1
00:00:00,000 --> 00:00:06,870


2
00:00:06,870 --> 00:00:07,370
Hi.

3
00:00:07,370 --> 00:00:09,100
Welcome back to recitation.

4
00:00:09,100 --> 00:00:11,550
In class, one of the things
you've talked about recently

5
00:00:11,550 --> 00:00:14,640
was computing surface areas
of solids of rotation.

6
00:00:14,640 --> 00:00:17,140
So I have a nice problem
relating to that here.

7
00:00:17,140 --> 00:00:21,880
So the circle with center
(R, 0) and radius little r--

8
00:00:21,880 --> 00:00:25,400
so this is center big
R, 0, and radius little

9
00:00:25,400 --> 00:00:29,660
r, which is less than big R--
is rotated around the y-axis.

10
00:00:29,660 --> 00:00:31,630
And the question is,
what's the surface

11
00:00:31,630 --> 00:00:33,180
area of the resulting solid?

12
00:00:33,180 --> 00:00:35,140
So we have here the circle.

13
00:00:35,140 --> 00:00:38,660
Its center is at
the point big R, 0,

14
00:00:38,660 --> 00:00:40,810
and its radius is
little r, so this

15
00:00:40,810 --> 00:00:42,980
is the equation of that circle.

16
00:00:42,980 --> 00:00:45,435
And we're going to rotate
this circle around this axis.

17
00:00:45,435 --> 00:00:46,810
So we're going to
spin it around.

18
00:00:46,810 --> 00:00:48,820
And what you're going
to get is a donut,

19
00:00:48,820 --> 00:00:50,900
or what mathematicians
call a torus.

20
00:00:50,900 --> 00:00:52,970
So here's a little
schematic of it

21
00:00:52,970 --> 00:00:56,400
here, with one dotted little
cross section corresponding

22
00:00:56,400 --> 00:00:58,540
to this circle.

23
00:00:58,540 --> 00:01:02,320
So the question is, what is
the surface area of this torus?

24
00:01:02,320 --> 00:01:04,410
So why don't you
pause the video,

25
00:01:04,410 --> 00:01:07,069
take a few minutes to work
this problem out yourself,

26
00:01:07,069 --> 00:01:08,860
come back, and we can
work it out together.

27
00:01:08,860 --> 00:01:17,270


28
00:01:17,270 --> 00:01:21,970
So when we solved surface
area problems in class,

29
00:01:21,970 --> 00:01:25,000
we took the curve that
we were going to rotate,

30
00:01:25,000 --> 00:01:28,830
and we cut it into lots of
little pieces with length ds.

31
00:01:28,830 --> 00:01:33,860
So let me draw that, just
very quickly, up here.

32
00:01:33,860 --> 00:01:36,270
So we had, you took
whatever your curve was,

33
00:01:36,270 --> 00:01:39,080
and you cut it into lots
of these little segments.

34
00:01:39,080 --> 00:01:43,090
And then for each segment,
you rotated it around an axis.

35
00:01:43,090 --> 00:01:47,200
And so if the segment has
this little length ds,

36
00:01:47,200 --> 00:01:50,610
a little piece of arc
length-- so in our case,

37
00:01:50,610 --> 00:01:54,370
we're going to rotate it around
the y-axis, so the length,

38
00:01:54,370 --> 00:01:57,630
the area that this thing
traces out as it spins around,

39
00:01:57,630 --> 00:02:01,550
is going to be this
little piece of area, dA,

40
00:02:01,550 --> 00:02:06,620
which is equal to 2*pi*x*ds.

41
00:02:06,620 --> 00:02:08,760
Now, this is a
little bit different

42
00:02:08,760 --> 00:02:11,110
than most of the examples
Professor Jerison

43
00:02:11,110 --> 00:02:14,900
did in class, because here we're
rotating around the y-axis, not

44
00:02:14,900 --> 00:02:16,000
around the x-axis.

45
00:02:16,000 --> 00:02:18,390
So if you rotated
around the x-axis,

46
00:02:18,390 --> 00:02:21,410
what you would get is 2*pi*y*ds.

47
00:02:21,410 --> 00:02:23,850
And here we get 2*pi*x*dx.

48
00:02:23,850 --> 00:02:26,510
The idea of the x and the
y in this formula, this x,

49
00:02:26,510 --> 00:02:28,320
it's just telling
you what the radius

50
00:02:28,320 --> 00:02:32,780
is between your little segment
and the axis around which

51
00:02:32,780 --> 00:02:34,020
you're rotating it.

