Page 304 - Calculus Demystified

Chapter 4
11 291
1 1
= −−
6 6
j=−12
= 8.

1 ln x e ln x
(c) Area = − dx + dx
1/2 x 1 x
! " 1 ! " e
2
2
ln x ln x
= − +
2 2
1/2 1
2
0 ln (1/2) 1 2 0 2
= − − − + −
2 2 2 2
≈ 0.7404.
0 4 3 4

3 x
3 x
(d) Area = −x e dx + x e dx
−3 0
! 4 " 0 ! 4 " 3
e x e x
= − +
4 4
−3 0
81 81 81
1 e e 1 e 1
= − − − + − = − .
4 4 4 4 2 2
2
2 x 4 3x 2

3
8. (a) Signed Area = x + 3xdx = +
−2 4 2 −2
4 2 4 2
2 3 · 2 (−2) 3 · (−2)
= + − + = 0.
4 2 4 2
2π 2π 1

(b) Signed Area = sin 3x cos 3xdx = sin 6xdx
−2π −2π 2
2π
1 cos 6x 1 1
= − =− − − = 0.
2 6 −2π 6 6
! " e
2
e ln x ln x

(c) Signed Area = dx =
1/2 x 2
1/2
2
1 2 ln (1/2)
= −
2 2
≈ 0.2598.