Thomas Calculus 13th [Solutions]

396 Chapter 5 Integration
65. a 0, b 2;
4 2 4
f ( x) g( x) 8 x x A (8 x x ) dx
0
2 5 2
8x
x 16 32 80 32 48
2 5
0
5 5 5
2
66. Limits of integration: x 2x x
2
x 3x
0
x( x 3) 0 a 0 and b 3;
f ( x) g( x) x
2
( x 2 x) 3x 2
x
3 2
A (3 x x ) dx 3x x 27 9 27 18 9
0 2 3
0
2 2 2
2 2 2
67. Limits of integration: x x 4x 2x 4x
0
2 x( x 2) 0 a 0 and b 2;
2 2 2
f ( x) g( x) ( x 4 x) x 2x 4x
A 2 2
0 ( 2x 4 x)
dx 2x
3 4x
2 2
3 2
0
16 16 32 48 8
3 2 6 3
2 3 3
2 2 2
68. Limits of integration: 7 2x
x 4 3x
3 0
3( x 1)( x 1) 0 a 1 and b 1;
2 2 2
f ( x) g( x) (7 2 x ) ( x 4) 3 3x
1 2
A
1 (3 3 x ) dx 3 x x 3 1 1 1 1
3
3 3
1
6 2 4
3
3 1
4
69. Limits of integration: x
2
4x 4
2
x
4
x
2
5x
4 0
2
( x
2
4)( x 1) 0 ( x 2)( x 2)( x 1)( x 1) 0
x 2, 1,1, 2; f ( x) g( x) 4
( x
2
4x 4)
2
x
4
x
2
5x
4 and
g( x) f ( x) 2
x
4
( x
2
4x 4)
4
x
2
5x
4
A
1 4 2 1 4
x 5x 4 dx ( x
2 1
2
5x 4) dx
2 4 2
1 ( 5x 4) dx
5 3 1 5 3 1 5 3 2
x 5x 4x
x 5x
4x
x 5x 4x
5 3
2
5 3
1
5 3
1
1 5 4 32 40 8 1 5 4 1 5 4 32 40 8 1 5 4 60 60 300 180
5 3 5 3 5 3 5 3 5 3 5 3 5 3 15
8
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