1
00:00:00,000 --> 00:00:09,900
PROFESSOR: Hi, welcome
back to recitation.

2
00:00:09,900 --> 00:00:12,250
I have a nice exercise here
for you that tests your

3
00:00:12,250 --> 00:00:14,460
knowledge of triple
integration.

4
00:00:14,460 --> 00:00:17,970
So in particular I've got
for you a cylinder.

5
00:00:17,970 --> 00:00:22,830
And my cylinder has height
h and it has radius b.

6
00:00:22,830 --> 00:00:26,170
And this is the kind
of cylinder I like.

7
00:00:26,170 --> 00:00:27,920
It's a constant density
cylinder.

8
00:00:27,920 --> 00:00:31,680
So its density is just
1 everywhere.

9
00:00:31,680 --> 00:00:35,180
So what I'd like you to do is
for the cylinder I'd like you

10
00:00:35,180 --> 00:00:37,390
to compute its moment
of inertia

11
00:00:37,390 --> 00:00:39,510
around its central axis.

12
00:00:39,510 --> 00:00:42,510
So why don't you pause the
video, have a go at that, come

13
00:00:42,510 --> 00:00:44,360
back, and you can check your
work against mine.

14
00:00:44,360 --> 00:00:52,740

15
00:00:52,740 --> 00:00:54,640
Hopefully you had some luck
working on this problem.

16
00:00:54,640 --> 00:00:56,350
Let's talk about it.

17
00:00:56,350 --> 00:00:58,980
So the first thing to notice
is that there aren't any

18
00:00:58,980 --> 00:01:01,020
coordinates in this problem.

19
00:01:01,020 --> 00:01:04,420
I've given you a cylinder but
it's up to you to choose

20
00:01:04,420 --> 00:01:07,810
coordinates, a way to arrange
your cylinder in space or a

21
00:01:07,810 --> 00:01:09,860
way to arrange your coordinates
with respect to

22
00:01:09,860 --> 00:01:10,590
your cylinder.

23
00:01:10,590 --> 00:01:15,060
So a convenient thing to do in
this case is going to be-- you

24
00:01:15,060 --> 00:01:18,690
know, we're working with respect
to the central axis of

25
00:01:18,690 --> 00:01:19,400
the cylinder.

26
00:01:19,400 --> 00:01:21,440
So let's make that
one of our axes.

27
00:01:21,440 --> 00:01:23,800
So in particular why don't
we make it our z-axis.

28
00:01:23,800 --> 00:01:26,210
That seems like a natural
sort of thing to do.

29
00:01:26,210 --> 00:01:29,160
So let me try and draw a
little picture here.

30
00:01:29,160 --> 00:01:30,655
So we've got our cylinder.

31
00:01:30,655 --> 00:01:38,720

32
00:01:38,720 --> 00:01:44,980
And there it is with our
3 coordinate axes.

33
00:01:44,980 --> 00:01:49,365
I guess it's got radius b
and it's got height h.

34
00:01:49,365 --> 00:01:53,350

35
00:01:53,350 --> 00:01:56,610
Now we've arranged it, now we
have coordinates, so now we

36
00:01:56,610 --> 00:01:59,290
want to see what it is we're
trying to do with it.

37
00:01:59,290 --> 00:02:01,830
So we're trying to compute
a moment of inertia.

38
00:02:01,830 --> 00:02:04,430
So we have to remember what
a moment of inertia means.

39
00:02:04,430 --> 00:02:06,270
So let me think.

40
00:02:06,270 --> 00:02:10,480
So a moment of inertia, when
you have a solid-- so your

41
00:02:10,480 --> 00:02:16,130
moment of inertia I with respect
to an axis is what you

42
00:02:16,130 --> 00:02:19,010
get when you take the
triple integral.

43
00:02:19,010 --> 00:02:28,470
So let's say your solid is D.
Your solid D. So you take D.

44
00:02:28,470 --> 00:02:32,400
So you take a triple integral
over D and you're integrating

45
00:02:32,400 --> 00:02:36,890
r squared with respect to
the element of mass.

46
00:02:36,890 --> 00:02:37,100
OK.

47
00:02:37,100 --> 00:02:41,210
So r squared here, this is the
distance from the axis around

48
00:02:41,210 --> 00:02:43,030
which you're computing the
moment of inertia.