52
00:02:34,020 --> 00:02:37,380
So here this is, you know,
2*pi*x is the circumference

53
00:02:37,380 --> 00:02:41,340
of this circle that it traces
out, and ds is its length,

54
00:02:41,340 --> 00:02:45,050
because it's giving you a
little ribbon as it goes around.

55
00:02:45,050 --> 00:02:48,740
So we have this formula,
dA equals 2*pi*x*ds.

56
00:02:48,740 --> 00:02:53,760
And so in order to get the
surface area, what we do,

57
00:02:53,760 --> 00:02:58,070
is we have to integrate this
over an appropriate region.

58
00:02:58,070 --> 00:03:00,250
So in order to do
that, we first need,

59
00:03:00,250 --> 00:03:04,510
you know, all the variables in
the integrand to be the same,

60
00:03:04,510 --> 00:03:07,190
so we need to write everything
in terms of x, or everything

61
00:03:07,190 --> 00:03:10,040
in terms of y, or everything
in terms of some variable

62
00:03:10,040 --> 00:03:12,010
that we can integrate against.

63
00:03:12,010 --> 00:03:18,310
So in our particular
case with this torus,

64
00:03:18,310 --> 00:03:20,630
I think we can take
advantage of a little bit

65
00:03:20,630 --> 00:03:24,890
of symmetry here, which
is that this, you know,

66
00:03:24,890 --> 00:03:27,280
torus is top-bottom
symmetric, right?

67
00:03:27,280 --> 00:03:30,580
As the top half of the
circle goes around,

68
00:03:30,580 --> 00:03:32,035
it traces out one surface.

69
00:03:32,035 --> 00:03:34,370
As the bottom half of
the circle goes around,

70
00:03:34,370 --> 00:03:35,767
it traces out another surface.

71
00:03:35,767 --> 00:03:37,850
But those two surfaces
have exactly the same area.

72
00:03:37,850 --> 00:03:39,558
They're just mirror
images of each other,

73
00:03:39,558 --> 00:03:42,060
because the circle is symmetric.

74
00:03:42,060 --> 00:03:45,540
So we can just
consider the problem

75
00:03:45,540 --> 00:03:49,230
of spinning the top half
of the circle around.

76
00:03:49,230 --> 00:03:51,350
And so for the top
half of the circle,

77
00:03:51,350 --> 00:03:54,020
we can write down
an equation for y

78
00:03:54,020 --> 00:03:57,030
in terms of x, and so then
we can integrate-- you know,

79
00:03:57,030 --> 00:03:59,950
that sets up a nice
integral with respect to x.

80
00:03:59,950 --> 00:04:03,490
So in order to do this, well
we're going to need two things.

81
00:04:03,490 --> 00:04:05,980
So we're going to need
to know what ds is.

82
00:04:05,980 --> 00:04:10,740
And so you had a couple
of different formulas

83
00:04:10,740 --> 00:04:12,240
for this in class.

84
00:04:12,240 --> 00:04:16,520
So you had ds-- so one
easy mnemonic that I

85
00:04:16,520 --> 00:04:21,310
like is to write ds equals
the square root of dx squared

86
00:04:21,310 --> 00:04:24,040
plus dy squared.

87
00:04:24,040 --> 00:04:27,600
So for me, this always--
I can remember this,

88
00:04:27,600 --> 00:04:29,750
because it's just the
Pythagorean theorem, right?

89
00:04:29,750 --> 00:04:32,900
So you have a little
dx horizontal distance,

90
00:04:32,900 --> 00:04:35,150
and a little dy
vertical distance,

91
00:04:35,150 --> 00:04:38,490
and so the ds is just a
hypotenuse of that triangle.

92
00:04:38,490 --> 00:04:39,740
So that's how I remember this.

93
00:04:39,740 --> 00:04:41,940
And so then you also have
the equivalent formula

94
00:04:41,940 --> 00:04:46,580
if you factor out
a dx from here.

95
00:04:46,580 --> 00:04:50,000
You know, divide through by
dx and multiply outside by dx.

96
00:04:50,000 --> 00:04:57,950
You can write this as the
square root of 1 plus dy by dx

97
00:04:57,950 --> 00:05:03,160
squared times, outside
the square root, dx.

98
00:05:03,160 --> 00:05:06,300
So this is our little piece ds.

99
00:05:06,300 --> 00:05:10,440
And OK, we have an x
in the formula already.

