--
So special cases

593
00:36:53,220 --> 00:36:56,560
that you should
always check, when

594
00:36:56,560 --> 00:36:59,220
you have a nice formula
like this, at least.

595
00:36:59,220 --> 00:37:00,730
But really with
anything in order

596
00:37:00,730 --> 00:37:03,250
to make sure that you've
got the right answer.

597
00:37:03,250 --> 00:37:05,710
If you take, for
example, the hemisphere.

598
00:37:05,710 --> 00:37:08,920
So you take 1/2 of this sphere.

599
00:37:08,920 --> 00:37:11,090
So that would be
starting at 0, sorry.

600
00:37:11,090 --> 00:37:14,300
And ending at a.

601
00:37:14,300 --> 00:37:17,270
So that's the
integral from 0 to a.

602
00:37:17,270 --> 00:37:21,890
So this is the case
x_1 = 0. x_2 = a.

603
00:37:21,890 --> 00:37:29,190
And what you're going
to get is a hemisphere.

604
00:37:29,190 --> 00:37:36,270
And the area is 2 pi a times a.

605
00:37:36,270 --> 00:37:38,150
Or in other words, 2 pi a^2.

606
00:37:42,410 --> 00:37:48,920
And if you take
the whole sphere,

607
00:37:48,920 --> 00:37:55,900
that's starting at x_1 = -a,
and x_2 = a, you're getting 2 pi

608
00:37:55,900 --> 00:37:58,330
a times (a - (-a)).

609
00:38:02,640 --> 00:38:06,380
Which is 4 pi a^2.

610
00:38:06,380 --> 00:38:09,230
That's the whole thing.

611
00:38:09,230 --> 00:38:10,140
Yeah, question.

612
00:38:10,140 --> 00:38:21,780
STUDENT: [INAUDIBLE]

613
00:38:21,780 --> 00:38:23,340
PROFESSOR: The
question is, would it

614
00:38:23,340 --> 00:38:27,730
be possible to rotate
around the y-axis?

615
00:38:27,730 --> 00:38:30,330
And the answer is yes.

616
00:38:30,330 --> 00:38:34,550
It's legal to rotate
around the y-axis.

617
00:38:34,550 --> 00:38:43,170
And there is-- If you use
vertical slices as we did here,

618
00:38:43,170 --> 00:38:45,490
that is, well they're
sort of tips of slices,

619
00:38:45,490 --> 00:38:46,790
it's a different idea.

620
00:38:46,790 --> 00:38:49,780
But anyway, it's using
dx as the integral

621
00:38:49,780 --> 00:38:52,670
of the variable of integration.

622
00:38:52,670 --> 00:38:55,310
So we're checking
each little piece,

623
00:38:55,310 --> 00:38:58,750
each little strip of that type.

624
00:38:58,750 --> 00:39:00,540
If we use dx here, we get this.

625
00:39:00,540 --> 00:39:02,710
If you did the same thing
rotated the other way,

626
00:39:02,710 --> 00:39:06,140
and use dy as the variable, you
get exactly the same answer.

627
00:39:06,140 --> 00:39:08,120
And it would be the
same calculation.

628
00:39:08,120 --> 00:39:14,080
Because they're parallel.

629
00:39:14,080 --> 00:39:14,870
So you're, yep.

630
00:39:14,870 --> 00:39:17,280
STUDENT: [INAUDIBLE]

631
00:39:17,280 --> 00:39:19,770
PROFESSOR: Can you do
surface area with shells?

632
00:39:19,770 --> 00:39:26,540
Well, the shell shape-- The
short answer is not quite.

633
00:39:26,540 --> 00:39:30,851
The shell shape is
a vertical shell

634
00:39:30,851 --> 00:39:32,600
which is itself already
three-dimensional,

635
00:39:32,600 --> 00:39:34,850
and it has a thickness.

636
00:39:34,850 --> 00:39:37,270
So this is just a matter
of terminology, though.

