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

1008 Chapter 15 Multiple Integrals
1–6 Evaluate the iterated integral.
1. y 5
3. y 1
5. y 1
1 yx 0
0 yy 0
0 ys2 0
s8x 2 2yd dy dx 2. y 2
xe y3 dx dy
0 yy2 0
4. y y2
y x
0 0
cosss 3 d dt ds 6. y 1
0 ye v
7–10 Evaluate the double integral.
7. y
D
y
x 2 1 1 dA,
8. y s2x 1 yd dA,
D
9. y e 2y 2 dA,
D
10. y ysx 2 2 y 2 dA,
D
x 2 y dx dy
x sin y dy dx
0 s1 1 ev dw dv
D − hsx, yd | 0 < x < 4, 0 < y < sx j
D − hsx, yd | 1 < y < 2, y 2 1 < x < 1j
D − hsx, yd | 0 < y < 3, 0 < x < yj
11. Draw an example of a region that is
(a) type I but not type II
(b) type II but not type I
12. Draw an example of a region that is
(a) both type I and type II
(b) neither type I nor type II
D − hsx, yd | 0 < x < 2, 0 < y < xj
13–14 Express D as a region of type I and also as a region of
type II. Then evaluate the double integral in two ways.
13. y x dA, D is enclosed by the lines y − x, y − 0, x − 1
D
14. y xy dA, D is enclosed by the curves y − x 2 , y − 3x
D
15–16 Set up iterated integrals for both orders of integration.
Then evaluate the double integral using the easier order and
explain why it’s easier.
15. y y dA, D is bounded by y − x 2 2, x − y 2
D
16. y y 2 e xy dA, D is bounded by y − x, y − 4, x − 0
D
;
17–22 Evaluate the double integral.
17. y x cos y dA, D is bounded by y − 0, y − x 2 , x − 1
D
18. y sx 2 1 2yd dA, D is bounded by y − x, y − x 3 , x > 0
D
19. y y 2 dA,
D
D is the triangular region with vertices (0, 1), (1, 2), s4, 1d
20. y xy dA, D is enclosed by the quarter-circle
D
y − s1 2 x 2 , x > 0, and the axes
21. y s2x 2 yd dA,
D
D is bounded by the circle with center the origin and radius 2
22. y y dA, D is the triangular region with vertices s0, 0d,
D
s1, 1d, and s4, 0d
23–32 Find the volume of the given solid.
23. Under the plane 3x 1 2y 2 z − 0 and above the region
enclosed by the parabolas y − x 2 and x − y 2
24. Under the surface z − 1 1 x 2 y 2 and above the region
enclosed by x − y 2 and x − 4
25. Under the surface z − xy and above the triangle with
vertices s1, 1d, s4, 1d, and s1, 2d
26. Enclosed by the paraboloid z − x 2 1 y 2 1 1 and the planes
x − 0, y − 0, z − 0, and x 1 y − 2
27. The tetrahedron enclosed by the coordinate planes and the
plane 2x 1 y 1 z − 4
28. Bounded by the planes z − x, y − x, x 1 y − 2, and z − 0
29. Enclosed by the cylinders z − x 2 , y − x 2 and the planes
z − 0, y − 4
30. Bounded by the cylinder y 2 1 z 2 − 4 and the planes x − 2y,
x − 0, z − 0 in the first octant
31. Bounded by the cylinder x 2 1 y 2 − 1 and the planes y − z,
x − 0, z − 0 in the first octant
32. Bounded by the cylinders x 2 1 y 2 − r 2 and y 2 1 z 2 − r 2
33. Use a graphing calculator or computer to estimate the
x-coordinates of the points of intersection of the curves
y − x 4 and y − 3x 2 x 2 . If D is the region bounded by these
curves, estimate yy D
x dA.
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