100
00:05:10,440 --> 00:05:14,970
So ds-- so we now have
ds with a dx here.

101
00:05:14,970 --> 00:05:17,160
So the only thing
we have left, is

102
00:05:17,160 --> 00:05:20,660
we need to figure out
what dy/dx is in order

103
00:05:20,660 --> 00:05:24,010
to put all this into the
formula, in order to integrate.

104
00:05:24,010 --> 00:05:24,510
So OK.

105
00:05:24,510 --> 00:05:25,817
So dy/dx.

106
00:05:25,817 --> 00:05:27,650
So OK, so that means I
need y in terms of x.

107
00:05:27,650 --> 00:05:28,230
Now let-- OK.

108
00:05:28,230 --> 00:05:32,950
So if we're focusing only on
the top half of this torus, that

109
00:05:32,950 --> 00:05:37,470
means y-- well, we can
solve this equation for y,

110
00:05:37,470 --> 00:05:38,970
but when we take
the square root,

111
00:05:38,970 --> 00:05:42,080
we're only taking the positive
square root, because we're only

112
00:05:42,080 --> 00:05:43,900
looking at the top
half of the torus,

113
00:05:43,900 --> 00:05:46,030
and then we'll just
double at the end

114
00:05:46,030 --> 00:05:48,290
to account for the
bottom half as well.

115
00:05:48,290 --> 00:05:51,980
So this is y equals
the square root

116
00:05:51,980 --> 00:05:56,990
of-- so you solve,
you subtract x minus r

117
00:05:56,990 --> 00:05:58,370
squared from both sides.

118
00:05:58,370 --> 00:06:04,710
So this is little r squared
minus x minus big R squared.

119
00:06:04,710 --> 00:06:06,290
So that's y.

120
00:06:06,290 --> 00:06:13,830
And now you can differentiate
to get dy by dx.

121
00:06:13,830 --> 00:06:14,330
All right.

122
00:06:14,330 --> 00:06:17,100
So now we have to do
our chain rule right.

123
00:06:17,100 --> 00:06:19,210
So we've got a 1/2 power.

124
00:06:19,210 --> 00:06:24,300
So this is 1/2
times-- well, it's

125
00:06:24,300 --> 00:06:27,220
going to be the inside
to the minus 1/2.

126
00:06:27,220 --> 00:06:31,830
So this is over the
square root of r squared

127
00:06:31,830 --> 00:06:36,990
minus x minus R quantity
squared, and now I

128
00:06:36,990 --> 00:06:39,950
need to multiply by the
derivative of the inside, which

129
00:06:39,950 --> 00:06:45,310
is minus 2 times x minus R. OK.

130
00:06:45,310 --> 00:06:48,110
And the 2's cancel
out a little bit,

131
00:06:48,110 --> 00:06:55,750
so we can rewrite this as
minus x minus R divided

132
00:06:55,750 --> 00:06:59,080
by the square root
of little r squared

133
00:06:59,080 --> 00:07:04,170
minus x minus R squared.

134
00:07:04,170 --> 00:07:05,470
OK.

135
00:07:05,470 --> 00:07:07,760
So that's dy/dx.

136
00:07:07,760 --> 00:07:09,490
The square root--
that's a little ugly,

137
00:07:09,490 --> 00:07:11,940
but it's OK, because we're
about to square it out again.

138
00:07:11,940 --> 00:07:12,780
So all right.

139
00:07:12,780 --> 00:07:13,750
So we've got dy/dx.

140
00:07:13,750 --> 00:07:16,750
So now we go back,
now we can compute ds,

141
00:07:16,750 --> 00:07:19,250
and then with ds, we can
go even further back,

142
00:07:19,250 --> 00:07:20,570
and we can compute dA.

143
00:07:20,570 --> 00:07:22,760
And once we've got
dA, to get a, we just

144
00:07:22,760 --> 00:07:25,907
integrate dA over the
appropriate bounds.

145
00:07:25,907 --> 00:07:27,990
So-- which we haven't
figured out yet, by the way.

146
00:07:27,990 --> 00:07:30,324
We will have to talk
about that at the very,

147
00:07:30,324 --> 00:07:33,390
you know, in a minute.

148
00:07:33,390 --> 00:07:33,890
OK.

149
00:07:33,890 --> 00:07:36,650
So we take this,
so we have dy/dx.

150
00:07:36,650 --> 00:07:40,870
So from dy/dx, we
get ds is equal

151
00:07:40,870 --> 00:07:44,740
to-- it's the square root of,
well, let's use this formula.

