25.
[
2
1, 2] by [
2
1, 2]
Use washer cross sections. A washer has inner radius
r
5
1,
outer radius
R
5
e
x
/2
, and area
p
(
R
2
2
r
2
)
5
p
(
e
x
2
1).
The volume is
E
ln 3
0
p
(
e
x
2
1)
dx
5
p
3
e
x
2
x
4
5
p
(3
2
ln 3
2
1)
5
p
(2
2
ln 3).
26.
Use cylindrical shells. Taking the hole to be vertical, a shell
has radius
x
and height 2
ˇ
2
w
2
w
2
w
x
w
2
w
. The volume of the
piece cut out is
E
ˇ
3
w
0
2
p
(
x
)(2
ˇ
2
w
2
w
2
w
x
w
2
w
)
dx
5
2
p
E
ˇ
3
w
0
2
x
ˇ
4
w
2
w
x
w
2
w
dx
5
2
p
3
2}
2
3
}
(4
2
x
2
)
3/2
4
52}
4
3
}
p
(1
2
8)
5 }
28
3
p
}
<
29.3215 ft
3
.
27.
The curve crosses the
x
axis at
x
56
3.
y
952
2
x
, so the
length is
E
3
2
3
ˇ
1
w
1
w
(
w
2
w
2
w
x
)
w
2
w
dx
5
E
3
2
3
ˇ
1
w
1
w
4
w
x
2
w
dx
, which
using NINT evaluates to
<
19.4942.
28.
[
2
2, 2] by [
2
2, 2]
The curves intersect at
x
5
0 and
x
56
1. Use the graphs’
x
 and
y
axis symmetry:
}
d
d
x
}
(
x
3
2
x
)
5
3
x
2
2
1, and the total perimeter is
4
E
1
0
ˇ
1
w
1
w
(
w
3
w
x
2
w
2
w
1
w
)
2
w
dx
, which using NINT evaluates to
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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