Thomas Calculus 13th [Solutions]

Section 5.6 Substitution and Area Between Curves 397
2 2
70. Limits of integration: x a x 0 x 0 or
2 2 2 2
a x 0 x 0 or a x 0 x a,0, a;
0 2 2 a 2 2
A x a x dx x a x dx
a
0
2 2 3/2
0
1 2 1 2 2 2 3/2
a
2 3 ( a x ) 2 3 ( a x )
a
0
2 3/2 2 3/2
3
1 ( a ) 1 ( a ) 2a
3 3 3
x, x 0
71. Limits of integration: y x
and
x, x 0
5y x 6 or y x 6 ; for x 0; x x 6
5 5
5 5
2
5 x x 6 25( x) x 12x
36
2
x 37x 36 0 ( x 1)( x 36) 0
x 1, 36 (but x 36 is not a solution);
2
for x 0 : 5 x x 6 25x x 12x
36
2
x 13x 36 0 ( x 4)( x 9) 0
x 4,9; there are three intersection points and
0
x 6
4
A
x dx x 6
9
x dx x x 6 dx
1 5
0 5
4 5
2 0 2 4
2 9
( x 6) 2 3/2 ( x 6) 2 3/2
x
x 2 3/2 ( x 6)
x
10 3 10 3
3 10
1 0
4
36 25 2 100 2 3/2
4 36 0 2 3/2 3/2
9 225 2 4 100 50 20 5
10 10 3 10 3 10 3 10 3 10 10 3 3
2
2 x 4, x 2 or x 2
72. Limits of integration: y | x 4|
2
4 x , 2 x 2
2
2
for x 2 and x 2 : x 4 x 4
2
2 2 2
2x 8 x 8 x 16 x 4; for 2 x 2 :
2
2 2 2
4 x
2
x 4 8 2x x 8 x 0 x 0; by
2
symmetry of the graph,
2 2
2
A 2 x
0 2
4 (4 x ) dx
2 4 2
x 2
2 2
4 x 4
3 2 3 4
dx 2 x
2 2 8 x x
0
6
2
2 8 0 2 32 64 16 8 40 56 64
2 6 6 3 3
Copyright
2014 Pearson Education, Inc.