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

Section 15.2 Double Integrals over General Regions 1009
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CAS
34. Find the approximate volume of the solid in the first octant
that is bounded by the planes y − x, z − 0, and z − x and
the cylinder y − cos x. (Use a graphing device to estimate
the points of intersection.)
35–38 Find the volume of the solid by subtracting two volumes.
35. The solid enclosed by the parabolic cylinders y − 1 2 x 2 ,
y − x 2 2 1 and the planes x 1 y 1 z − 2,
2x 1 2y 2 z 1 10 − 0
36. The solid enclosed by the parabolic cylinder y − x 2 and the
planes z − 3y, z − 2 1 y
37. The solid under the plane z − 3, above the plane z − y, and
between the parabolic cylinders y − x 2 and y − 1 2 x 2
38. The solid in the first octant under the plane z − x 1 y, above
the surface z − xy, and enclosed by the surfaces x − 0,
y − 0, and x 2 1 y 2 − 4
39–40 Sketch the solid whose volume is given by the iterated
integral.
39. y 1
0 y12x 0
s1 2 x 2 yd dy dx 40. y 1
0 y12x2 0
s1 2 xd dy dx
41–44 Use a computer algebra system to find the exact volume
of the solid.
41. Under the surface z − x 3 y 4 1 xy 2 and above the region
bounded by the curves y − x 3 2 x and y − x 2 1 x
for x > 0
42. Between the paraboloids z − 2x 2 1 y 2 and
z − 8 2 x 2 2 2y 2 and inside the cylinder x 2 1 y 2 − 1
43. Enclosed by z − 1 2 x 2 2 y 2 and z − 0
44. Enclosed by z − x 2 1 y 2 and z − 2y
45–50 Sketch the region of integration and change the order of
integration.
45. y 1
0 yy 0
f sx, yd dx dy 46. y 2
0 y4 f sx, yd dy dx
x2 53. y 1
54. y 2
55. y 1
56. y 8
0 y1 sx
sy 3 1 1 dy dx
0 y1 yy2 y cossx 3 2 1d dx dy
0 yy2 arcsin y cos x s1 1 cos 2 x dx dy
0 y2 s 3 y
e x 4 dx dy
57–58 Express D as a union of regions of type I or type II and
evaluate the integral.
57. y x 2 dA
D
D
y
1
_1 0 1
_1
(1, 1)
x
58. y y dA
D
y=(x+1)@
_1
y
1
0
_1
x=y-Á
59–60 Use Property 11 to estimate the value of the integral.
59. y s4 2 x 2 y 2 dA, S − hsx, yd | x 2 1 y 2 < 1, x > 0j
S
60. y y sin 4 sx 1 yd dA, T is the triangle enclosed by the lines
T
y − 0, y − 2x, and x − 1
61–62 Find the averge value of f over the region D.
61. f sx, yd − xy, D is the triangle with vertices s0, 0d, s1, 0d,
and s1, 3d
62. f sx, yd − x sin y, D is enclosed by the curves y − 0,
y − x 2 , and x − 1
x
47. y y2
y cos x
0 0
f sx, yd dy dx
48. y 2 f sx, yd dx dy
22 ys42y2 0
63. Prove Property 11.
49. y 2
x
1 yln 0
f sx, yd dy dx 50. y 1
0 yy4 f sx, yd dy dx
arctan x
51–56 Evaluate the integral by reversing the order of integration.
51. y 1
0 y3 e x 2 dx dy 52. y 1
3y 0 y1 sy sin y dy dx
x2 64. In evaluating a double integral over a region D, a sum of
iterated integrals was obtained as follows:
y f sx, yd dA − y 1
D
0 y2y 0
f sx, yd dx dy 1 y 3
1 y32y 0
f sx, yd dx dy
Sketch the region D and express the double integral as an
iterated integral with reversed order of integration.
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