Thomas Calculus 13th [Solutions]

402 Chapter 5 Integration
91.
c 0, d ; f ( y) g( y)
2
3 sin y cos y 0 3sin y cos y
/2 3/2
/2
A 3 2
0 sin y cos y dy 3 3
(cos y ) 0
2(0 1) 2
92. a 1, b 1; f ( x) g( x)
2 1/3
sec x x
3
1 2 1/3
A
1 sec x x dx
3
1
3 3 4/3
tan x x
3 4 1
3 3 3 3 6 3
3 3
4 4
93. A A1 A2
3
3
Limits of integration: x y and x y y y
3
y y 0 y( y 1)( y 1) 0 c1 1, d1
0
3
and c 2 0, d 2 1; f 1 ( y) g 1 ( y)
y y and
3
f2 ( y) g2
( y)
y y by symmetry
about the origin, A1 A2 2A2
A
1 3
2 ( y y ) dy 2 y y 2 1 1 1
0
2 4 2 4 2
0
2 4 1
94. A A1 A2
3
5 3 5
Limits of integration: y x and y x x x
5 3 3
x x 0 x ( x 1)( x 1) 0 a1 1, b1
0
3 5
and a 2 0, b 2 1; f 1 ( x) g 1 ( x)
x x and
5 3
f2 ( x) g2
( x)
x x by symmetry about the
1 3 5
origin, A1 A2 2A2
A 2 ( x x ) dx
0
4 6 1
2 x x 2 1 1 1
4 6 0
4 6 6
95. A A1 A2
Limits of integration: y x and y 1 x 1 , x 0
2 2
x x
3
x 1 x 1, f1( x) g1( x) x 0 x
1
2
A 1
1 x dx x 1 1
0 2 2 ; f2 ( x ) g2
( x ) 0 2
0
x
2
2 2
x A 2
1 x dx 1
2
1 1 1 ;
x 1 2 2
A A 1 1
1 A2
1
2 2
Copyright
2014 Pearson Education, Inc.