964
||||
CHAPTER 15 MULTIPLE INTEGRALS
EXAMPLE 5 If R 苷 关0, 兾2兴 关0, 兾2兴, then, by Equation 5,
yy sin x cos y dA 苷 y
兾2
0
sin x dx y
兾2
0
cos y dy
R
[
兾2
0
兾2
0
] [sin y]
苷 cos x
苷1ⴢ1苷1
z
The function f 共x, y兲 苷 sin x cos y in
Example 5 is positive on R, so the integral represents the volume of the solid that lies above R
and below the graph of f shown in Figure 6.
N
0
y
x
FIGURE 6
15.2
EXERCISES
1–2 Find x05 f 共x, y兲 dx and x01 f 共x, y兲 dy.
1. f 共x, y兲 苷 12x 2 y 3
18.
2. f 共x, y兲 苷 y xe y
2
ⱍ
R 苷 兵共x, y兲 0 x 1, 0 y 1其
dA,
R
19.
3–14 Calculate the iterated integral.
1 x2
yy 1 y
yy x sin共x y兲 dA,
R 苷 关0, 兾6兴 关0, 兾3兴
R
3.
3
yy
1
5.
7.
2
yy
0
2
1
yy
4
1
11.
0
yy
1
0
2
1
yy
共1 4x y兲 dx dy
兾2
0
0
9.
1
0
1
0
x sin y dy dx
共2x y兲 dx dy
冉 冊
x
y
y
x
dy dx
2
1
5
兾6
1
1
yy
10.
yy
12.
r sin 2 d dr
14.
20.
兾2
8.
共u v兲5 du dv
共4x 3 9x 2 y 2 兲 dy dx
y y
6.
8
1
0
yy
4.
0
2
1
1
3
0
0
1
1
0
0
1
1
0
0
yy
x
R
cos y dx dy
xe x
dy dx
y
21.
e x3y dx dy
22.
yy xye
x2y
dA,
R 苷 关0, 1兴 关0, 2兴
R
yy x
2
R
xysx 2 y 2 dy dx
R 苷 关0, 1兴 关0, 1兴
yy 1 xy dA,
x
dA,
y2
R 苷 关1, 2兴 关0, 1兴
23–24 Sketch the solid whose volume is given by the iterated
integral.
13.
2
yy
0
0
yy
ss t ds dt
15–22 Calculate the double integral.
15.
yy 共6x
2
y 3 5y 4 兲 dA,
24.
ⱍ
yy cos共x 2y兲 dA,
yy
R
1
0
1
1
0
0
yy
共4 x 2y兲 dx dy
共2 x 2 y 2 兲 dy dx
25. Find the volume of the solid that lies under the plane
ⱍ
R 苷 兵共x, y兲 0 x , 0 y 兾2其
R
17.
1
0
yy
R 苷 兵共x, y兲 0 x 3, 0 y 1其
R
16.
23.
xy 2
dA,
2
x 1
ⱍ
R 苷 兵共x, y兲 0 x 1, 3 y 3其
3x 2y z 苷 12 and above the rectangle
R 苷 兵共x, y兲 0 x 1, 2 y 3其.
ⱍ
26. Find the volume of the solid that lies under the hyperbolic
paraboloid z 苷 4 x 2 y 2 and above the square
R 苷 关1, 1兴 关0, 2兴.
M
SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS
27. Find the volume of the solid lying under the elliptic
CAS
paraboloid x 2兾4 y 2兾9 z 苷 1 and above the rectangle
R 苷 关1, 1兴 关2, 2兴.
34. Graph the solid that lies between the surfaces
2
ⱍ ⱍ
z 苷 1 e x sin y and the planes x 苷 1, y 苷 0, y 苷 ,
and z 苷 0.
ⱍ ⱍ
35–36 Find the average value of f over the given rectangle.
35. f 共x, y兲 苷 x 2 y,
29. Find the volume of the solid enclosed by the surface
z 苷 x sec 2 y and the planes z 苷 0, x 苷 0, x 苷 2, y 苷 0,
and y 苷 兾4.
R has vertices 共1, 0兲, 共1, 5兲, 共1, 5兲, 共1, 0兲
36. f 共x, y兲 苷 e ysx e y ,
30. Find the volume of the solid in the first octant bounded by
the cylinder z 苷 16 x 2 and the plane y 苷 5.
CAS
1
y y
0
z 苷 2 x 2 共 y 2兲2 and the planes z 苷 1, x 苷 1, x 苷 1,
y 苷 0, and y 苷 4.
1
0
xy
dy dx
共x y兲3
1
yy
and
0
1
0
xy
dx dy
共x y兲3
Do the answers contradict Fubini’s Theorem? Explain what
is happening.
; 32. Graph the solid that lies between the surface
z 苷 2xy兾共x 2 1兲 and the plane z 苷 x 2y and is bounded
by the planes x 苷 0, x 苷 2, y 苷 0, and y 苷 4. Then find its
volume.
R 苷 关0, 4兴 关0, 1兴
37. Use your CAS to compute the iterated integrals
31. Find the volume of the solid enclosed by the paraboloid
38. (a) In what way are the theorems of Fubini and Clairaut
similar?
(b) If f 共x, y兲 is continuous on 关a, b兴 关c, d 兴 and
33. Use a computer algebra system to find the exact value of the
t共x, y兲 苷 y
integral xxR x 5y 3e x y dA, where R 苷 关0, 1兴 关0, 1兴. Then use
the CAS to draw the solid whose volume is given by the
integral.
15.3
965
z 苷 ex cos共x 2 y 2 兲 and z 苷 2 x 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.
28. Find the volume of the solid enclosed by the surface
CAS
||||
x
a
y
y
c
f 共s, t兲 dt ds
for a x b, c y d, show that txy 苷 tyx 苷 f 共x, y兲.
DOUBLE INTEGRALS OVER GENERAL REGIONS
For single integrals, the region over which we integrate is always an interval. But for
double integrals, we want to be able to integrate a function f not just over rectangles but
also over regions D of more general shape, such as the one illustrated in Figure 1. We suppose that D is a bounded region, which means that D can be enclosed in a rectangular
region R as in Figure 2. Then we define a new function F with domain R by
F共x, y兲 苷
1
再
f 共x, y兲 if 共x, y兲 is in D
0
if 共x, y兲 is in R but not in D
y
y
R
D
0
FIGURE 1
D
x
0
FIGURE 2
x
972
||||
CHAPTER 15 MULTIPLE INTEGRALS
EXAMPLE 6 Use Property 11 to estimate the integral
xxD e sin x cos y dA, where D is the disk
with center the origin and radius 2.
SOLUTION Since 1 sin x 1 and 1 cos y 1, we have 1 sin x cos y 1 and
therefore
e1 e sin x cos y e 1 苷 e
Thus, using m 苷 e 1 苷 1兾e, M 苷 e, and A共D兲 苷 共2兲2 in Property 11, we obtain
4
yy e sin x cos y dA 4 e
e
D
15.3
EXERCISES
1–6 Evaluate the iterated integral.
1.
3.
5.
4
yy
0
sy
0
1
yy
0
M
x
x2
17.
xy 2 dx dy
2.
