Chapter 5
Section 5.1
1. Answers will vary.
2. False
3. 
4. True
5. (a) Area is 
. 
(b) Area is 
. 
(c) 
7. (a) 3
(b) 6
9. [1, 2], [2, 3], [3, 4]
11. 
13. (a) 14
(b) 48
15. (a) 
(b) 
(c) See Student Solutions Manual.
(d) See Student Solutions Manual.
(e) See Student Solutions Manual.
17. (a) s4 = 40, s8 = 44
(b) S4 = 56, S8 = 52
19. (a) s4 = 34, s8 = 
(b) S4 = 50, S8 = 
21. (a) s4 = 
(b) s4 = 
23. sn = 
25. (a) A = 
(b) A = 
(c) Answers will vary.
27. (a) A = 
(b) A = 
(c) Answers will vary.
29. (a) A = 
(b) A = 
(c) Answers will vary.
31. (a) A = 
(b) A = 
(c) Answers will vary.
33. 10
35. 18
37. 
39. A ≈ 25,994
41. A ≈ 1.693
43. (a) 
(b) 
.
(c) See Student Solutions Manual.
(d) See Student Solutions Manual.
(e) 
(f) 5.500 ≤ A ≤ 5.590
45. See Student Solutions Manual.
47. See Student Solutions Manual.
Section 5.2
1. Partition
2. (c) 2
3. True
4. Lower limit of integration; upper limit of integration; integral sign; integrand
5. 0
6. False
7. True
8. (b) 10
9.
11. 
13. (a) [−4,−1], [−1, 0], [0, 1], [1, 3], [3, 5], [5, 6].
(b) 0
(c) 13
15. 
17. 
19. 
21. 
23. 7e
25. 3π
27. 0
29. 
31. 
33. Yes; answers will vary.
35. No; answers will vary.
37. No; answers will vary.
39. (a) 
(b) 
(c) 
41. (a) 
(b) 
(c) 
43. (a)
(b) 
(c) e2 − 1
45. 
47. (a) 
(b) ≈14.787
49. (a) 
(b)
51. 
53. newton-meters
55. meters
57. (a) 
(b) 
(c) See Student Solutions Manual.
59. (a)
(b) 
(c) 
(d) 
61. See Student Solutions Manual.
63. See Student Solutions Manual.
65. See Student Solutions Manual.
Section 5.3
1. f (x)
2. False
3. False
4. False
5. 
7. (3 + t2)3/2
9. ln x
11. 
13. 5x4 sec(x5)
15. −sin(x2)
17. −10x(30x2)2/3
19. 5
21. 
23. 
25. 
27. 
29. 
31. 1
33. 
35. 
37. 
39. 
41. e2 − 1
43. 160
45. 
47. 
49. 
which goes to infinity as r →∞
51. The corporation has 23 million dollars in sales over two years.
53. (a)
(b) cm3
(c) 100cm3 of helium leaked out in the first 5 hours.
55. (a) 
(b) 132.5m
57. P(x) = 3x2+6x+6
59. (a)
(b) Even
(c) Answers will vary.
61. (a) 1 and 2
(b) Even
(c) Answers will vary.
63. 
65. F′(c)
67. e4−1
69. 4
71. See Student Solutions Manual.
73. (a) See Student Solutions Manual.
(b) See Student Solutions Manual.
(c) Answers will vary.
75. 
77. (a)
(b) Yes
(c) Yes
Section 5.4
1. True
2. False
3. True
4. −2
5. Average value
6. False
7. 7
9. 31
11. 17
13. 
15. 4
17. −3
19. 
21. 
23. 
25. −3
27. 
29. 
31. 
33. 
35. 
37. 
39. 
41. 
43. See Student Solutions Manual.
45. See Student Solutions Manual.
47. 12 to 32
49. 
51. 1 to 
53. 1 to e
55. u = 
57. u = 
59. u = 
61. e – 1
63. 
65. 
67. 
69. 
71. (a) 12
(b) 4
(c) Answers will vary.
73. (a) 
(b) 
(c) Answers will vary.
75. 9
77. 
79. 37.5 °C
81. (a)
= 117.1875 N
(b) 4 m
83. ≈1000.345 kg/m3
85. 
87. (a) 2
(b) 
89. 12 m/s
91. (a)
(b) Answers will vary.
