Thomas Calculus 13th [Solutions]

Section 6.2 Volumes Using Cylindrical Shells 443
b 2 2 2 2
f ( x) 2 dx b f ( x) dx 10 6 b f ( x) 4 f ( x) 4 f ( x) dx 4
a a a
b b b b b
(4 f ( x) 4) dx 4 4 f ( x) dx 4 dx 4 f ( x) dx ( b a) 1 f ( x) dx 1 b a
a a a a a
6.2 VOLUMES USING CYLINDRICAL SHELLS
1. For the sketch given, a 0, b 2;
b
2 2 2 3 2 4 2
shell shell
V 2 4 16
radius height dx 2 x 1 x dx 2 x x dx 2 x x 2
a
0 4 0 4 2 16
0
2 16
2 3 6
2. For the sketch given, a 0, b 2;
b
2 2 2 3 4 2
shell shell 2
V 2 radius height dx
0 2 x 2 x dx
4 2 0 2 x x dx
4 2 x x
a
16
2 4 1 6
0
3. For the sketch given, c 0, d 2;
d
2 shell shell 2 2 2 3
radius height 0 2 2 0
2 y
V dy y y dy y dy
c
4
2
0
4 2
4. For the sketch given, c 0, d 3;
d shell shell 3 2 3 3 y
V 2 9
radius height dy 2 y 3 3 y dy 2 y dy 2
c
0 0
4 2
0
4 3
5. For the sketch given, a 0, b 3;
b shell shell 3
2
V 2 radius height dx 2 x x 1 dx;
a
0
2
u x 1 du 2 x dx; x 0 u 1, x 3 u 4
4 1/2 3/2
4
V u du 2 2 3/2 2 14
1 3 u
1 3 4 1 3 (8 1)
3
6. For the sketch given, a 0, b 3;
V b shell shell 3
2 2 9x
;
a
dx x dx
radius height 0 3
x 9
3 2 2
[ u x 9 du 3x dx 3 du 9 x dx; x 0 u 9, x 3 u 36]
36 1/2 1/2
36
V 2 3u du 6 2u
12 36 9 36
9 9
7. a 0, b 2;
b shell shell 2
V 2 radius height dx 2 x x x dx
a
0
2
2 2 2 2 3
2
3
0 2 x dx
2 0 3 x dx x 8 0
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