152
00:07:44,740 --> 00:07:49,150
It's the square root of 1
plus, now, dy/dx squared.

153
00:07:49,150 --> 00:07:49,650
OK.

154
00:07:49,650 --> 00:07:51,024
So I put this in
and I square it.

155
00:07:51,024 --> 00:07:56,820
So I get x minus
R squared on top.

156
00:07:56,820 --> 00:07:59,750
You know, you square the
minus sign, gives you 1.

157
00:07:59,750 --> 00:08:02,250
And on the bottom, we
square the square root,

158
00:08:02,250 --> 00:08:15,660
so we get little r squared
minus x minus R squared, OK, dx.

159
00:08:15,660 --> 00:08:16,230
Great.

160
00:08:16,230 --> 00:08:16,730
Good.

161
00:08:16,730 --> 00:08:18,730
So now we add these two
things together, right?

162
00:08:18,730 --> 00:08:20,840
I mean, well we
want to-- I should

163
00:08:20,840 --> 00:08:24,050
say-- we want to simplify
this to a usable form.

164
00:08:24,050 --> 00:08:25,470
And so to put it
in a usable form,

165
00:08:25,470 --> 00:08:26,844
well we're going
to manipulate it

166
00:08:26,844 --> 00:08:29,630
until it looks nice, or as
nice as we could hope for.

167
00:08:29,630 --> 00:08:30,660
And in this case, right.

168
00:08:30,660 --> 00:08:33,700
So we can, there's an
obvious simplification,

169
00:08:33,700 --> 00:08:35,530
or simplifying step,
which is we can

170
00:08:35,530 --> 00:08:36,990
add this 1 into the fraction.

171
00:08:36,990 --> 00:08:37,820
OK.

172
00:08:37,820 --> 00:08:41,640
And so this is little r
squared minus x minus big R

173
00:08:41,640 --> 00:08:44,500
squared, over little r squared
minus x minus big R squared,

174
00:08:44,500 --> 00:08:48,050
and so when you add it
to x minus big R squared

175
00:08:48,050 --> 00:08:52,220
over little r squared minus
x minus big R squared,

176
00:08:52,220 --> 00:08:53,300
this part cancels.

177
00:08:53,300 --> 00:08:55,900
And so you're left with-- OK.

178
00:08:55,900 --> 00:08:59,430
And now I'm just going to pass
the square root through right

179
00:08:59,430 --> 00:09:05,320
away, so this is little r over
the square root of little r

180
00:09:05,320 --> 00:09:13,940
squared minus x minus
big R squared dx.

181
00:09:13,940 --> 00:09:14,440
OK.

182
00:09:14,440 --> 00:09:16,023
So just a little bit
of algebra there.

183
00:09:16,023 --> 00:09:18,130
So this is ds.

184
00:09:18,130 --> 00:09:21,180
Now, OK, so now we're going--
we want to compute surface area,

185
00:09:21,180 --> 00:09:26,730
so we need dA, and dA
is 2*pi*x times ds.

186
00:09:26,730 --> 00:09:27,530
So-- OK.

187
00:09:27,530 --> 00:09:31,080
So that's easy to write
down now that I've got ds.

188
00:09:31,080 --> 00:09:46,300
So this is dA equal to 2*pi*r*x
over r squared minus x minus

189
00:09:46,300 --> 00:09:52,730
big R squared, square
root of that, dx.

190
00:09:52,730 --> 00:09:53,230
All right.

191
00:09:53,230 --> 00:09:54,850
This is dA.

192
00:09:54,850 --> 00:09:56,950
You know, we haven't done
any calculus, actually.

193
00:09:56,950 --> 00:09:58,741
Oh, I guess we took a
derivative somewhere.

194
00:09:58,741 --> 00:10:01,460
We haven't done any
integration yet.

195
00:10:01,460 --> 00:10:05,072
Now, to compute the surface
area, we just integrate this.

196
00:10:05,072 --> 00:10:10,350
So we get to, you know, the
calculus step of this problem.

197
00:10:10,350 --> 00:10:12,140
So integrate this.

198
00:10:12,140 --> 00:10:14,670
But of course, you know,
I'm expecting a number out

199
00:10:14,670 --> 00:10:15,170
at the end.

200
00:10:15,170 --> 00:10:17,020
I'm taking a definite integral.

201
00:10:17,020 --> 00:10:18,650
So I need bounds.

202
00:10:18,650 --> 00:10:21,190
Well, what bounds do I need?

