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