共1 2y兲 dy dx
兾2
cos 0
0
y y
4.
e sin dr d
6.
1
yy
0
2
2x
2y
0
y
yy
1
yy
0
v
0
D
共x y兲 dy dx
2
D is bounded by the circle with center the origin and radius 2
18.
xy dx dy
yy 共2x y兲 dA,
yy 2xy dA,
D is the triangular region with vertices 共0, 0兲,
D
s1 v 2 du dv
共1, 2兲, and 共0, 3兲
19–28 Find the volume of the given solid.
7–18 Evaluate the double integral.
7.
yy y
2
dA,
ⱍ
D 苷 兵共x, y兲 1 y 1, y 2 x y其
D
8.
yy
D
9.
y
dA,
x5 1
yy x dA,
yy x
3
dA,
yy
12.
yy
13.
3x 2y z 苷 6
ⱍ
24. Bounded by the planes z 苷 x, y 苷 x, x y 苷 2, and z 苷 0
ⱍ
x sy 2 x 2 dA, D 苷 兵共x, y兲 0 y 1, 0 x y其
yy x cos y dA,
D is bounded by y 苷 0, y 苷 x , x 苷 1
yy 共x y兲 dA,
yy y
D is bounded by y 苷 sx and y 苷 x
3
x 苷 0, z 苷 0 in the first octant
dA,
D is the triangular region with vertices (0, 2), (1, 1), 共3, 2兲
yy xy
D
x 苷 0, z 苷 0 in the first octant
28. Bounded by the cylinders x 2 y 2 苷 r 2 and y 2 z 2 苷 r 2
D
16.
z 苷 0, y 苷 4
27. Bounded by the cylinder x 2 y 2 苷 1 and the planes y 苷 z,
2
D
15.
25. Enclosed by the cylinders z 苷 x 2, y 苷 x 2 and the planes
26. Bounded by the cylinder y 2 z 2 苷 4 and the planes x 苷 2y,
2
D
14.
21. Under the surface z 苷 xy and above the triangle with vertices
23. Bounded by the coordinate planes and the plane
D 苷 兵共x, y兲 0 y 4, 0 x y其
D
by x 苷 y 2 and x 苷 y 3
y 苷 1, y 苷 x, z 苷 0
ⱍ
D
20. Under the surface z 苷 2x y 2 and above the region bounded
22. Enclosed by the paraboloid z 苷 x 2 3y 2 and the planes x 苷 0,
D 苷 兵共x, y兲 1 x e, 0 y ln x其
y 2e xy dA,
bounded by y 苷 x and y 苷 x 4
共1, 1兲, 共4, 1兲, and 共1, 2兲
ⱍ
D
11.
ⱍ
D 苷 兵共x, y兲 0 x 1, 0 y x 2 其
D 苷 兵共x, y兲 0 x , 0 y sin x其
D
10.
19. Under the plane x 2y z 苷 0 and above the region
2
dA,
D is enclosed by x 苷 0 and x 苷 s1 y 2
; 29. Use a graphing calculator or computer to estimate the
x-coordinates of the points of intersection of the curves y 苷 x 4
and y 苷 3x x 2. If D is the region bounded by these curves,
estimate xxD x dA.
SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS
; 30. Find the approximate volume of the solid in the first octant
||||
973
51–52 Express D as a union of regions of type I or type II and
that is bounded by the planes y 苷 x, z 苷 0, and z 苷 x and
the cylinder y 苷 cos x. (Use a graphing device to estimate
the points of intersection.)
evaluate the integral.
51.
yy x
2
dA
52.
D
yy y dA
D
31–32 Find the volume of the solid by subtracting two volumes.
y
31. The solid enclosed by the parabolic cylinders
1
y 苷 1 x 2, y 苷 x 2 1 and the planes x y z 苷 2,
2x 2y z 10 苷 0
y
1
(1, 1)
x=y-Á
y=(x+1)@
D
_1
_1
32. The solid enclosed by the parabolic cylinder y 苷 x 2 and the
0
1
x
0
x
planes z 苷 3y, z 苷 2 y
_1
_1
33–34 Sketch the solid whose volume is given by the iterated
integral.
33.
1
yy
0
1x
共1 x y兲 dy dx
0
1
y y
34.
0
1x 2
0
53–54 Use Property 11 to estimate the value of the integral.
共1 x兲 dy dx
53.
共x 2 y 2 兲2
yy e
Q
dA, Q is the quarter-circle with center the origin
1
and radius 2 in the first quadrant
CAS
35–38 Use a computer algebra system to find the exact volume
54.
of the solid.
4
55–56 Find the average value of f over region D.
36. Between the paraboloids z 苷 2x 2 y 2 and
z 苷 8 x 2 2y 2 and inside the cylinder x 2 y 2 苷 1
55. f 共x, y兲 苷 xy,
and 共1, 3兲
37. Enclosed by z 苷 1 x y and z 苷 0
2
y 苷 x 2, and x 苷 1
integration.
4
0
sx
0
58. In evaluating a double integral over a region D, a sum of
f 共x, y兲 dy dx
40.
1
4
0
4x
yy
iterated integrals was obtained as follows:
f 共x, y兲 dy dx
1
yy f 共x, y兲 dA 苷 y y
0
41.
43.
3
s9y 2
0
s9y 2
yy
2
yy
1
ln x
f 共x, y兲 dx dy
f 共x, y兲 dy dx
0
42.
44.
3
s9y
0
0
yy
1
yy
0
兾4
arctan x
47.
49.
50.
1
yy
3
0
3y
4
2
0
sx
yy
1
yy
0
8
1
dy dx
y3 1
兾2
arcsin y
yy
0
2
e x dx dy
2
3
sy
46.
48.
cos x s1 cos 2 x dx dy
4
e x dx dy
s
y y
0
s
y
1
1
0
x
yy
2y
0
f 共x, y兲 dx dy y
3
1
y
3y
0
f 共x, y兲 dx dy
D
f 共x, y兲 dx dy
Sketch the region D and express the double integral as an
iterated integral with reversed order of integration.
f 共x, y兲 dy dx
59. Evaluate xxD 共x 2 tan x y 3 4兲 dA, where
ⱍ
D 苷 兵共x, y兲 x 2 y 2 2其. [Hint: Exploit the fact that
D is symmetric with respect to both axes.]
60. Use symmetry to evaluate xxD 共2 3x 4y兲 dA, where D
45–50 Evaluate the integral by reversing the order of integration.
45.
D is enclosed by the curves y 苷 0,
57. Prove Property 11.
39– 44 Sketch the region of integration and change the order of
yy
D is the triangle with vertices 共0, 0兲, 共1, 0兲,
56. f 共x, y兲 苷 x sin y,
38. Enclosed by z 苷 x 2 y 2 and z 苷 2y
39.
T is the triangle enclosed by the lines
y 苷 0, y 苷 2x, and x 苷 1
2
bounded by the curves y 苷 x 3 x and y 苷 x 2 x for x 0
2
4
T
35. Under the surface z 苷 x y xy and above the region
3
yy sin 共x y兲 dA,
is the region bounded by the square with vertices 共5, 0兲
and 共0, 5兲.
cos共x 2 兲 dx dy
61. Compute xxD s1 x 2 y 2 dA, where D is the disk
x 2 y 2 1, by first identifying the integral as the volume
of a solid.
e x兾y dy dx
CAS
62. Graph the solid bounded by the plane x y z 苷 1 and
the paraboloid z 苷 4 x 2 y 2 and find its exact volume.