637
00:39:37,270 --> 00:39:41,210
This thickness is this dx,
when we do this rotation here.

638
00:39:41,210 --> 00:39:43,820
And then there are
two other dimensions.

639
00:39:43,820 --> 00:39:46,190
If we have a curved
surface, there's

640
00:39:46,190 --> 00:39:56,460
no other dimension
left to form a shell.

641
00:39:56,460 --> 00:39:59,421
But basically, you can chop
things up into any bits

642
00:39:59,421 --> 00:40:00,670
that you can actually measure.

643
00:40:00,670 --> 00:40:04,020
That you can figure
out what the area is.

644
00:40:04,020 --> 00:40:08,270
That's the main point.

645
00:40:08,270 --> 00:40:10,160
Now, I said we were
going to, we've

646
00:40:10,160 --> 00:40:12,910
just launched into
three-dimensional space.

647
00:40:12,910 --> 00:40:21,650
And I want to now move on to
other space-like phenomena.

648
00:40:21,650 --> 00:40:26,780
But we're going to do this.

649
00:40:26,780 --> 00:40:31,510
So this is also a
preparation for 18.02,

650
00:40:31,510 --> 00:40:34,840
where you'll be doing
this a tremendous amount.

651
00:40:34,840 --> 00:40:49,530
We're going to talk now
about parametric equations.

652
00:40:49,530 --> 00:40:57,102
Really just parametric curves.

653
00:40:57,102 --> 00:40:58,810
So you're going to
see this now and we're

654
00:40:58,810 --> 00:41:00,539
going to interpret
it a couple of times,

655
00:41:00,539 --> 00:41:02,580
and we're going to think
about polar coordinates.

656
00:41:02,580 --> 00:41:05,920
These are all preparation for
thinking in more variables,

657
00:41:05,920 --> 00:41:08,570
and thinking in a
different way than you've

658
00:41:08,570 --> 00:41:09,720
been thinking before.

659
00:41:09,720 --> 00:41:11,970
So I want you to
prepare your brain

660
00:41:11,970 --> 00:41:13,975
to make a transition here.

661
00:41:13,975 --> 00:41:15,600
This is the beginning
of the transition

662
00:41:15,600 --> 00:41:21,580
to multivariable thinking.

663
00:41:21,580 --> 00:41:26,350
We're going to consider
curves like this.

664
00:41:26,350 --> 00:41:29,850
Which are described with x
being a function of t and y

665
00:41:29,850 --> 00:41:31,410
being a function of t.

666
00:41:31,410 --> 00:41:35,630
And this letter t is
called the parameter.

667
00:41:35,630 --> 00:41:38,080
In this case you should
think of it-- the easiest way

668
00:41:38,080 --> 00:41:39,700
to think of it is as time.

669
00:41:39,700 --> 00:41:43,840
And what you have is
what's called a trajectory.

670
00:41:43,840 --> 00:41:48,090
So this is also
called a trajectory.

671
00:41:48,090 --> 00:41:54,590
And its location, let's say, at
time 0, is this location here.

672
00:41:54,590 --> 00:41:58,310
(x(0), y(0)), that's
a point in the plane.

673
00:41:58,310 --> 00:42:01,380
And then over here, for
instance, maybe it's

674
00:42:01,380 --> 00:42:04,410
(x(1), y(1)).

675
00:42:04,410 --> 00:42:06,540
And I drew arrows
along here to indicate

676
00:42:06,540 --> 00:42:10,110
that we're going from this
place over to that place.

677
00:42:10,110 --> 00:42:16,190
These are later times. t = 1
is a later time than t = 0.

678
00:42:16,190 --> 00:42:20,170
So that's just a
very casual, it's

679
00:42:20,170 --> 00:42:22,680
just the way we use
these notations.

680
00:42:22,680 --> 00:42:31,670
Now let me give you the first
example, which is x = a cos

681
00:42:31,670 --> 00:42:40,340
t, y = a sin t.

