James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

400 Chapter 5 Integrals
15. y − y 3x12
1
17. y − y y4
t
1 1 t 3 dt 16. y − y x4
sx
tan d 18. y − y 1
19–44 Evaluate the integral.
19. y 3
1
21. y 2
0
sx 2 1 2x 2 4d dx 20. y 1 x 100 dx
21
( 4 5 t 3 2 3 4 t 2 1 2 5 t) dt 22. y 1
23. y 9
sx dx 24. y 8
x 22y3 dx
1 1
25. y sin d 26. y 5 e dx
y6 25
0
0 cos2 d
sin x s1 1 t 2 dt
(1 2 8v 3 1 16v 7 ) dv
;
48. y − 2x 2 x 2 , y − 0
49–52 Use a graph to give a rough estimate of the area of
the region that lies beneath the given curve. Then find the
exact area.
49. y − s 3 x , 0 < x < 27
50. y − x 24 , 1 < x < 6
51. y − sin x, 0 < x <
52. y − sec 2 x, 0 < x < y3
53–54 Evaluate the integral and interpret it as a difference of
areas. Illustrate with a sketch.
53. y 2 x 3 dx
21
54. y 2
cos x dx
y6
;
27. y 1
su 1 2dsu 2 3d du 28. y 4
s4 2 tdst dt
0
29. y 4 2 1 x 2
1
sx
31. y y2
csc t cot t dt
y6
0
dx 30. y 2 s3u 2 2dsu 1 1d du
21
32. y y3
csc 2 d
33. y 1
s1 1 0 rd3 dr 34. y 3
s2 sin x 2 e x d dx
0
35. y 2 v 3 1 3v 6
18
dv 36.
1 v y Î 3 4 1 z dz
y4
37. y 1
sx e 1 e x d dx 38. y 1
cosh t dt
0
39. y s3 8
1ys3 1 1 x dx 40. y 3 y 3 2 2y 2 2 y
dy
2 1 y 2
41. y 4
2 s ds 42.
y 1ys2
0
43. y
0
44. y 2 22
f sxd dx where f sxd −H sin x
cos x
0
1y2
4
s1 2 x 2 dx
if 0 < x , y2
if y2 < x <
f sxd dx where f sxd −H 2 4 2 x 2 if 22 < x < 0
if 0 , x < 2
45–48 Sketch the region enclosed by the given curves and
calculate its area.
45. y − sx , y − 0, x − 4
46. y − x 3 , y − 0, x − 1
47. y − 4 2 x 2 , y − 0
; 55–58 What is wrong with the equation?
1
55. y 1 x 24 dx − x23
2
22 23G22− 3 8
56. y 2 21
2G
4
x dx − 2 2 3 x
2
21− 3 2
57. y sec tan d − sec g − 23
y3
y3
58. y
0
sec 2 x dx − tan xg 0
− 0
59–63 Find the derivative of the function.
59. tsxd − y 3x
2x
F Hint: y 3x
2x
u 2 2 1
u 2 1 1 du
f sud du − y 0
60. tsxd − y 112x
t sin t dt
122x
61. Fsxd − y x2
e t 2 dt
x
63. y − y sin x
lns1 1 2vd dv
cos x
2x
f sud du 1 y 3x
f sud duG
62. Fsxd − y 2x
64. If f sxd − y x 0 s1 2 t 2 de t 2 dt, on what interval is f
increasing?
65. On what interval is the curve
y − y x
concave downward?
0
t 2
t 2 1 t 1 2 dt
0
sx
arctan t dt
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