(Use your CAS to do the graphing, to find the equations of
the boundary curves of the region of integration, and to evaluate the double integral.)
978
||||
CHAPTER 15 MULTIPLE INTEGRALS
Find the volume of the solid that lies under the paraboloid z 苷 x 2 y 2,
above the xy-plane, and inside the cylinder x 2 y 2 苷 2x.
y
V EXAMPLE 4
(x-1)@+¥=1
(or r=2 cos ¨)
SOLUTION The solid lies above the disk D whose boundary circle has equation
x 2 y 2 苷 2x or, after completing the square,
D
0
1
共x 1兲2 y 2 苷 1
x
2
(See Figures 9 and 10.) In polar coordinates we have x 2 y 2 苷 r 2 and x 苷 r cos , so
the boundary circle becomes r 2 苷 2r cos , or r 苷 2 cos . Thus the disk D is given by
D 苷 兵共r, 兲 兾2 兾2, 0 r 2 cos 其
ⱍ
FIGURE 9
z
and, by Formula 3, we have
V 苷 yy 共x y 兲 dA 苷 y
2
2
兾2
D
苷4y
兾2
兾2
苷2y
x
兾2
0
y
cos 4 d 苷 8 y
y
0
兾2
0
[1 2 cos 2 2 cos r r dr d 苷
2
cos 4 d 苷 8 y
兾2
0
1
2
y
兾2
兾2
冉
冋册
r4
4
2 cos d
0
1 cos 2
2
冊
2
d
]
共1 cos 4 兲 d
兾2
苷 2[ 32 sin 2 18 sin 4]0 苷 2
FIGURE 10
15.4
兾2
冉 冊冉 冊
3
2
2
苷
3
2
M
EXERCISES
1– 4 A region R is shown. Decide whether to use polar coordinates
5–6 Sketch the region whose area is given by the integral and eval-
or rectangular coordinates and write xxR f 共x, y兲 dA as an iterated
integral, where f is an arbitrary continuous function on R.
uate the integral.
1.
y
2.
5.
y
4
1
2
y y
7
4
r dr d
6.
兾2
y y
0
4 cos 0
r dr d
y=1-≈
7–14 Evaluate the given integral by changing to polar coordinates.
7.
0
4
x
_1
3.
4.
y
0
where D is the disk with center the origin and radius 3
x
1
y
6
8.
xxR 共x y兲 dA,
9.
xxR cos共x 2 y 2 兲 dA,
1
10.
3
0
_1
0
1
x
x
xxD xy dA,
where R is the region that lies to the left of the
y-axis between the circles x 2 y 2 苷 1 and x 2 y 2 苷 4
where R is the region that lies above the
x-axis within the circle x 2 y 2 苷 9
xxR s4 x 2 y 2 dA,
where R 苷 兵共x, y兲 x 2 y 2 4, x 0其
ⱍ
x 2 y 2
11.
dA, where D is the region bounded by the
xxD e
semicircle x 苷 s4 y 2 and the y-axis
12.
xxR ye x dA,
where R is the region in the first quadrant enclosed
by the circle x 2 y 2 苷 25
SECTION 15.4 DOUBLE INTEGRALS IN POLAR COORDINATES
13.
xxR arctan共 y兾x兲 dA,
where R 苷 兵共x, y兲 1 x 2 y 2 4, 0 y x其
14.
xxD x dA,
ⱍ
where D is the region in the first quadrant that lies
between the circles x 2 y 2 苷 4 and x 2 y 2 苷 2x
15–18 Use a double integral to find the area of the region.
15. One loop of the rose r 苷 cos 3
16. The region enclosed by the curve r 苷 4 3 cos 17. The region within both of the circles r 苷 cos and r 苷 sin 18. The region inside the cardioid r 苷 1 cos and outside the
circle r 苷 3 cos is constant along east-west lines and increases linearly from
2 ft at the south end to 7 ft at the north end. Find the volume of
water in the pool.
34. An agricultural sprinkler distributes water in a circular pattern
of radius 100 ft. It supplies water to a depth of er feet per hour
at a distance of r feet from the sprinkler.
(a) If 0 R 100, what is the total amount of water supplied
per hour to the region inside the circle of radius R centered
at the sprinkler?
(b) Determine an expression for the average amount of water
per hour per square foot supplied to the region inside the
circle of radius R.
35. Use polar coordinates to combine the sum
1
1兾s2
19–27 Use polar coordinates to find the volume of the given solid.
19. Under the cone z 苷 sx 2 y 2 and above the disk x 2 y 2 4
20. Below the paraboloid z 苷 18 2x 2y and above the
2
2
x
s1x 2
xy dy dx y
s2
1
y
x
xy dy dx y
0
I 苷 yy e共x y 兲 dA 苷 y
al
cylinder x y 苷 4
2
24. Bounded by the paraboloid z 苷 1 2x 2 2y 2 and the
共x 2y 2 兲
yy e
共x 2y 2 兲
yy
0
y
ex dx y
共x y兲 dx dy
32.
a
0
0
sa 2 y 2
2
s2xx 2
0
0
yy
共x 2y 2 兲
yy e
dA
Sa
2
ey dy 苷 (c) Deduce that
y
2
ex dx 苷 s
(d) By making the change of variable t 苷 s2 x, show that
y
yy
2
dA 苷 lim
2
coordinates.
31.
2
e共x y 兲 dA 苷 al
y
29–32 Evaluate the iterated integral by converting to polar
30.
2
where Sa is the square with vertices 共a, a兲. Use this to
show that
28. (a) A cylindrical drill with radius r 1 is used to bore a hole
sin共x 2 y 2 兲 dy dx
2
e共x y 兲 dy dx
dA
⺢2
4x 2 4y 2 z 2 苷 64
s2y 2
yy e
27. Inside both the cylinder x 2 y 2 苷 4 and the ellipsoid
s9x 2
(b) An equivalent definition of the improper integral in part (a)
is
z 苷 4 x2 y2
1
y y
26. Bounded by the paraboloids z 苷 3x 2 3y 2 and
0
y
Da
plane z 苷 7 in the first octant
3
xy dy dx
where Da is the disk with radius a and center the origin.
Show that
23. A sphere of radius a
3
苷 lim
22. Inside the sphere x 2 y 2 z 2 苷 16 and outside the
y y
2
⺢2
plane z 苷 2
through the center of a sphere of radius r 2 . Find the volume
of the ring-shaped solid that remains.
(b) Express the volume in part (a) in terms of the height h of
the ring. Notice that the volume depends only on h, not
on r 1 or r 2 .
s4x 2
0
36. (a) We define the improper integral (over the entire plane ⺢ 2 兲
2
x 2 y 2 z2 苷 1
y
into one double integral. Then evaluate the double integral.