49
00:02:43,030 --> 00:02:45,860
And in our case, so in
any case, this little

50
00:02:45,860 --> 00:02:47,870
moment of mass is--

51
00:02:47,870 --> 00:02:49,560
sorry, little element
of mass--

52
00:02:49,560 --> 00:02:53,000
is density times a little
element of volume.

53
00:02:53,000 --> 00:02:56,810
So we can also write this as the
triple integral over our

54
00:02:56,810 --> 00:03:03,590
region of r squared times
delta times dV.

55
00:03:03,590 --> 00:03:06,060
OK, so this is what this
is in general.

56
00:03:06,060 --> 00:03:07,800
So now let's think about
it in our case.

57
00:03:07,800 --> 00:03:12,930
Well, in our case, we've
cleverly chosen our central

58
00:03:12,930 --> 00:03:14,940
axis to be the z-axis.

59
00:03:14,940 --> 00:03:20,100
So this r is just the distance
from the z-axis.

60
00:03:20,100 --> 00:03:24,100
This I was kind enough to give
you a constant density

61
00:03:24,100 --> 00:03:26,560
cylinder, so this
delta is just 1.

62
00:03:26,560 --> 00:03:27,590
That's going to be easy.

63
00:03:27,590 --> 00:03:29,640
And then we have this
triple integral

64
00:03:29,640 --> 00:03:30,900
over the whole cylinder.

65
00:03:30,900 --> 00:03:33,250
So we want a triple integral
over the whole cylinder of

66
00:03:33,250 --> 00:03:35,800
some integrand that
involves r.

67
00:03:35,800 --> 00:03:40,370
So a natural thing to do
in this situation--

68
00:03:40,370 --> 00:03:43,930
the natural sort of thing to do
when you have a cylinder or

69
00:03:43,930 --> 00:03:46,780
anything that's rotationally
symmetric, and you have an

70
00:03:46,780 --> 00:03:50,880
integrand that behaves nicely
with respect to rotation, that

71
00:03:50,880 --> 00:03:54,130
can be written easily in terms
of r, or r and theta--

72
00:03:54,130 --> 00:03:56,575
is to do cylindrical
coordinates.

73
00:03:56,575 --> 00:03:59,450
Is to think of cylindrical
coordinates.

74
00:03:59,450 --> 00:04:02,830
So in our case that means we
just need to figure out-- at

75
00:04:02,830 --> 00:04:05,390
this point-- we need to figure
out how do we integrate over

76
00:04:05,390 --> 00:04:07,390
the cylinder in cylindrical
coordinates?

77
00:04:07,390 --> 00:04:10,450

78
00:04:10,450 --> 00:04:11,700
So let's do it.

79
00:04:11,700 --> 00:04:21,350

80
00:04:21,350 --> 00:04:23,950
So in our case, so it doesn't
matter too much what order we

81
00:04:23,950 --> 00:04:24,430
do things in.

82
00:04:24,430 --> 00:04:28,910
So we need dV. We need to write
that in terms of the

83
00:04:28,910 --> 00:04:29,680
cylindrical coordinates.

84
00:04:29,680 --> 00:04:31,850
So that's dz, dr, and d theta.

85
00:04:31,850 --> 00:04:35,220
And so we know that dV
is r dz dr d theta.

86
00:04:35,220 --> 00:04:38,170
You might want some other order
there, but that's a

87
00:04:38,170 --> 00:04:39,270
good, nice order.

88
00:04:39,270 --> 00:04:43,100
It usually is the simplest
order to consider.

89
00:04:43,100 --> 00:04:47,170
So this moment of inertia in our
case is going to be this

90
00:04:47,170 --> 00:04:49,680
triple integral.

91
00:04:49,680 --> 00:04:53,130
OK, so we said r squared
delta, r squared times

92
00:04:53,130 --> 00:04:54,620
density, density is 1.

93
00:04:54,620 --> 00:04:56,100
So that's just r squared.

94
00:04:56,100 --> 00:04:59,610
And r, the distance to
the axis is r, the

95
00:04:59,610 --> 00:05:01,520
distance to the z-axis.

96
00:05:01,520 --> 00:05:11,830
So that's just r squared
times r dz dr d theta.

97
00:05:11,830 --> 00:05:15,880
So this is the integral we're
trying to compute, but we need

98
00:05:15,880 --> 00:05:16,690
bounds, right?

