Thomas Calculus 13th [Solutions]

452 Chapter 6 Applications of Definite Integrals
35. (a)
(b)
d shell shell 1
2
V 2 radius height dy 2 y 8y y dy
c
0
2 3/2 3 4 2 5/2 y
2 2 2y y dy 2 y
0
5 4
5
4 2 2 4 3
2 2 2 4 2 4 4
5 4 5 4
2 4 8 1 8 (8 5) 24
5 5 5
4 2
b shell shell 4 4 3/2 2 5/2
V 2 radius height dx
0 2 x x x dx
8 2 x x dx
0 8 2 x x
a
5 32
0
5 4 6 8
0
2 3 4 4
7 9 4
2 2 2 4 2 2 2 2 (32 20) 2 3 2 3 48
5 32 5 32 160 160 5 5
36. (a)
b shell shell 1
2
V 2 radius height dx 2 x 2x x x dx
a
0
1 2 1 2 3
2 x x x dx 2 x x dx
0 0
3 4
2 x x
1 2 1 1
3 4
0
3 4 6
(b)
b shell shell 1 2 1
2
V 2 radius height dx 2 1 x 2x x x dx 2 1 x x x dx
a
0 0
1 2 3 3
2 x 2x x dx 2 x 2 x x 2 1 2 1 2 (6 8 3)
0 2 3 4
0
2 3 4 12 6
2 4 1
37. (a)
b 2 2 1 1/2
V R( x) r( x) dx x 1 dx
a
1/16
1/2
1
2 x x (2 1) 2 1 1
1/16
4 16
1
7 9
16 16
(b) V d shell shell 1
2 1 1
radius height 2
c
dy 0
y 4
y 16
dy
2 3 y
1 2 y
2 y dy 2 y
1 16 2 32
1
2 2
2 1 1 1 2 1
8 2 32 32
2 9 (8 1)
38. (a) V d 2 2 2
( ) ( )
1 1
c
R y r y dy 1
4
y 16
dy
2
1 3 y
y
1 1 1 1
3 16
1
24 8 3 16
( 2 6 16 3) 11
48 48
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