203
00:10:21,190 --> 00:10:23,580
Well, I'm integrating
with respect to x.

204
00:10:23,580 --> 00:10:28,310
So I need to integrate over
the relevant values of x.

205
00:10:28,310 --> 00:10:29,972
What are the
relevant values of x?

206
00:10:29,972 --> 00:10:31,680
Well, come back to
our picture over here.

207
00:10:31,680 --> 00:10:34,320


208
00:10:34,320 --> 00:10:35,310
We have this circle.

209
00:10:35,310 --> 00:10:39,840
Its center is at x
equals R, y equals 0.

210
00:10:39,840 --> 00:10:41,280
And it has radius little r.

211
00:10:41,280 --> 00:10:45,470
So, you know, the
relevant values of x

212
00:10:45,470 --> 00:10:47,930
are just from the leftmost
point of the circle

213
00:10:47,930 --> 00:10:49,770
to the rightmost
point of the circle.

214
00:10:49,770 --> 00:10:51,640
And so this leftmost
point is just-- well,

215
00:10:51,640 --> 00:10:53,460
the radius is
little r, so this is

216
00:10:53,460 --> 00:10:56,280
big R minus-- x equals
big R minus little r,

217
00:10:56,280 --> 00:11:00,520
and the rightmost point is x
equals big R plus little r.

218
00:11:00,520 --> 00:11:07,130
So the bounds-- so I'm going
to go up here-- so I have area

219
00:11:07,130 --> 00:11:09,645
is what I get when
I integrate dA.

220
00:11:09,645 --> 00:11:14,720
And I want to integrate it from
x equals big R minus little

221
00:11:14,720 --> 00:11:19,320
r to big R plus little
r, and dA is this thing

222
00:11:19,320 --> 00:11:20,840
I found, just a moment ago.

223
00:11:20,840 --> 00:11:22,660
So this is-- well, OK.

224
00:11:22,660 --> 00:11:27,970
So 2*pi little r is a constant.

225
00:11:27,970 --> 00:11:30,350
I'm just going to factor
that out in front.

226
00:11:30,350 --> 00:11:39,890
So this is 2*pi little r times
x over the square root of little

227
00:11:39,890 --> 00:11:51,130
r squared minus big R minus--
sorry-- x minus big R squared.

228
00:11:51,130 --> 00:11:53,500
That's all under
the square root.

229
00:11:53,500 --> 00:11:55,791
ds.

230
00:11:55,791 --> 00:11:56,290
OK.

231
00:11:56,290 --> 00:11:58,840
So now we have to figure out
how to integrate this thing.

232
00:11:58,840 --> 00:12:00,510
Right?

233
00:12:00,510 --> 00:12:04,730
So this is a little ugly.

234
00:12:04,730 --> 00:12:05,970
It's not horrible, though.

235
00:12:05,970 --> 00:12:06,470
Right?

236
00:12:06,470 --> 00:12:08,550
So down here we have
something that's

237
00:12:08,550 --> 00:12:13,270
really reminiscent of one of
these trig integral things

238
00:12:13,270 --> 00:12:13,790
we've done.

239
00:12:13,790 --> 00:12:14,290
Right?

240
00:12:14,290 --> 00:12:16,400
We've got a square root
of a something squared

241
00:12:16,400 --> 00:12:18,280
minus a something else squared.

242
00:12:18,280 --> 00:12:21,510
So that, reminds, you
know, what does that

243
00:12:21,510 --> 00:12:23,740
remind, maybe some
sine substitution.

244
00:12:23,740 --> 00:12:24,950
Something like that.

245
00:12:24,950 --> 00:12:27,300
There's some there's some
trig substitution waiting

246
00:12:27,300 --> 00:12:30,240
to happen here.

247
00:12:30,240 --> 00:12:30,960
But, so OK.

248
00:12:30,960 --> 00:12:36,040
We could sort of shoot to do
it all in one substitution.

249
00:12:36,040 --> 00:12:38,040
I like, my life
always feels simpler

250
00:12:38,040 --> 00:12:40,500
when I do one little
substitution at a time.

251
00:12:40,500 --> 00:12:42,470
And so one little
substitution I could do

252
00:12:42,470 --> 00:12:46,700
is to simplify this
x minus r thing.

253
00:12:46,700 --> 00:12:48,600
I could just shift this by r.

254
00:12:48,600 --> 00:12:51,560
So I'm just going to do a
little linear substitution.