99
00:05:16,690 --> 00:05:20,430
It's a triple integral, it's a
definite integral, we need to

100
00:05:20,430 --> 00:05:22,670
figure out what the bounds
are to evaluate it

101
00:05:22,670 --> 00:05:23,760
as an iterated integral.

102
00:05:23,760 --> 00:05:27,050
So let's go look at this
little picture we drew.

103
00:05:27,050 --> 00:05:31,670
So I guess I didn't discuss
this, but I made a choice just

104
00:05:31,670 --> 00:05:34,270
to put the bottom of the
cylinder in the xy plane and

105
00:05:34,270 --> 00:05:36,060
the top at height h.

106
00:05:36,060 --> 00:05:37,560
It's not going to matter.

107
00:05:37,560 --> 00:05:38,920
If you had made some
other choice, it

108
00:05:38,920 --> 00:05:41,130
would work out fine.

109
00:05:41,130 --> 00:05:43,610
So that means the z is
going from 0 to h,

110
00:05:43,610 --> 00:05:45,580
regardless of r and theta.

111
00:05:45,580 --> 00:05:47,980
So z is going from 0 to h.

112
00:05:47,980 --> 00:05:49,775
That's nice, let's put
that over here.

113
00:05:49,775 --> 00:05:51,470
z is the inside one.

114
00:05:51,470 --> 00:05:53,070
It's going from 0 to h.

115
00:05:53,070 --> 00:05:54,730
Then r is next.

116
00:05:54,730 --> 00:05:56,620
Well, this is also--

117
00:05:56,620 --> 00:06:00,090
you know, cylinders are great
for cylindrical coordinates.

118
00:06:00,090 --> 00:06:01,610
Shocker, right, given the name.

119
00:06:01,610 --> 00:06:03,110
I know.

120
00:06:03,110 --> 00:06:06,630
So r is going from 0 to
what's the radius?

121
00:06:06,630 --> 00:06:07,700
Our radius was b.

122
00:06:07,700 --> 00:06:09,830
So r goes from 0 to
b, and that's true

123
00:06:09,830 --> 00:06:11,640
regardless of theta.

124
00:06:11,640 --> 00:06:16,790
So back over here, if we have r
going from 0 to b and theta

125
00:06:16,790 --> 00:06:20,620
is just going from 0 to 2 pi,
we're doing a full rotation

126
00:06:20,620 --> 00:06:22,070
all the way around
the cylinder.

127
00:06:22,070 --> 00:06:24,400
So this is what our moment of
inertia is, and now we just

128
00:06:24,400 --> 00:06:25,840
have to compute it.

129
00:06:25,840 --> 00:06:29,200
So we've got our inner
integral here is

130
00:06:29,200 --> 00:06:30,670
with respect to z.

131
00:06:30,670 --> 00:06:39,150
So the inner integral is
the integral from 0 to

132
00:06:39,150 --> 00:06:43,620
h of r cubed dz.

133
00:06:43,620 --> 00:06:46,140
And r cubed doesn't have
any z's in it.

134
00:06:46,140 --> 00:06:46,920
Fabulous.

135
00:06:46,920 --> 00:06:52,940
So that's just going to
be r cubed z, where z

136
00:06:52,940 --> 00:06:55,190
goes from 0 to h.

137
00:06:55,190 --> 00:06:58,050
So that's h r cubed minus 0.

138
00:06:58,050 --> 00:06:59,490
So hr cubed.

139
00:06:59,490 --> 00:06:59,750
All right.

140
00:06:59,750 --> 00:07:00,570
So that's the inner.

141
00:07:00,570 --> 00:07:01,820
Now let's look at the
middle integral.

142
00:07:01,820 --> 00:07:05,900

143
00:07:05,900 --> 00:07:08,140
So this is going to be
the integral as--

144
00:07:08,140 --> 00:07:11,810
that's our r integral.

145
00:07:11,810 --> 00:07:14,710
So that's going from 0 to b.

146
00:07:14,710 --> 00:07:15,590
And what are we integrating?

147
00:07:15,590 --> 00:07:17,410
We're integrating the
inner integral.

148
00:07:17,410 --> 00:07:20,130
So the inner integral
was h r cubed.

149
00:07:20,130 --> 00:07:24,960
So we're integrating h r
cubed dr from 0 to b.

150
00:07:24,960 --> 00:07:26,730
All right, this is not
quite as easy.

151
00:07:26,730 --> 00:07:30,210
But h as a constant, we're
integrating r cubed r.