682
00:42:40,340 --> 00:42:43,630
And the first thing to figure
out is what kind of curve

683
00:42:43,630 --> 00:42:45,490
this is.

684
00:42:45,490 --> 00:42:47,880
And to do that, we
want to figure out

685
00:42:47,880 --> 00:42:52,000
what equation it satisfies
in rectangular coordinates.

686
00:42:52,000 --> 00:42:53,650
So to figure out
what curve this is,

687
00:42:53,650 --> 00:42:59,730
we recognize that if we square
and add-- So we add x^2 to y^2,

688
00:42:59,730 --> 00:43:02,670
we're going to get something
very nice and clean.

689
00:43:02,670 --> 00:43:08,700
We're going to get a^2
cos^2 t + a^2 sin^2 t.

690
00:43:11,990 --> 00:43:13,500
Yeah that's right, OK.

691
00:43:13,500 --> 00:43:19,230
Which is just a ^2 a^2 (cos^2 +
sin^2), or in other words a^2.

692
00:43:19,230 --> 00:43:23,450
So lo and behold, what
we have is a circle.

693
00:43:23,450 --> 00:43:27,840
And then we know what
shape this is now.

694
00:43:27,840 --> 00:43:33,240
And the other thing I'd
like to keep track of

695
00:43:33,240 --> 00:43:35,990
is which direction we're
going on the circle.

696
00:43:35,990 --> 00:43:40,020
Because there's more to this
parameter then just the shape.

697
00:43:40,020 --> 00:43:42,570
There's also where
we are at what time.

698
00:43:42,570 --> 00:43:46,610
This would be, think of it like
the trajectory of a planet.

699
00:43:46,610 --> 00:43:51,600
So here, I have to do this
by plotting the picture

700
00:43:51,600 --> 00:43:53,250
and figuring out what happens.

701
00:43:53,250 --> 00:43:59,980
So at t = 0, we have
(x, y) is equal to,

702
00:43:59,980 --> 00:44:06,190
plug in here (a cos 0, a sin 0).

703
00:44:06,190 --> 00:44:10,570
Which is just a * 1
+ a * 0, so (a, 0).

704
00:44:10,570 --> 00:44:11,830
And that's here.

705
00:44:11,830 --> 00:44:14,660
That's the point (a, 0).

706
00:44:14,660 --> 00:44:18,474
We know that it's the
circle of radius a.

707
00:44:18,474 --> 00:44:19,890
So we know that
the curve is going

708
00:44:19,890 --> 00:44:22,140
to go around like this somehow.

709
00:44:22,140 --> 00:44:26,390
So let's see what
happens at t = pi / 2.

710
00:44:26,390 --> 00:44:30,950
So at that point, we have (x,
y) = (a cos(pi/2), a sin(pi/2)).

711
00:44:37,310 --> 00:44:41,580
Which is (0, a), because
sine of pi / 2 is 1.

712
00:44:41,580 --> 00:44:43,190
So that's up here.

713
00:44:43,190 --> 00:44:46,100
So this is what
happens at t = 0.

714
00:44:46,100 --> 00:44:49,220
This is what happens
at t = pi / 2.

715
00:44:49,220 --> 00:44:51,390
And the trajectory
clearly goes this way.

716
00:44:51,390 --> 00:44:54,800
In fact, this turns
out to be t = pi, etc.

717
00:44:54,800 --> 00:44:58,330
And it repeats at t = 2pi.

718
00:44:58,330 --> 00:45:00,610
So the other feature
that we have,

719
00:45:00,610 --> 00:45:11,560
which is qualitative feature,
is that it's counterclockwise.

720
00:45:11,560 --> 00:45:17,020
Now the last little bit is
going to be the arc length.

721
00:45:17,020 --> 00:45:19,030
Keeping track of the arc length.

722
00:45:19,030 --> 00:45:23,390
And we'll do that next time.