255
00:12:51,560 --> 00:12:57,600
I'm going to do u
equals x minus big R,

256
00:12:57,600 --> 00:12:59,970
or I'm going to want to
substitute the other way,

257
00:12:59,970 --> 00:13:06,050
so that's the same thing as
saying x equals u plus big R.

258
00:13:06,050 --> 00:13:11,460
And so du equals dx.

259
00:13:11,460 --> 00:13:14,110
This is a simple
little substitution.

260
00:13:14,110 --> 00:13:15,980
And I'm going to have
to move the bounds,

261
00:13:15,980 --> 00:13:22,680
so when x is big R minus little
r, then u is minus little r,

262
00:13:22,680 --> 00:13:25,600
and this top bound here, u is
going to be equal to little r.

263
00:13:25,600 --> 00:13:26,620
Positive little r.

264
00:13:26,620 --> 00:13:30,810
So let me just make
that substitution.

265
00:13:30,810 --> 00:13:35,340
So area is 2 pi little
r, integral from

266
00:13:35,340 --> 00:13:37,880
minus little r to plus little r.

267
00:13:37,880 --> 00:13:44,390
So x is u plus big
R divided by, now

268
00:13:44,390 --> 00:13:48,930
this thing becomes square root
of little r squared minus u

269
00:13:48,930 --> 00:13:51,510
squared du.

270
00:13:51,510 --> 00:13:54,697


271
00:13:54,697 --> 00:13:55,280
OK, all right.

272
00:13:55,280 --> 00:14:01,990
So now-- this is really kind
of two separate pieces, right,

273
00:14:01,990 --> 00:14:04,190
for purposes of
difficulty of integrating.

274
00:14:04,190 --> 00:14:07,154
There's the u over the
square root of r squared

275
00:14:07,154 --> 00:14:08,570
minus u squared
piece, and there's

276
00:14:08,570 --> 00:14:10,280
the big R over the
square root of r

277
00:14:10,280 --> 00:14:12,640
squared minus u squared piece.

278
00:14:12,640 --> 00:14:15,440
So let's think about
them separately.

279
00:14:15,440 --> 00:14:18,050
So for this first piece,
the u over the square root

280
00:14:18,050 --> 00:14:21,550
of little r squared
minus u squared,

281
00:14:21,550 --> 00:14:23,610
this is something
you can integrate.

282
00:14:23,610 --> 00:14:27,260
This is a-- you don't need a
trig substitution to do that.

283
00:14:27,260 --> 00:14:31,230
But it's actually-- you don't
need to do any work to do that.

284
00:14:31,230 --> 00:14:35,460
Because that function u
divided by square root

285
00:14:35,460 --> 00:14:38,280
of little r squared
minus u squared,

286
00:14:38,280 --> 00:14:40,110
that's an odd function.

287
00:14:40,110 --> 00:14:40,860
Right?

288
00:14:40,860 --> 00:14:44,820
This part, the denominator
is even, u is odd,

289
00:14:44,820 --> 00:14:46,870
and we're integrating
over an interval that's

290
00:14:46,870 --> 00:14:48,260
symmetric around the origin.

291
00:14:48,260 --> 00:14:51,610
So when we integrate this u
divided by this denominator

292
00:14:51,610 --> 00:14:57,700
part between u from minus r to
r, that's just going to give 0.

293
00:14:57,700 --> 00:15:00,410
So I can forget
about that entirely.

294
00:15:00,410 --> 00:15:02,450
So this is nice.

295
00:15:02,450 --> 00:15:04,000
And then I'll have
a constant here,

296
00:15:04,000 --> 00:15:05,740
so I'm going to factor
that out as well.

297
00:15:05,740 --> 00:15:09,780
So I get 2 pi little
r big R, integral

298
00:15:09,780 --> 00:15:17,890
from minus r to r of du over the
square root of little r squared

299
00:15:17,890 --> 00:15:18,780
minus u squared.

300
00:15:18,780 --> 00:15:21,400


301
00:15:21,400 --> 00:15:23,820
OK.

302
00:15:23,820 --> 00:15:24,320
Good.

303
00:15:24,320 --> 00:15:26,700
So far, so good.

304
00:15:26,700 --> 00:15:28,770
So we've got this
nice simplified

305
00:15:28,770 --> 00:15:29,710
form for the integral.

306
00:15:29,710 --> 00:15:34,090
So now this is screaming
out trig substitution to me.

307
00:15:34,090 --> 00:15:34,590
Right?

