FDWK_3ed_Ch05_pp262-319

Section 5.2 Definite Integrals 283
In Exercises 13–22, use the graph of the integrand and areas to
evaluate the integral.
13. 4
32
x 3
2 ) dx 21 14. 2x 4 dx 2
2
(
15. 3
x
39 2 dx 9 p 16. 0
16 x 2 dx
2
4
17. 1
x dx 5 1
2 18. 1 x dx 1
2
19. 1
2 x dx 3 20. 1
1 1 x 2 dx 2 p 2
1
21. 2
u du 3p 2
2
12
1
1
52
22. 2
rdr 24
In Exercises 23–28, use areas to evaluate the integral.
23. b
xdx, b 0 1 2 b2 24. b
4x dx, b 0 2b 2
0
25. b
2s ds, 0 a b b 2 a 2 26. b
3t dt,
a
27. 2a
xdx, a 0 3 2 a2 28. a
a
0
a
3a
0 a b
4p
xdx, a 0 a 2
In Exercises 29–32, express the desired quantity as a definite integral
and evaluate the integral using Theorem 2.
29. Find the distance traveled by a train moving at 87 mph from
8:00 A.M. to 11:00 A.M. See page 284.
30. Find the output from a pump producing 25 gallons per minute
during the first hour of its operation. See page 284.
31. Find the calories burned by a walker burning 300 calories per
hour between 6:00 P.M. and 7:30 P.M. See page 284.
32. Find the amount of water lost from a bucket leaking 0.4 liters
per hour between 8:30 A.M. and 11:00 A.M. See page 284.
In Exercises 33–36, use NINT to evaluate the expression.
33. 5
3
x
0 x 2 dx 0.9905 34. 3 2 tan xdx 4.3863
4
35. Find the area enclosed between the x-axis and the graph of
y 4 x 2 from x 2 to x 2. 3 2
3
36. Find the area enclosed between the x-axis and the graph of
y x 2 e x from x 1 to x 3. 1.8719
In Exercises 37–40, (a) find the points of discontinuity of the
integrand on the interval of integration, and (b) use area to evaluate
the integral.
(a) 5, 4. 3, 2. 1, 0, 1, 2, 3, 4, 5 (b) 88
37. 3
2
x
x
dx (a) 0 (b) 1 38. 5
2 int x 3 dx
39. 4
x 2
1
dx 40. 6
9 x2
dx
x 1
3 5
x 3
(a) 1 (b) 7 2 (a) 3 (b) 7 7
2
6
0
3 2 (b2 a 2 )
Standardized Test Questions
You should solve the following problems without using a
graphing calculator.
41. True or False If b
a
f x dx 0, then f (x) is positive for all x
in [a, b]. Justify your answer. See page 284.
42. True or False If f (x) is positive for all x in [a, b], then
b
a
f x dx 0. Justify your answer. See page 284.
43. Multiple Choice If 5 2 f x dx 18, then 5 2
f x 4 dx
E
(A) 20 (B) 22 (C) 23 (D) 25 (E) 30
44. Multiple Choice 4 4
4 x dx D
(A) 0 (B) 4 (C) 8 (D) 16 (E) 32
45. Multiple Choice If the interval [0, ] is divided into n
subintervals of length n and c k is chosen from the kth
subinterval, which of the following is a Riemann sum C
n
∞
n
(A) sin(c k ) (B) sin(c k ) (C) sin(c k ) n
k1
k1
n
n
(D) sin n (c k) (E)
k1
sin(c k ) k
k1
46. Multiple Choice Which of the following quantities would not
be represented by the definite integral 8 0
70 dt A
(A) The distance traveled by a train moving at 70 mph for
8 minutes.
(B) The volume of ice cream produced by a machine making 70
gallons per hour for 8 hours.
(C) The length of a track left by a snail traveling at 70 cm per
hour for 8 hours.
(D) The total sales of a company selling $70 of merchandise per
hour for 8 hours.
(E) The amount the tide has risen 8 minutes after low tide if it
rises at a rate of 70 mm per minute during that period.
Explorations
In Exercises 47–56, use graphs, your knowledge of area, and the fact
that
1
x 3 dx 1
0
4
to evaluate the integral.
47. 1
x 3 dx 0 48. 1
x 3 3 dx 1 3
4
1
49. 3
x 2 3 dx 1 1
4 50.
2
0
x 3 dx
1
k1
51. 1
1 x 3 dx 3 2
4 52. x 1 3 dx
0
1
53. 2
x dx
2
)3
1 8
2 54. x 3 dx 0
0
(
55. 1
x 3 1 dx 3 1
4 56. 3 x dx
0
8
0
3 4
1 2
1 4