5128_Ch08_434-471

Section 8.4 Improper Integrals 467
Thus,
V plim
b→ [
]
2x
1
2x2 4e2x
b
0
2b
1
plim
b→ [ 2b2
4e2b
1 4 ] p 4 ,
and the volume of the solid is p4. Now try Exercise 55.
Quick Review 8.4 (For help, go to Sections 1.2, 5.3, and 8.2.)
In Exercises 1–4, evaluate the integral.
In Exercises 7 and 8, confirm the inequality.
1. 3
dx
ln 2 2.
0 x
1
x dx
3
1 x 2
7.
0
1
co s
x
x
2
12 , x Because 1 cos x 1 for all x
x
1
dx
3. x
2
1 4 2 x tan1 C 4. 2
d 8. 1 x
x 4 1 3 x3 C
x
2
1 x , x 1 Because x2 1 x 2 x for x 1
In Exercises 9 and 10, show that the functions f and g grow at the
same rate as x→.
In Exercises 5 and 6, find the domain of the function.
1
1
5. gx (3, 3) 6. hx
9 x
2 x 9. f x 4e x 5, gx 3e x 7 lim 4 ex
5
1 (1, ∞) x→∞ 3ex
4 7 3
10. f x 2x 1, gx x 3
lim 2 x 1 2
x→∞
x3
Section 8.4 Exercises
In Exercises 1–4, (a) express the improper integral as a limit of
definite integrals, and (b) evaluate the integral.
1.
(a) lim
2x
0 x
2
dx
2.
b→∞
b 2x
0 x
2
dx
1 dx
(a) lim
b→
b dx
1
x
1/3
1
1 x
1/3
(b) ∞, diverges
(b) , diverges
3. 2x
(x
2
1) 2 dx
4. dx (a) lim
b→
b
dx
1
x
1 x
(b) , diverges
In Exercises 5–24, evaluate the improper integral or state that it
diverges.
5.
d x
x4 1/3 6.
2 dx
1
x3
1
7. dx
3
diverges 8.
1
dx
4
diverges
x
1 x
1
9. d 0
x
x2 1 10.
(x
dx2) 3 1/8
2
2dx
11.
x
2
ln(3)
12. 3dx
1
2 x
2
3 ln(2)
x
13.
0
dx
2dx
1
x 2 ln(2) 14.
5x 6
x 2 ln(3)
4x 3
3. (a) lim
b→−
0 2x
b (x
2
1) 2 dx lim
(b) 0, converges
b→
b
1
2x
0 (x
2
1) 2 dx
15.
5x
6
2dx
1 x
2
dx diverges 16.
2x
2 x
2
ln(2)
2x
17.
0
xe 2x dx (3/4) e 2 18. x 2 e x dx 2
1
19.
x ln(x) dx diverges 20.
(x 1)e x dx 2
1
21. ex dx 2 22. 2xex2 dx 0
23. dx
e x
e x p/2 24. e2x dx diverges
In Exercises 25–30, (a) state why the integral is improper. Then
(b) evaluate the integral or state that it diverges.
2
25. 0 1
dxx 2 See page 468.
1
27. 0
1
29. 0
0
1
26. 0
4
x 1
dx
28.
x x
2 See page 468.
2
0
4
x ln(x)dx See page 468. 30. 1
dx
1
x See page 468.
2
x
e dx See page 468.
x
d
x
x
See page 468.