308
00:15:34,590 --> 00:15:35,839
There's nothing else I can do.

309
00:15:35,839 --> 00:15:37,650
So there's sort of two
things you could do.

310
00:15:37,650 --> 00:15:42,550
One is you could recognize
this as an integral related

311
00:15:42,550 --> 00:15:45,080
to arcsine, just because
you remember that.

312
00:15:45,080 --> 00:15:47,850
The other is, you have this r
squared minus u squared thing.

313
00:15:47,850 --> 00:15:50,890
And so r squared
minus u squared, that

314
00:15:50,890 --> 00:15:52,860
needs a trig substitution
of some sort,

315
00:15:52,860 --> 00:15:56,000
and the relevant trig
substitution that you

316
00:15:56,000 --> 00:15:57,530
would want to do,
is you would want

317
00:15:57,530 --> 00:16:07,290
to set u equal to r sine t.

318
00:16:07,290 --> 00:16:07,790
Why?

319
00:16:07,790 --> 00:16:11,630
Because then down here we'll
have r squared minus r squared

320
00:16:11,630 --> 00:16:15,380
sine t, or if you factor out
the little r, that's just 1

321
00:16:15,380 --> 00:16:18,310
minus sine squared t,
which is cosine squared t,

322
00:16:18,310 --> 00:16:20,810
and then you take a square root,
and you're all good, right?

323
00:16:20,810 --> 00:16:22,300
Square root of cosine squared t.

324
00:16:22,300 --> 00:16:23,820
OK.

325
00:16:23,820 --> 00:16:26,660
So we find this
trig substitution.

326
00:16:26,660 --> 00:16:30,220
So if we do u equals
r sine t, that's good.

327
00:16:30,220 --> 00:16:38,400
So du, then, is r
cosine t dt, and I

328
00:16:38,400 --> 00:16:43,620
need to change the bounds, so
minus u is equal to minus r

329
00:16:43,620 --> 00:16:46,800
when sine of t is
equal to minus 1.

330
00:16:46,800 --> 00:16:48,700
So that's at minus pi over 2.

331
00:16:48,700 --> 00:16:51,810
And u is equal to r when
sine of t is equal to 1.

332
00:16:51,810 --> 00:16:53,360
So that's a pi over 2.

333
00:16:53,360 --> 00:17:00,270
So this is equal to 2 pi
little r big R integral

334
00:17:00,270 --> 00:17:12,060
from minus pi over 2 to
pi over 2 of r cosine t dt

335
00:17:12,060 --> 00:17:12,980
divided by-- OK.

336
00:17:12,980 --> 00:17:16,590
So then down here, we
have the square root

337
00:17:16,590 --> 00:17:22,420
of r squared minus r
squared sine squared t.

338
00:17:22,420 --> 00:17:25,030


339
00:17:25,030 --> 00:17:29,440
And as we said, so r squared
minus r squared sine squared t,

340
00:17:29,440 --> 00:17:31,950
this is r squared
cosine squared t.

341
00:17:31,950 --> 00:17:34,970
And then you take a square root,
and you just get r cosine t.

342
00:17:34,970 --> 00:17:38,280
And so r cosine t
cancels r cosine t.

343
00:17:38,280 --> 00:17:42,040
So the integrand here
is actually 1, or 1 dt.

344
00:17:42,040 --> 00:17:43,670
So that's really
easy to integrate.

345
00:17:43,670 --> 00:17:45,934
You integrate a constant,
you just get-- well,

346
00:17:45,934 --> 00:17:47,850
the constant times the
length of the integral.

347
00:17:47,850 --> 00:17:55,740
So this is equal to 2*pi*r*R
times the constant is 1,

348
00:17:55,740 --> 00:17:59,250
times the length of the
integral, which is another pi,

349
00:17:59,250 --> 00:18:00,580
times pi.

350
00:18:00,580 --> 00:18:01,080
OK.

351
00:18:01,080 --> 00:18:10,890
So this is equal to 2 pi squared
little r big R. But remember,

352
00:18:10,890 --> 00:18:13,850
so far we've only
computed-- this is just

353
00:18:13,850 --> 00:18:16,910
the top half of the torus, is
all we've ever talked about.

354
00:18:16,910 --> 00:18:20,210
So we want to get the whole
torus, you just double this.

355
00:18:20,210 --> 00:18:23,900
So you double this, and you
get the area of the whole torus

356
00:18:23,900 --> 00:18:30,330
is 4 pi squared little r big
R, which is a nice formula.