21. Enclosed by the hyperboloid x 2 y 2 z 2 苷 1 and the
25. Above the cone z 苷 sx 2 y 2 and below the sphere
2
s2
xy-plane
29.
979
33. A swimming pool is circular with a 40-ft diameter. The depth
y y
2
||||
2
ex 兾2 dx 苷 s2
(This is a fundamental result for probability and statistics.)
x 2 y dx dy
37. Use the result of Exercise 36 part (c) to evaluate the following
sx 2 y 2 dy dx
integrals.
(a)
y
0
2
x 2ex dx
(b)
y
0
sx ex dx
988
||||
CHAPTER 15 MULTIPLE INTEGRALS
EXAMPLE 8 A factory produces (cylindrically shaped) roller bearings that are sold as
having diameter 4.0 cm and length 6.0 cm. In fact, the diameters X are normally distributed with mean 4.0 cm and standard deviation 0.01 cm while the lengths Y are normally
distributed with mean 6.0 cm and standard deviation 0.01 cm. Assuming that X and Y are
independent, write the joint density function and graph it. Find the probability that a
bearing randomly chosen from the production line has either length or diameter that
differs from the mean by more than 0.02 cm.
SOLUTION We are given that X and Y are normally distributed with 1 苷 4.0, 2 苷 6.0, and
1 苷 2 苷 0.01. So the individual density functions for X and Y are
f1共x兲 苷
1
2
e共x4兲 兾0.0002
0.01 s2
f2共y兲 苷
1
2
e共 y6兲 兾0.0002
0.01s2
1500
Since X and Y are independent, the joint density function is the product:
1000
500
0
5.95
f 共x, y兲 苷 f1共x兲 f2共y兲 苷
3.95
y
4
6
6.05
x
4.05
苷
FIGURE 9
Graph of the bivariate normal joint
density function in Example 8
1
2
2
e共x4兲 兾0.0002e共y6兲 兾0.0002
0.0002
5000 5000关共x4兲2共 y6兲2兴
e
A graph of this function is shown in Figure 9.
Let’s first calculate the probability that both X and Y differ from their means by less
than 0.02 cm. Using a calculator or computer to estimate the integral, we have
P共3.98 X 4.02, 5.98 Y 6.02兲 苷 y
4.02
3.98
苷
y
6.02
5.98
5000
f 共x, y兲 dy dx
4.02
y y
3.98
6.02
5.98
2
2
e5000关共x4兲 共 y6兲 兴 dy dx
⬇ 0.91
Then the probability that either X or Y differs from its mean by more than 0.02 cm is
approximately
1 0.91 苷 0.09
15.5
M
EXERCISES
1. Electric charge is distributed over the rectangle 1 x 3,
0 y 2 so that the charge density at 共x, y兲 is
共x, y兲 苷 2x y y 2 (measured in coulombs per square meter).
Find the total charge on the rectangle.
2. Electric charge is distributed over the disk x 2 y 2 4 so
that the charge density at 共x, y兲 is 共x, y兲 苷 x y x 2 y 2
(measured in coulombs per square meter). Find the total charge
on the disk.
3–10 Find the mass and center of mass of the lamina that occupies
the region D and has the given density function .
ⱍ
3. D 苷 兵共x, y兲 0 x 2, 1 y 1其 ; 共x, y兲 苷 xy 2
ⱍ
4. D 苷 兵共x, y兲 0 x a, 0 y b其 ; 共x, y兲 苷 cxy
5. D is the triangular region with vertices 共0, 0兲, 共2, 1兲, 共0, 3兲;
共x, y兲 苷 x y
6. D is the triangular region enclosed by the lines x 苷 0, y 苷 x,
and 2x y 苷 6; 共x, y兲 苷 x 2
7. D is bounded by y 苷 e x, y 苷 0, x 苷 0, and x 苷 1; 共x, y兲 苷 y
8. D is bounded by y 苷 sx , y 苷 0, and x 苷 1; 共x, y兲 苷 x
ⱍ
9. D 苷 兵共x, y兲 0 y sin共 x兾L兲, 0 x L其 ; 共x, y兲 苷 y
10. D is bounded by the parabolas y 苷 x 2 and x 苷 y 2;
共x, y兲 苷 sx
SECTION 15.5 APPLICATIONS OF DOUBLE INTEGRALS
11. A lamina occupies the part of the disk x 2 y 2 1 in the
and Y is
f 共x, y兲 苷
12. Find the center of mass of the lamina in Exercise 11 if the
density at any point is proportional to the square of its
distance from the origin.
y 苷 s1 x 2 and y 苷 s4 x 2 together with the portions
of the x-axis that join them. Find the center of mass of the
lamina if the density at any point is proportional to its distance from the origin.
f 共x, y兲 苷
15. Find the center of mass of a lamina in the shape of an isos-
celes right triangle with equal sides of length a if the density
at any point is proportional to the square of the distance from
the vertex opposite the hypotenuse.
function
f 共x, y兲 苷
2
but outside the circle x 2 y 2 苷 1. Find the center of mass
if the density at any point is inversely proportional to its distance from the origin.
Exercise 7.
of 1000 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density
function with mean 苷 1000, find the probability that
both of the lamp’s bulbs fail within 1000 hours.
(b) Another lamp has just one bulb of the same type as in
part (a). If one bulb burns out and is replaced by a bulb
of the same type, find the probability that the two bulbs
fail within a total of 1000 hours.
Exercise 12.
19. Find the moments of inertia I x , I y , I 0 for the lamina of
Exercise 15.
20. Consider a square fan blade with sides of length 2 and the
lower left corner placed at the origin. If the density of the
blade is 共x, y兲 苷 1 0.1x, is it more difficult to rotate the
blade about the x-axis or the y-axis?
CAS
22. D is enclosed by the cardioid r 苷 1 cos ;
共x, y兲 苷 sx 2 y 2
CAS
23–26 A lamina with constant density 共x, y兲 苷 occupies the
given region. Find the moments of inertia I x and I y and the radii
of gyration x and y.
23. The rectangle 0 x b, 0 y h
24. The triangle with vertices 共0, 0兲, 共b, 0兲, and 共0, h兲
25. The part of the disk x 2 y 2 a 2 in the first quadrant
26. The region under the curve y 苷 sin x from x 苷 0 to x 苷 0.1e共0.5x0.2y兲 if x 0, y 0
0
otherwise
30. (a) A lamp has two bulbs of a type with an average lifetime
18. Find the moments of inertia I x , I y , I 0 for the lamina of
共x, y兲 苷 xy
再
(a) Verify that f is indeed a joint density function.
(b) Find the following probabilities.
(i) P共Y 1兲
(ii) P共X 2, Y 4兲
(c) Find the expected values of X and Y.
17. Find the moments of inertia I x , I y , I 0 for the lamina of
ⱍ
4xy if 0 x 1, 0 y 1
0
otherwise
29. Suppose X and Y are random variables with joint density
16. A lamina occupies the region inside the circle x y 苷 2y
21. D 苷 兵共x, y兲 0 y sin x, 0 x 其 ;
再
is a joint density function.