93. See Student Solutions Manual.
95. k = 
97. ≈8.296
99. Only (c) need not be true.
101. 
103. (a)
(b) 2 g
105. (a) (−∞,∞)
(b) [0,∞)
(c) (−∞,∞)
(d) All x ≠ 1.
(e)
107. F(x) = 
109. 
111. See Student Solutions Manual.
113. (a) See Student Solutions Manual.
(b) Answers will vary.
115. See Student Solutions Manual.
Section 5.5
1. f (x)
2. False
3. 
4. True
5. 
7. sin−1 x + C
9. 
11. 
13. 
15. 
17. 
19. 
21. 
23. 
25. 
27. 
29. 
31. 8 tan−1 x + C
33. 
35. sec x + C
37. 
39. y = ex + 3
41. y = 
43. y = 
45. y = x
47. sin y = 
49. 
= 4ex − 2
51. ≈6944 mosquitoes
53. (a)
= kP
(b) P = P0ekt
(c) P = 
(d) ≈25, 398 people
55. (a)
= kA
(b) A = A0ekt
(c) 
(d) ≈7.679 g
57. (a)
= 0.00409P
(b) P = P0e0.00409t
(c) (1.2359 × 109)e0.00409t
(d) ≈1.287 × 109 people
59. ≈52.67% remains.
61. ≈0.262 mol and ≈394.7 min
63. 
65 See Student Solutions Manual.
67. See Student Solutions Manual.
69. See Student Solutions Manual.
71. (a) y′ = sec x
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.
73. (a) y′ = 
(b) See Student Solutions Manual.
75. (a) 
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.
(d) See Student Solutions Manual.
Section 5.6
1. 
2. False
3. (c) 0
4. True
5. 
7. 
9. 
11. 
13. 
15. −e1/x + C
17. 
19. 
21. 
23. 4(ex − 1) 3/4 + C
25. 
27. 
29. 
31. sec x+C
33. −ecos x +C
35. 
37. 
39. 
41. 
43. 
45. (a-b) 
(c) Answers will vary.
47. (a-b) 
(c) Answers will vary.
49. (a-b) 
(c) Answers will vary.
51. (a-b) 
(c) Answers will vary.
53. 
55. 
57. 
59. sin eπ − sin 1
61. 
63. 
65. 0
67. 
69. 50
71. 
73. 
75. 
77. 
79. 
81. 
83. 
85. (ln 10) ln | ln x| + C
87. (ln 10) ln 2
89. (ln 2) ln 3
91. b = 
93. (a)
(b) 
95. (a) 0
(b) 0
(c) 0
(d) See Student Solutions Manual.
97. 
99. 
101. a2 sinh 1 + a(b − a)
103. 
105. 8
107. 4
109. (a) 
(b) u = −26ekt + 30
(c) ≈16.507°C
111. (a)
(b) u = (u0 − T)ekt + T
(c) u = 25e−0.159t + 12
(d) ≈ 8:50 a.m.
(e) It takes about 13 hr, 20 min to cool to 15°C.
113. v(z) = 
115. (a) v(t) = 
(b) s(t) = 
(c) 
(d) See Student Solutions Manual.
117. See Student Solutions Manual.
119. See Student Solutions Manual.
121. (a) 5
(b) 5
(c) 5
(d) a = 2, b = 4
(e) a = k, b = k + 2
123. Hint: Split integral at 0; see Student Solutions Manual.
125. c = 1
127. See Student Solutions Manual.
129. 
131. 
133. 
135. See Student Solutions Manual.
Review Exercises
1. s4 = 16, S4 = 24, s8 = 18, S8 = 22
3. A = \(\lim\limits_{n\rightarrow \infty }\) sn = 18
5. (a) 14
(b) 
(c) 
(d)
7. x2/3 sin x
9. −2x tan x2
11. 
13. 
15. 4
17. 
19. 2ex + ln |x| + C
21. The train traveled a distance of 460 km in 16 hours.
23. 
25. 0
27. 
29. 
31. 0
33. 
35. 
37. 
39. 
41. 
43. 
45. 
47. 
49. 
51. 
53. 
55. 5
57. 15,552 gallons
59. (a)
(b) A = A0ekt
(c) A = 
(d) ≈8.842 g