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

1000 chapter 15 Multiple Integrals
17. y 1
0 y2 1
18. y y6
y y2
0 0
sx 1 e 2y d dx dy
ssin x 1 sin yd dy dx
19. y 3 23 yy2 sy 1 y 2 cos xd dx dy 20. y 3
0
1 y5 1
21. y 4
23. y 3
24. y 1
25. y 1
1 y2 1
S
0 yy2 0
0 y1 0
0 y1 0
x y 1 y xD dy dx 22. y 1
t 2 sin 3 d dt
xysx 2 1 y 2 dy dx
vsu 1 v 2 d 4 du dv 26. y 1
27–34 Calculate the double integral.
0 y2 0
0 y1 0
ln y
xy
dy dx
ye x2y dx dy
ss 1 t ds dt
27. y x sec 2 y dA, R − hsx, yd | 0 < x < 2, 0 < y < y4j
R
28. y sy 1 xy 22 d dA, R − hsx, yd | 0 < x < 2, 1 < y < 2j
R
29. y
R
30. y
R
xy 2
x 2 1 1 dA, R − hsx, yd | 0 < x < 1, 23 < y < 3j
tan
s1 2 t 2
dA, R − hs, td | 0 < < y3, 0 < t < 1 2 j
31. y x sinsx 1 yd dA, R − f0, y6g 3 f0, y3g
R
32. y
R
x
dA, R − f0, 1g 3 f0, 1g
1 1 xy
33. y ye 2xy dA, R − f0, 2g 3 f0, 3g
R
34. y
R
1
dA, R − f1, 3g 3 f1, 2g
1 1 x 1 y
35–36 Sketch the solid whose volume is given by the iterated
integral.
35. y 1
36. y 1
0 y1 0
0 y1 0
s4 2 x 2 2yd dx dy
s2 2 x 2 2 y 2 d dy dx
37. Find the volume of the solid that lies under the plane
4x 1 6y 2 2z 1 15 − 0 and above the rectangle
R − hsx, yd | 21 < x < 2, 21 < y < 1j.
38. Find the volume of the solid that lies under the hyperbolic
paraboloid z − 3y 2 2 x 2 1 2 and above the rectangle
R − f21, 1g 3 f1, 2g.
;
CAS
CAS
CAS
39. Find the volume of the solid lying under the elliptic
paraboloid x 2 y4 1 y 2 y9 1 z − 1 and above the rectangle
R − f21, 1g 3 f22, 2g.
40. Find the volume of the solid enclosed by the surface
z − x 2 1 xy 2 and the planes z − 0, x − 0, x − 5,
and y − 62.
41. Find the volume of the solid enclosed by the surface
z − 1 1 x 2 ye y and the planes z − 0, x − 61, y − 0,
and y − 1.
42. Find the volume of the solid in the first octant bounded by
the cylinder z − 16 2 x 2 and the plane y − 5.
43. Find the volume of the solid enclosed by the paraboloid
z − 2 1 x 2 1 sy 2 2d 2 and the planes z − 1, x − 1,
x − 21, y − 0, and y − 4.
44. Graph the solid that lies between the surface
z − 2xyysx 2 1 1d and the plane z − x 1 2y and is bounded
by the planes x − 0, x − 2, y − 0, and y − 4. Then find its
volume.
45. Use a computer algebra system to find the exact value of the
integral yy R
x 5 y 3 e xy dA, where R − f0, 1g 3 f0, 1g. Then use
the CAS to draw the solid whose volume is given by the
integral.
46. Graph the solid that lies between the surfaces
z − e 2x 2 cossx 2 1 y 2 d and z − 2 2 x 2 2 y 2 for | x |
| < 1,
y |
< 1. Use a computer algebra system to approximate the
volume of this solid correct to four decimal places.
47–48 Find the average value of f over the given rectangle.
47. f sx, yd − x 2 y,
R has vertices s21, 0d, s21, 5d, s1, 5d, s1, 0d
48.
49. y y
R
f sx, yd − e y sx 1 e y , R − f0, 4g 3 f0, 1g
49–50 Use symmetry to evaluate the double integral.
xy
1 1 x dA, R − hsx, yd | 21 < x < 1, 0 < y < 1j
4
50. y s1 1 x 2 sin y 1 y 2 sin xd dA, R − f2, g 3 f2, g
R
51. Use a CAS to compute the iterated integrals
y 1 x 2 y
0 y1 0 sx 1 yd dy dx and y 1 x 2 y
3 0 y1
3
dx dy
0 sx 1 yd
Do the answers contradict Fubini’s Theorem? Explain what
is happening.
52. (a) In what way are the theorems of Fubini and Clairaut
similar?
(b) If f sx, yd is continuous on fa, bg 3 fc, dg and
tsx, yd − y x
a yy c
f ss, td dt ds
for a , x , b, c , y , d, show that t xy − t yx − f sx, yd.
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