357
00:18:30,330 --> 00:18:34,750
So quickly, just to
summarize what we've done.

358
00:18:34,750 --> 00:18:37,220
Standard setup.

359
00:18:37,220 --> 00:18:41,390
Here we did it, we're rotating
around the y-axis instead

360
00:18:41,390 --> 00:18:43,100
of the x-axis.

361
00:18:43,100 --> 00:18:46,940
So this formula for
dA looks a little bit

362
00:18:46,940 --> 00:18:50,340
different than what you
saw in class, mostly.

363
00:18:50,340 --> 00:18:55,180
But the thing to remember
is just that what goes here

364
00:18:55,180 --> 00:18:58,820
is the radius that this,
that your little segment is

365
00:18:58,820 --> 00:19:00,057
traveling.

366
00:19:00,057 --> 00:19:02,140
And this is the circumference
that it's traveling,

367
00:19:02,140 --> 00:19:04,899
and so this is the area
of that little ribbon

368
00:19:04,899 --> 00:19:05,690
that it traces out.

369
00:19:05,690 --> 00:19:06,930
So dA-- OK.

370
00:19:06,930 --> 00:19:10,490
And then we just did, you
know, the sort of usual thing.

371
00:19:10,490 --> 00:19:12,300
For your formula,
you need-- you know,

372
00:19:12,300 --> 00:19:14,440
you remember the formula for ds.

373
00:19:14,440 --> 00:19:18,230
Then you needed to find this
derivative and plug it in.

374
00:19:18,230 --> 00:19:21,460
And so after you've done all
that preparatory work then

375
00:19:21,460 --> 00:19:24,700
you have your integrand set up.

376
00:19:24,700 --> 00:19:26,330
And once you set
up your integrand,

377
00:19:26,330 --> 00:19:31,070
you do whatever integration
tools you can to hit it with.

378
00:19:31,070 --> 00:19:35,450
So in our case, that was
simplifying substitution,

379
00:19:35,450 --> 00:19:39,660
and then a nice observation that
we used here to simplify this,

380
00:19:39,660 --> 00:19:41,330
that this part was odd.

381
00:19:41,330 --> 00:19:43,690
I mean, you could've-- you
didn't need that observation.

382
00:19:43,690 --> 00:19:46,050
You could have done the
problem perfectly well

383
00:19:46,050 --> 00:19:51,810
with another substitution there
to kill off that first part.

384
00:19:51,810 --> 00:19:54,970
And then, OK, and then
a trig substitution,

385
00:19:54,970 --> 00:19:56,525
remembering that
this is an arcsine,

386
00:19:56,525 --> 00:19:59,960
or related to an arcsine,
to finish it off.

387
00:19:59,960 --> 00:20:03,170


388
00:20:03,170 --> 00:20:07,520
And is there anything
else I want to say?

389
00:20:07,520 --> 00:20:08,784
I think that's-- oh, yes.

390
00:20:08,784 --> 00:20:10,450
There's one other
thing I wanted to say,

391
00:20:10,450 --> 00:20:13,340
which is that we could have
done this slightly differently.

392
00:20:13,340 --> 00:20:16,960
Which is, here we solved for
y explicitly in terms of x.

393
00:20:16,960 --> 00:20:19,520
And it would have been
possible to carry this

394
00:20:19,520 --> 00:20:22,690
through using implicit
differentiation instead.

395
00:20:22,690 --> 00:20:24,600
It would have
actually simplified--

396
00:20:24,600 --> 00:20:27,400
I would have had to write
this square root of little r

397
00:20:27,400 --> 00:20:31,340
squared minus x minus big R
quantity squared fewer times

398
00:20:31,340 --> 00:20:33,690
if I'd done implicit
differentiation, starting

399
00:20:33,690 --> 00:20:37,137
from this implicit
equation relating x and y.

400
00:20:37,137 --> 00:20:38,720
You have to be a
little careful there,

401
00:20:38,720 --> 00:20:42,100
whether you're doing the whole
circle all at once, or just

402
00:20:42,100 --> 00:20:43,630
the top half or the bottom half.

403
00:20:43,630 --> 00:20:45,870
But you could do it with
implicit differentiation

404
00:20:45,870 --> 00:20:48,150
instead, and maybe save
yourself a little bit

405
00:20:48,150 --> 00:20:52,640
of messy-looking arithmetic.

406
00:20:52,640 --> 00:20:54,622
So I think I'll end there.

407
00:20:54,622 --> 00:20:55,122