(b) If X and Y are random variables whose joint density function is the function f in part (a), find
(i) P (X 12 )
(ii) P (X 12 , Y 12 )
(c) Find the expected values of X and Y.
density at any point is inversely proportional to its distance
from the origin.
of mass, and moments of inertia of the lamina that occupies the
region D and has the given density function.
Cx共1 y兲 if 0 x 1, 0 y 2
0
otherwise
28. (a) Verify that
14. Find the center of mass of the lamina in Exercise 13 if the
21–22 Use a computer algebra system to find the mass, center
再
(a) Find the value of the constant C.
(b) Find P共X 1, Y 1兲.
(c) Find P共X Y 1兲.
13. The boundary of a lamina consists of the semicircles
CAS
989
27. The joint density function for a pair of random variables X
first quadrant. Find its center of mass if the density at any
point is proportional to its distance from the x-axis.
2
||||
31. Suppose that X and Y are independent random variables,
where X is normally distributed with mean 45 and standard
deviation 0.5 and Y is normally distributed with mean 20 and
standard deviation 0.1.
(a) Find P共40 X 50, 20 Y 25兲.
(b) Find P共4共X 45兲2 100共Y 20兲2 2兲.
32. Xavier and Yolanda both have classes that end at noon and
they agree to meet every day after class. They arrive at the
coffee shop independently. Xavier’s arrival time is X and
Yolanda’s arrival time is Y, where X and Y are measured in
minutes after noon. The individual density functions are
f1共x兲 苷
再
ex if x 0
0
if x 0
f2共 y兲 苷
再
y if 0 y 10
0
otherwise
1
50
(Xavier arrives sometime after noon and is more likely to
arrive promptly than late. Yolanda always arrives by 12:10 PM
and is more likely to arrive late than promptly.) After Yolanda
arrives, she’ll wait for up to half an hour for Xavier, but he
won’t wait for her. Find the probability that they meet.
998
||||
CHAPTER 15 MULTIPLE INTEGRALS
15.6
EXERCISES
20. The solid bounded by the cylinder y 苷 x 2 and the planes
1. Evaluate the integral in Example 1, integrating first with
z 苷 0, z 苷 4, and y 苷 9
respect to y, then z , and then x.
2. Evaluate the integral xxxE 共xz y 兲 dV, where
21. The solid enclosed by the cylinder x 2 y 2 苷 9 and the
3
planes y z 苷 5 and z 苷 1
E 苷 兵共x, y, z兲 1 x 1, 0 y 2, 0 z 1其
ⱍ
22. The solid enclosed by the paraboloid x 苷 y 2 z 2 and the
plane x 苷 16
using three different orders of integration.
3– 8 Evaluate the iterated integral.
1
z
xz
0
0
3.
yyy
5.
yyy
7.
y yy
8.
0
3
0
1
s1z 2
0
0
兾2
y
x
0
0
0
s
x
xz
y yy
0
6xz dy dx dz
0
ze y dx dz dy
23. (a) Express the volume of the wedge in the first octant that is
1
2x
y
0
x
0
1
z
y
0
0
0
4.
yy y
6.
yyy
CAS
2
zey dx dy dz
cos共x y z兲 dz dx dy
0
cut from the cylinder y 2 z 2 苷 1 by the planes y 苷 x and
x 苷 1 as a triple integral.
(b) Use either the Table of Integrals (on Reference Pages 6–10)
or a computer algebra system to find the exact value of the
triple integral in part (a).
2xyz dz dy dx
24. (a) In the Midpoint Rule for triple integrals we use a triple
Riemann sum to approximate a triple integral over a box
B, where f 共x, y, z兲 is evaluated at the center 共 x i , yj , zk 兲
of the box Bijk . Use the Midpoint Rule to estimate
xxxB sx 2 y 2 z 2 dV, where B is the cube defined by
0 x 4, 0 y 4, 0 z 4. Divide B into eight
cubes of equal size.
(b) Use a computer algebra system to approximate the integral
in part (a) correct to the nearest integer. Compare with the
answer to part (a).
2
x sin y dy dz dx
9–18 Evaluate the triple integral.
9.
xxxE 2x dV, where
CAS
ⱍ
E 苷 {共x, y, z兲 0 y 2, 0 x s4 y 2 , 0 z y}
10.
xxxE yz cos共x 5 兲 dV, where
ⱍ
E 苷 兵共x, y, z兲 0 x 1, 0 y x, x z 2x其
11.
where E lies under the plane z 苷 1 x y
and above the region in the xy-plane bounded by the curves
y 苷 sx , y 苷 0, and x 苷 1
xxxE 6xy dV,
25–26 Use the Midpoint Rule for triple integrals (Exercise 24) to
estimate the value of the integral. Divide B into eight sub-boxes of
equal size.
25.
1
dV, where
ln共1 x y z兲
B 苷 兵共x, y, z兲 0 x 4, 0 y 8, 0 z 4其
xxxB
ⱍ
where E is bounded by the planes x 苷 0, y 苷 0,
z 苷 0, and 2x 2y z 苷 4
12.
xxxE y dV,
13.
xxxE x 2e y dV,
14.
xxxE xy dV,
where E is bounded by the parabolic cylinders
y 苷 x 2 and x 苷 y 2 and the planes z 苷 0 and z 苷 x y
27–28 Sketch the solid whose volume is given by the iterated
15.
xxxT x 2 dV,
integral.
where E is bounded by the parabolic cylinder
z 苷 1 y 2 and the planes z 苷 0, x 苷 1, and x 苷 1
where T is the solid tetrahedron with vertices
共0, 0, 0兲, 共1, 0, 0兲, 共0, 1, 0兲, and 共0, 0, 1兲
16.
xxxT xyz dV,
17.
xxxE x dV, where E is bounded by the paraboloid
where T is the solid tetrahedron with vertices
共0, 0, 0兲, 共1, 0, 0兲, 共1, 1, 0兲, and 共1, 0, 1兲
x 苷 4y 2 4z 2 and the plane x 苷 4
18.
where E is bounded by the cylinder y 2 z 2 苷 9
and the planes x 苷 0, y 苷 3x, and z 苷 0 in the first octant
xxxE z dV,
26.
xxxB sin共xy 2z 3兲 dV, where
ⱍ
B 苷 兵共x, y, z兲 0 x 4, 0 y 2, 0 z 1其
27.
1
1x
0
0
yy y
22z
0
dy dz dx
2
2y
y y y
0
0
4y 2
0
dx dz dy
29–32 Express the integral xxxE f 共x, y, z兲 dV as an iterated integral
in six different ways, where E is the solid bounded by the given
surfaces.
29. y 苷 4 x 2 4z 2,
30. y z 苷 9,
2
2
19–22 Use a triple integral to find the volume of the given solid.
31. y 苷 x ,
19. The tetrahedron enclosed by the coordinate planes and the
32. x 苷 2,
plane 2x y z 苷 4
28.
2
y苷0
x 苷 2, x 苷 2
z 苷 0,
y 苷 2,
y 2z 苷 4
z 苷 0, x y 2z 苷 2
SECTION 15.6 TRIPLE INTEGRALS
||||
999
40. E is the tetrahedron bounded by the planes x 苷 0, y 苷 0,
33. The figure shows the region of integration for the integral
z 苷 0, x y z 苷 1; 共x, y, z兲 苷 y
1
1
1y
0
sx
0
y y y
f 共x, y, z兲 dz dy dx
41– 44 Assume that the solid has constant density k.
