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30.
[
2
1, 4] by [
2
1, 3]
A cross section has radius
r
5
y
3/2
and area
A
(
y
)
5
p
r
2
5
p
y
3
. The volume is
E
2
0
p
y
3
dy
5
p
3
}
1
4
}
y
4
4
5
4
p
.
31.
[
2
1.2, 3.5] by [
2
1, 2.1]
Use washer cross sections. A washer has inner radius
r
5
1,
outer radius
R
5
y
1
1, and area
A
(
y
)
5
p
(
R
2
2
r
2
)
5
p
[(
y
1
1)
2
2
1]
5
p
(
y
2
1
2
y
). The
volume is
E
1
0
p
(
y
2
1
2
y
)
dy
5
p
3
}
1
3
}
y
3
1
y
2
4
5 }
4
3
}
p
.
32.
[
2
1.7, 3] by [
2
1, 2.1]
Use cylindrical shells: a shell has radius
x
and height
x
. The
volume is
E
1
0
2
p
(
x
)(
x
)
dx
5
2
p
3
}
1
3
}
x
3
4
5 }
2
3
}
p
.
33.
[
2
2, 4] by [
2
1, 5]
Use cylindrical shells: A shell has radius
x
and height
x
2
.
The volume is
E
2
0
2
p
(
x
)(
x
2
)
dx
5
2
p
3
}
1
4
}
x
4
4
5
8
p
.
34.
[
2
0.5, 1.5] by [
2
0.5, 1.5]
The curves intersect at
x
5
0 and
x
5
1. Use cylindrical
shells: a shell has radius
x
and height
ˇ
x
w
2
x
. The volume
is
E
1
0
2
p
(
x
)(
ˇ
x
w
2
x
)
dx
5
2
p
3
}
2
5
}
x
5/2
2 }
1
3
}
x
3
4
5 }
2
1
p
5
}
.
35.
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 Spring '08
 GERMAN

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