Thomas Calculus 13th [Solutions]

450 Chapter 6 Applications of Definite Integrals
(b)
(c)
(d)
b shell shell 4 4 4 2
V 2 radius height dx 2 x x 2 x dx 2 x 2 x dx 2 2x x dx
a
0 2 0 2 0 2
3 4
2
2 x x 2 16 64 32
6
0
6 3
b shell shell 4 4 4
2
V 2 radius height dx 2 (4 x) x 2 x dx 2 (4 x) 2 x dx 2 8 4x x dx
a
0 2 0 2 0
2
3 4
2
2 8x
2x
x 2 32 32 64 64
6
0
6 3
b 2 2 4 2
2 4
2
2
V R( x) r( x) dx (8 x) 6 x dx 64 16x x 36 6x x dx
a
0 2 0
4
4
3 2 2
x
0 4 10 x 28 dx x
4
5 x 28 x [16 (5) (16) (7)(16)] (3) (16) 48
0
3 4
31. (a)
(b)
(c)
(d)
32. (a)
(b)
(c)
d shell shell 2
V 2 radius height dy 2 y( y 1) dy
c
1
2 2
2 y y dy 2
1
3 2 2
y y
3 2
2 8 4 1 1
3 2 3 2
2 7 2 1 (14 12 3) 5
3 2 3 3
1
V
b
2 shell shell 2 2 2 2
radius height dx
1 x (2 x ) dx 2 1 x x dx 2 x x
a
3
1
2 4 8 1 1 2 12 8 3 1 2 4 2 4
3 3 3 3 3 3 3
V
b shell shell 2
10
2
2 20 16
radius height dx 2 x (2 x) dx 2
x
a
1 3 1 3 3
2
x dx
2 2 3
2
20 x 8 x 1 x
40 32 8 20 8 1 3
3 3 3 1 3 3 3 3 3 3 3
2
d shell shell 2 2 2 ( y 1)
V 2 2
radius height dy 2 ( y 1)( y 1) dy 2 ( y 1) 2
c
1 1
3 3
1
d shell shell 2 2
V 2 radius height dy 2 y y 0 dy
c
0
4 2
4
2 2 3 2
0
2 y
y dy
4 2 4
8
0
b shell shell
V 2
a radius height dx
4 4 3/2
2 x 2 x dx 2 2x x dx
0 0
2 5/2
4
5
2 x 2 x 2 16 2 2
5 0
5
2 16 64 2 (80 64) 32
5 5 5
V
b shell shell 4 4
2 radius height dx 2 (4 x) 2 x dx 2 8
a
0 0
1/2
4x 2x 3/2
x dx
2 8x 3/2 2 5/2
4
8 x x 2 x 2 32 64 16 64 2 (240 320 192) 2 (112) 224
3 5 0
3 5 15 15 15
3 2
3 2
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