Page 305 - Calculus Demystified

Solutions to Exercises
292
! " 3
3 4 e x e 81 e 81
4
3 x
(d) Signed Area = x e dx = = − = 0.
−3 4 4 4
−3

1 1
2
2
2
9. (a) Area = [−3x + 10]−[2x − 4] dx = −5x + 14 dx
−1 −1
1
3
−5x −5 5 74
= + 14x = + 14 − − 14 = .
3 −1 3 3 3
1 x 3 x 4
1
3
2
(b) Area = x − x dx = −
0 3 4 0

1 1 0 0 1
= − − − = .
3 4 3 4 12
1 x 3
1
2
(c) Area = [−x + 3]− 2xdx = − + 3x − x 2
−3 3 −3

1 2 −27 2
= − + 3 · 1 − 1 − − + 3 · (−3) − (−3)
3 3
32
= .
3
e x 2
e
(d) Area = x − ln xdx = −[x ln x − x]
1 2 1
2 2 2
e 1 e − 3
= −[e · 1 − e] − −[1 · 0 − 1] = .
2 2 2
π/4 x 2
π/4
(e) Area = x − sin xdx = + cos x
0 2 0
√ √
π 2 2 0 π 2 2
= + − + 1 = + − 1.
32 2 2 32 2
3

3 x 2
x
x
(f) Area = e − xdx = e −
0 2 0

3 9 0 0 3 11
= e − − e − = e − .
2 2 2
1

1 x 2 x 3 1 1 0 0
2
10. (a) Area = x − x dx = − = − − −
0 2 3 0 2 3 2 3
1
= .
6