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

1062 chapter 15 Multiple Integrals
EXERCISES
1. A contour map is shown for a function f on the square
R − f0, 3g 3 f0, 3g. Use a Riemann sum with nine terms to
estimate the value of yy R
f sx, yd dA. Take the sample points to
be the upper right corners of the squares.
y
3
2
1
0
1
2
3
10
9
8
7
4 5 6
1 2 3
2. Use the Midpoint Rule to estimate the integral in Exercise 1.
3–8 Calculate the iterated integral.
3. y 2
5. y 1
7. y
1 y2 0
0 yx 0
sy 1 2xe y d dx dy 4. y 1
cossx 2 d dy dx 6. y 1
0 y1 0 ys12y2 0
y sin x dz dy dx 8. y 1
0 y1 0
0 yex x
0 yy 0 y1 x
ye xy dx dy
3xy 2 dy dx
x
6xyz dz dx dy
9–10 Write yy R
f sx, yd dA as an iterated integral, where R is the
region shown and f is an arbitrary continuous function on R.
9. y
10.
4
_4
_2
2
R
0 2 4
x
11. The cylindrical coordinates of a point are (2s3 , y3, 2). Find
the rectangular and spherical coordinates of the point.
12. The rectangular coordinates of a point are s2, 2, 21d. Find the
cylindrical and spherical coordinates of the point.
13. The spherical coordinates of a point are s8, y4, y6d. Find
the rectangular and cylindrical coordinates of the point.
_4
14. Identify the surfaces whose equations are given.
(a) − y4
(b) − y4
y
4
0
R
4
x
15. Write the equation in cylindrical coordinates and in spherical
coordinates.
(a) x 2 1 y 2 1 z 2 − 4 (b) x 2 1 y 2 − 4
16. Sketch the solid consisting of all points with spherical coordinates
s, , d such that 0 < < y2, 0 < < y6, and
0 < < 2 cos .
17. Describe the region whose area is given by the integral
y y2 sin 2
y
0 0
r dr d
18. Describe the solid whose volume is given by the integral
y y2
0
y y2
0
y 2
1 2 sin d d d
and evaluate the integral.
19–20 Calculate the iterated integral by first reversing the order of
integration.
19. y 1
0 y1 x
cossy 2 d dy dx 20. y 1
ye x2
0 y1 sy x 3
dx dy
21–34 Calculate the value of the multiple integral.
21. yy R
ye xy dA, where R − hsx, yd | 0 < x < 2, 0 < y < 3j
22. yy D
xy dA, where D − hsx, yd | 0 < y < 1, y 2 < x < y 1 2j
y
23. y
1 1 x dA, 2 D
where D is bounded by y − sx , y − 0, x − 1
1
24. y y 2
dA, where D is the triangular region with
1 1 x
D
vertices s0, 0d, s1, 1d, and s0, 1d
25. yy D
y dA, where D is the region in the first quadrant bounded by
the parabolas x − y 2 and x − 8 2 y 2
26. yy D
y dA, where D is the region in the first quadrant that lies
above the hyperbola xy − 1 and the line y − x and below the
line y − 2
27. yy D
sx 2 1 y 2 d 3y2 dA, where D is the region in the first
quad rant bounded by the lines y − 0 and y − s3 x and the
circle x 2 1 y 2 − 9
28. yy D
x dA, where D is the region in the first quadrant that lies
between the circles x 2 1 y 2 − 1 and x 2 1 y 2 − 2
29. yyy E
xy dV, where
E − hsx, y, zd | 0 < x < 3, 0 < y < x, 0 < z < x 1 yj
30. yyy T
xy dV, where T is the solid tetrahedron with vertices
s0, 0, 0d, s 1 3 , 0, 0d, s0, 1, 0d, and s0, 0, 1d
31. yyy E
y 2 z 2 dV, where E is bounded by the paraboloid
x − 1 2 y 2 2 z 2 and the plane x − 0
32. yyy E
z dV, where E is bounded by the planes y − 0, z − 0,
x 1 y − 2 and the cylinder y 2 1 z 2 − 1 in the first octant
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