Thomas Calculus 13th [Solutions]

444 Chapter 6 Applications of Definite Integrals
8. a 0, b 1;
b shell shell 1
V 2 radius height dx 2 x 2x x dx
a
0
2
1
3
1 2 3
0 2 x dx
2
0 3 x dx x
0
2 1
9. a 0, b 1;
b shell shell 1
2
V 2 radius height dx 2 x (2 x)
x dx
a
0
2 1 2 3 2
0 2 x x x dx 2 x x x
3 4
0
2 1 1 1 2 12 4 3 10 5
3 4 12 12 6
10. a 0, b 1;
3 4 1
b shell shell 1
2 2
V 2 radius height dx 2 x 2 x x dx
a
0
1 2 1 3
2 x 2 2x dx 4 x x dx
0 0
2 4
4 x x
1 4 1 1
2 4
0
2 4
11. a 0, b 1;
V
b shell shell 1
2 radius height dx 2
a
0
x x (2x 1) dx
2
1 3/2 2 5/2 3 2
1
x 2x x dx 2 2 x 2 x 1 x
0 5 3 2 0
2 2 2 1 2 12 20 15 7
5 3 2 30 15
12. a 1, b 4;
b shell shell 4
3 1/2
V 2 radius height dx 2 x x dx
a
1 2
4 1/2 3/2
4
3 x dx 3 2 x 2 4 3/2
1
1 3 1
2 (8 1) 14
13. (a)
x sin x , 0 x sin x, 0 x
x f ( x) x
x f ( x) ;
x, x 0
0, x 0
since sin 0
0 we have
sin x, 0 x
x f ( x) x f ( x) sin x, 0 x
sin x, x 0
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