41. Find the moments of inertia for a cube with side length L if
Rewrite this integral as an equivalent iterated integral in the
five other orders.
one vertex is located at the origin and three edges lie along the
coordinate axes.
z
42. Find the moments of inertia for a rectangular brick with dimen1
sions a, b, and c and mass M if the center of the brick is situated at the origin and the edges are parallel to the coordinate
axes.
z=1-y
43. Find the moment of inertia about the z-axis of the solid cylin0
der x 2 y 2 a 2, 0 z h.
1
y
44. Find the moment of inertia about the z-axis of the solid cone
y=œ„
x
sx 2 y 2 z h.
x
45– 46 Set up, but do not evaluate, integral expressions for
34. The figure shows the region of integration for the integral
1
1x 2
0
0
yy
y
1x
0
(a) the mass, (b) the center of mass, and (c) the moment of inertia
about the z -axis.
f 共x, y, z兲 dy dz dx
45. The solid of Exercise 21;
46. The hemisphere x 2 y 2 z 2 1, z 0;
Rewrite this integral as an equivalent iterated integral in the
five other orders.
z
共x, y, z兲 苷 sx 2 y 2 z 2
CAS
z=1-≈
0
x
1
y
y=1-x
CAS
35–36 Write five other iterated integrals that are equal to the
given iterated integral.
35.
1
0
36.
1
yyy
1
y
x2
y
yy y
0
0
f 共x, y, z兲 dz dx dy
0
y
0
37– 40 Find the mass and center of mass of the solid E with the
given density function .
共x, y, z兲 苷 2
共x, y, z兲 苷 x 2 y 2, find the following quantities, correct
to three decimal places.
(a) The mass
(b) The center of mass
(c) The moment of inertia about the z-axis
38. E is bounded by the parabolic cylinder z 苷 1 y and the
planes x z 苷 1, x 苷 0, and z 苷 0;
f 共x, y, z兲 苷 Cxyz if 0 x 2, 0 y 2, 0 z 2, and
f 共x, y, z兲 苷 0 otherwise.
(a) Find the value of the constant C.
(b) Find P共X 1, Y 1, Z 1兲.
(c) Find P共X Y Z 1兲.
50. Suppose X , Y , and Z are random variables with joint density
2
共x, y, z兲 苷 4
39. E is the cube given by 0 x a, 0 y a, 0 z a ;
共x, y, z兲 苷 x 2 y 2 z 2
48. If E is the solid of Exercise 18 with density function
49. The joint density function for random variables X , Y , and Z is
f 共x, y, z兲 dz dy dx
37. E is the solid of Exercise 11;
47. Let E be the solid in the first octant bounded by the cylinder
x 2 y 2 苷 1 and the planes y 苷 z, x 苷 0, and z 苷 0 with the
density function 共x, y, z兲 苷 1 x y z. Use a computer
algebra system to find the exact values of the following quantities for E.
(a) The mass
(b) The center of mass
(c) The moment of inertia about the z-axis
1
1
共x, y, z兲 苷 sx 2 y 2
function f 共x, y, z兲 苷 Ce共0.5x0.2y0.1z兲 if x 0, y 0, z 0,
and f 共x, y, z兲 苷 0 otherwise.
(a) Find the value of the constant C.
(b) Find P共X 1, Y 1兲.
(c) Find P共X 1, Y 1, Z 1兲.
1004
||||
15.7
CHAPTER 15 MULTIPLE INTEGRALS
EXERCISES
20. Evaluate xxxE x dV, where E is enclosed by the planes z 苷 0
1–2 Plot the point whose cylindrical coordinates are given. Then
and z 苷 x y 5 and by the cylinders x 2 y 2 苷 4 and
x 2 y 2 苷 9.
find the rectangular coordinates of the point.
1. (a) 共2, 兾4, 1兲
(b) 共4, 兾3, 5兲
2. (a) 共1, , e兲
(b) 共1, 3兾2, 2兲
21. Evaluate xxxE x 2 dV, where E is the solid that lies within the
cylinder x 2 y 2 苷 1, above the plane z 苷 0, and below the
cone z 2 苷 4x 2 4y 2.
3– 4 Change from rectangular to cylindrical coordinates.
3. (a) 共1, 1, 4兲
(b) (1, s3 , 2)
4. (a) (2 s3, 2, 1)
(b) 共4, 3, 2兲
22. Find the volume of the solid that lies within both the cylinder
x 2 y 2 苷 1 and the sphere x 2 y 2 z 2 苷 4.
23. (a) Find the volume of the region E bounded by the parabo-
loids z 苷 x 2 y 2 and z 苷 36 3x 2 3y 2.
(b) Find the centroid of E (the center of mass in the case
where the density is constant).
5–6 Describe in words the surface whose equation is given.
5. 苷 兾4
6. r 苷 5
24. (a) Find the volume of the solid that the cylinder r 苷 a cos 7– 8 Identify the surface whose equation is given.
7. z 苷 4 r 2
cuts out of the sphere of radius a centered at the origin.
(b) Illustrate the solid of part (a) by graphing the sphere and
the cylinder on the same screen.
;
8. 2r 2 z 2 苷 1
25. Find the mass and center of mass of the solid S bounded by
the paraboloid z 苷 4x 2 4y 2 and the plane z 苷 a 共a 0兲 if
S has constant density K.
9–10 Write the equations in cylindrical coordinates.
9. (a) z 苷 x 2 y 2
(b) x 2 y 2 苷 2y
10. (a) 3x 2y z 苷 6
26. Find the mass of a ball B given by x 2 y 2 z 2 a 2 if the
(b) x 2 y 2 z 2 苷 1
density at any point is proportional to its distance from the
z -axis.
11–12 Sketch the solid described by the given inequalities.
11. 0 r 2,
兾2 兾2, 0 z 1
12. 0 兾2,
27–28 Evaluate the integral by changing to cylindrical coordinates.
rz2
13. A cylindrical shell is 20 cm long, with inner radius 6 cm and
outer radius 7 cm. Write inequalities that describe the shell
in an appropriate coordinate system. Explain how you have
positioned the coordinate system with respect to the shell.
; 14. Use a graphing device to draw the solid enclosed by the
paraboloids z 苷 x 2 y 2 and z 苷 5 x 2 y 2.
15–16 Sketch the solid whose volume is given by the integral
and evaluate the integral.
15.
4
2
yy y
0
0
4
r
r dz d dr
16.
兾2
2
y yy
0
0
9r 2
0
r dz dr d
17–26 Use cylindrical coordinates.
2
s4y 2
2
s4y 2
27.
y y
28.
y y
3
3
s9x 2
0
y
2
sx 2y 2
y
9x 2y 2
0
xz dz dx dy
sx 2 y 2 dz dy dx
29. When studying the formation of mountain ranges, geologists
estimate the amount of work required to lift a mountain from
sea level. Consider a mountain that is essentially in the shape
of a right circular cone. Suppose that the weight density of
the material in the vicinity of a point P is t共P兲 and the height
is h共P兲.