152
00:07:30,210 --> 00:07:31,770
I've done worse.

153
00:07:31,770 --> 00:07:33,040
You've done worse.

154
00:07:33,040 --> 00:07:38,410
So that's going to be h
r to the fourth over 4

155
00:07:38,410 --> 00:07:40,130
between 0 and b.

156
00:07:40,130 --> 00:07:40,440
So OK.

157
00:07:40,440 --> 00:07:44,770
So that's h b to the fourth over
4 minus h times 0 to the

158
00:07:44,770 --> 00:07:45,550
fourth over 4.

159
00:07:45,550 --> 00:07:46,710
So the second term's 0.

160
00:07:46,710 --> 00:07:52,550
So this is just equal to h times
b to the fourth over 4.

161
00:07:52,550 --> 00:07:57,590
And finally, we have our
outer most integral.

162
00:07:57,590 --> 00:07:59,010
So what was that integral?

163
00:07:59,010 --> 00:08:03,440
Well, that was the integral from
0 to 2 pi d theta of the

164
00:08:03,440 --> 00:08:04,820
second integral.

165
00:08:04,820 --> 00:08:06,600
So this is of the
middle integral.

166
00:08:06,600 --> 00:08:13,660
So it's the integral from 0 to
2 pi d theta of the middle

167
00:08:13,660 --> 00:08:18,740
integral which was h b
to the fourth over 4.

168
00:08:18,740 --> 00:08:20,800
And this is just a
constant again.

169
00:08:20,800 --> 00:08:21,940
Great.

170
00:08:21,940 --> 00:08:27,950
So this is hb to the fourth
over 4 times 2 pi.

171
00:08:27,950 --> 00:08:30,170
So what does that work out to?

172
00:08:30,170 --> 00:08:37,640
That's h b to the fourth
pi over 2.

173
00:08:37,640 --> 00:08:38,270
All right.

174
00:08:38,270 --> 00:08:39,160
So there you go.

175
00:08:39,160 --> 00:08:44,970
Now if you wanted to, you could
also rewrite this a

176
00:08:44,970 --> 00:08:46,740
little bit, because you
could note that

177
00:08:46,740 --> 00:08:50,660
this is pi h b squared.

178
00:08:50,660 --> 00:08:54,650
That's your volume
of your cylinder.

179
00:08:54,650 --> 00:08:56,760
And in fact, it's not just your
volume, it's your mass of

180
00:08:56,760 --> 00:08:58,690
your cylinder, because it
had constant density 1.

181
00:08:58,690 --> 00:09:04,770
So you also could've written
this as mass times a

182
00:09:04,770 --> 00:09:07,470
squared over 2.

183
00:09:07,470 --> 00:09:09,610
Sorry. b squared over 2.

184
00:09:09,610 --> 00:09:11,910
I don't know where
a came from.

185
00:09:11,910 --> 00:09:13,080
Mass times b squared.

186
00:09:13,080 --> 00:09:15,470
So you have some other options
for how you could write this

187
00:09:15,470 --> 00:09:21,020
answer by involving the volume
and mass and so on.

188
00:09:21,020 --> 00:09:23,860
So let's just recap
very quickly.

189
00:09:23,860 --> 00:09:24,800
Why we did what we did.

190
00:09:24,800 --> 00:09:26,200
We had a cylinder.

191
00:09:26,200 --> 00:09:31,940
And so really, given a cylinder,
it was a natural

192
00:09:31,940 --> 00:09:34,360
choice to look at cylindrical
coordinates.

193
00:09:34,360 --> 00:09:36,450
And once we had cylindrical
coordinates,

194
00:09:36,450 --> 00:09:37,270
everything was easy.

195
00:09:37,270 --> 00:09:41,060
So we just took our general form
of the moment of inertia,

196
00:09:41,060 --> 00:09:43,960
took the region in question, in
cylindrical coordinates it

197
00:09:43,960 --> 00:09:46,970
was very, very easy to describe
this entire region.

198
00:09:46,970 --> 00:09:50,710
And then our integrals were
pretty easy to compute after

199
00:09:50,710 --> 00:09:53,140
we made that choice.

200
00:09:53,140 --> 00:09:56,190
After we made that choice they
were nice and easy to compute.

201
00:09:56,190 --> 00:09:59,000
So I'll stop there.

202
00:09:59,000 --> 00:09:59,666