(a) Find a definite integral that represents the total work done
in forming the mountain.
(b) Assume that Mount Fuji in Japan is in the shape of a right
circular cone with radius 62,000 ft, height 12,400 ft, and
density a constant 200 lb兾ft3 . How much work was done
in forming Mount Fuji if the land was initially at sea level?
17. Evaluate xxxE sx 2 y 2 dV, where E is the region that lies
inside the cylinder x 2 y 2 苷 16 and between the planes
z 苷 5 and z 苷 4.
18. Evaluate xxxE 共x 3 xy 2 兲 dV, where E is the solid in the first
octant that lies beneath the paraboloid z 苷 1 x y .
2
2
19. Evaluate xxxE e z dV, where E is enclosed by the paraboloid
z 苷 1 x 2 y 2, the cylinder x 2 y 2 苷 5, and the xy-plane.
P
1010
||||
15.8
CHAPTER 15 MULTIPLE INTEGRALS
EXERCISES
1–2 Plot the point whose spherical coordinates are given. Then
19–20 Set up the triple integral of an arbitrary continuous function
find the rectangular coordinates of the point.
f 共x, y, z兲 in cylindrical or spherical coordinates over the solid
shown.
1. (a) 共1, 0, 0兲
(b) 共2, 兾3, 兾4兲
2. (a) 共5, , 兾2兲
(b) 共4, 3兾4, 兾3兲
z
19.
z
20.
3
3– 4 Change from rectangular to spherical coordinates.
3. (a) (1, s3 , 2s3 )
(b) 共0, 1, 1兲
4. (a) (0, s3 , 1)
(b) (1, 1, s6 )
5–6 Describe in words the surface whose equation is given.
5. 苷 兾3
6. 苷 3
2
y
x
1
x
2
y
21–34 Use spherical coordinates.
2
2
2 2
21. Evaluate xxxB 共x y z 兲 dV, where B is the ball with
center the origin and radius 5.
7– 8 Identify the surface whose equation is given.
8. 2 共sin 2 sin 2 cos2兲 苷 9
7. 苷 sin sin 9–10 Write the equation in spherical coordinates.
9. (a) z 2 苷 x 2 y 2
(b) x 2 z 2 苷 9
10. (a) x 2 2x y 2 z 2 苷 0
(b) x 2y 3z 苷 1
11–14 Sketch the solid described by the given inequalities.
0 兾2,
11. 2,
12. 2 3,
0 兾2
兾2 13. 1,
3兾4 14. 2,
csc 15. A solid lies above the cone z 苷 sx 2 y 2 and below the
sphere x 2 y 2 z 2 苷 z. Write a description of the solid in
terms of inequalities involving spherical coordinates.
16. (a) Find inequalities that describe a hollow ball with diameter
30 cm and thickness 0.5 cm. Explain how you have
positioned the coordinate system that you have chosen.
(b) Suppose the ball is cut in half. Write inequalities that
describe one of the halves.
17–18 Sketch the solid whose volume is given by the integral and
evaluate the integral.
17.
18.
兾6
兾2
3
0
0
0
y y y
2
y y y
0
兾2
2
1
2 sin d d d
2 sin d d d
22. Evaluate xxxH 共9 x 2 y 2 兲 dV, where H is the solid
hemisphere x 2 y 2 z 2 9, z 0.
23. Evaluate xxxE z dV, where E lies between the spheres
x 2 y 2 z 2 苷 1 and x 2 y 2 z 2 苷 4 in the first octant.
24. Evaluate xxxE e sx y z dV, where E is enclosed by the sphere
2
2
2
x y z 苷 9 in the first octant.
2
2
2
25. Evaluate xxxE x 2 dV, where E is bounded by the x z-plane
and the hemispheres y 苷 s9 x 2 z 2 and
y 苷 s16 x 2 z 2 .
26. Evaluate xxxE xyz dV, where E lies between the spheres
苷 2 and 苷 4 and above the cone 苷 兾3.
27. Find the volume of the part of the ball a that lies between
the cones 苷 兾6 and 苷 兾3.
28. Find the average distance from a point in a ball of radius a to
its center.
29. (a) Find the volume of the solid that lies above the cone
苷 兾3 and below the sphere 苷 4 cos .
(b) Find the centroid of the solid in part (a).
30. Find the volume of the solid that lies within the sphere
x 2 y 2 z 2 苷 4, above the xy-plane, and below the cone
z 苷 sx 2 y 2 .
31. Find the centroid of the solid in Exercise 25.
32. Let H be a solid hemisphere of radius a whose density at any
point is proportional to its distance from the center of the base.
(a) Find the mass of H .
(b) Find the center of mass of H .
(c) Find the moment of inertia of H about its axis.
33. (a) Find the centroid of a solid homogeneous hemisphere of
radius a.
(b) Find the moment of inertia of the solid in part (a) about a
diameter of its base.
SECTION 15.8 TRIPLE INTEGRALS IN SPHERICAL COORDINATES
34. Find the mass and center of mass of a solid hemisphere of
radius a if the density at any point is proportional to its
distance from the base.
CAS
||||
1011
43. The surfaces 苷 1 5 sin m sin n have been used as
1
models for tumors. The “bumpy sphere” with m 苷 6 and
n 苷 5 is shown. Use a computer algebra system to find the
volume it encloses.
35–38 Use cylindrical or spherical coordinates, whichever seems
more appropriate.
35. Find the volume and centroid of the solid E that lies
above the cone z 苷 sx 2 y 2 and below the sphere
x 2 y 2 z 2 苷 1.
36. Find the volume of the smaller wedge cut from a sphere of
radius a by two planes that intersect along a diameter at an
angle of 兾6.
CAS
37. Evaluate xxxE z dV, where E lies above the paraboloid
z 苷 x 2 y 2 and below the plane z 苷 2y. Use either the
Table of Integrals (on Reference Pages 6 –10) or a computer
algebra system to evaluate the integral.
38. (a) Find the volume enclosed by the torus 苷 sin .
(b) Use a computer to draw the torus.
;
39– 40 Evaluate the integral by changing to spherical coordinates.
39.
40.
1
yy
0
s1x 2
0
y
s2x 2y 2
a
sa 2y 2
a
sa 2y 2
y y
sx 2y 2
y
xy dz dy dx
sa 2x 2y 2
sa 2x 2y 2
y y y
2
sx 2 y 2 z 2 e共x y
2
z 2 兲
dx dy dz 苷 2
(The improper triple integral is defined as the limit of a
triple integral over a solid sphere as the radius of the sphere
increases indefinitely.)
45. (a) Use cylindrical coordinates to show that the volume of
共x z y z z 兲 dz dx dy
2
44. Show that
2
3
the solid bounded above by the sphere r 2 z 2 苷 a 2 and
below by the cone z 苷 r cot 0 (or 苷 0 ), where
0 0 兾2, is
; 41. Use a graphing device to draw a silo consisting of a cylinder
V苷
with radius 3 and height 10 surmounted by a hemisphere.
42. The latitude and longitude of a point P in the Northern Hemi-
sphere are related to spherical coordinates , , as follows.
We take the origin to be the center of the earth and the positive z -axis to pass through the North Pole. The positive x-axis
passes through the point where the prime meridian (the
meridian through Greenwich, England) intersects the equator.
Then the latitude of P is 苷 90 and the longitude is
苷 360 . Find the great-circle distance from Los
Angeles (lat. 34.06 N, long. 118.25 W) to Montréal (lat.
45.50 N, long. 73.60 W). Take the radius of the earth to be
3960 mi. (A great circle is the circle of intersection of a
sphere and a plane through the center of the sphere.)
2 a 3
共1 cos 0 兲
3
(b) Deduce that the volume of the spherical wedge given by
1 2 , 1 2 , 1 2 is
V 苷
23 13
共cos 1 cos 2 兲共 2 1 兲
3
(c) Use the Mean Value Theorem to show that the volume in
part (b) can be written as
苲
V 苷 苲 2 sin 苲
where 苲 lies between 1 and 2 , lies between 1 and
2 , 苷 2 1 , 苷 2 1 , and 苷 2 1 .
A122
||||
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
Minimum f 共4, 1兲 苷 11
Maximum f 共1, 1兲 苷 1; saddle points (0, 0), (0, 3), (3, 0)
Maximum f 共1, 2兲 苷 4, minimum f 共2, 4兲 苷 64
Maximum f 共1, 0兲 苷 2, minima f 共1, 1兲 苷 3,
saddle points 共1, 1兲, 共1, 0兲
59. Maximum f (s2兾3, 1兾s3 ) 苷 2兾(3 s3 ),
minimum f (s2兾3, 1兾s3 ) 苷 2兾(3 s3 )
61. Maximum 1, minimum 1
63. (31兾4, 31兾4s2, 31兾4 ), (31兾4, 31兾4s2, 31兾4 )
65. P(2 s3 ), P(3 s3 )兾6, P(2 s3 3)兾3
EXERCISES 15.3
51.
53.
55.
57.
1. 32
N
PAGE 972
5. e 1
3
10
3.
13. 共1 cos 1兲
15.
128
15
27.
1
2
23. 6
25.
33.
7.
147
20
1
3
17. 0
2 1
4
2
2
N
(b) Yes
64
3
(0, 1, 0)
y
37. 兾2
35. 13,984,735,616兾14,549,535
f 共x, y兲 dx dy
2 4
0 y2
x x
39.
41.
x33 x0s9x
y
N
31.
x
CHAPTER 15
EXERCISES 15.1
21.
17
2
31
8
(0, 0, 1)
(1, 0, 0)
3. (a) x 苷 w兾3, base 苷 w兾3
1. L W , L W
7. s6兾2, 3 s2兾2
1
z
PAGE 948
2
19.
11. 2 e 16 7
18
29. 0, 1.213, 0.713
0
PROBLEMS PLUS
9. 4
3
PAGE 958
y
3
y=œ„
x
2
1. (a) 288
(b) 144
3. (a) 2兾2 ⬇ 4.935
(b) 0
5. (a) 6
(b) 3.5
7. U V L
9. (a) ⬇248
(b) 15.5
11. 60
13. 3
15. 1.141606, 1.143191, 1.143535, 1.143617, 1.143637, 1.143642
f 共x, y兲 dy dx
2
≈+¥=9
x=4
0
43.
y=0
x
4
0
–3
x0ln 2 xe2 f 共x, y兲 dx dy
y
y
y=0
3
x
y=ln x or x=e†
ln 2
EXERCISES 15.2
3
N
PAGE 964
2
1. 500y , 3x
3. 10
13. 15.
11. 0
1
1
19. 2 (s3 1) 12 23.
21
2
x=2
5. 2
7. 261,632兾45
17. 9 ln 2
1
21. 2 共e 2 3兲
9.
21
2
y=0
ln 2
z
0
45. 6 共e 9 1兲
47. 3 ln 9
53. 共兾16兲e1兾16 xxQ e共x y
1
1
2
4
59. 8
0
1
N
51. 1
3
4
PAGE 978
x03兾2 x04 f 共r cos , r sin 兲r dr d
5.
y
1
1
3
61. 2兾3
EXERCISES 15.4
1.
(2 s2 1)
55.
dA 兾16
49.
兲
2 2
x
2
1
3.
1
x1
x0共x1兲兾2 f 共x, y兲 dy dx
33 兾2
y
x
4
25. 47.5
27.
33. 21e 57
166
27
29. 2
31.
64
3
7
0
x
R
2
z
7. 0
0
0
y
5
6
1 1
x
0
35.
37. Fubini’s Theorem does not apply. The integrand has an infinite
discontinuity at the origin.
9. 2 sin 9
11. 共兾2兲共1 e4 兲
1
15. 兾12
17. 共 2兲
23. a
25. 共2兾3兲 1 (1兾s2 )
4
3
1
8
19.
[
3
16
3
]
13.
3
64
2
21. 4
3
27. 共8兾3兲(64 24 s3 )
29. 2 共1 cos 9兲
31. 2s2兾3
33. 1800 ft
15
16
1
3
35.
37. (a) s 兾4
(b) s 兾2
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
EXERCISES 15.5
1.
64
3
PAGE 988
N
3. , ( , 0)
4
3
C
7. 4 共e 2 1兲,
1
冉
3 3
e 1 4共e 1兲
,
2共e 2 1兲 9共e 2 1兲
2
3
冊
2
3
冉
冊
2
1 16
, Ix 苷 3 2兾64,
,
3
9
1
Iy 苷 16 共 4 3 2 兲, I0 苷 4兾16 9 2兾64
23. bh 3兾3, b 3h兾3; b兾s3, h兾s3
25. a 4兾16, a 4兾16; a兾2, a兾 2
1
(b) 0.375
(c) 485 ⬇ 0.1042
27. (a) 2
0.2
29. (b) (i) e
⬇ 0.8187
(ii) 1 e1.8 e0.8 e1 ⬇ 0.3481
(c) 2, 5
(b) ⬇0.632
31. (a) ⬇0.500
33. (a) xxD 共k兾20兲[20 s共x x0 兲2 共 y y0 兲2 ] dA, where D is
the disk with radius 10 mi centered at the center of the city
8
(b) 200k兾3 ⬇ 209k, 200(兾2 9 )k ⬇ 136k, on the edge
21. m 苷 2兾8, 共 x, y 兲 苷
x01 xsx1 x01y f 共x, y, z兲 dz dy dx
苷 x01 x0y x01y f 共x, y, z兲 dz dx dy
苷 x01 x01z x0y f 共x, y, z兲 dx dy dz
苷 x01 x01y x0y f 共x, y, z兲 dx dz dy
1z
f 共x, y, z兲 dy dz dx
苷 x01 x01sx xsx
1 共1z兲 1z
苷 x0 x0
xsx f 共x, y, z兲 dy dx dz
2
9. L兾4, 共L兾2, 16兾共9兲兲
11. ( 8 , 3兾16)
13. 共0, 45兾共14兲兲
15. 共2a兾5, 2a兾5兲 if vertex is (0, 0) and sides are along positive axes
1
1
1
17. 16 共e 4 1兲, 8 共e 2 1兲, 16 共e 4 2e 2 3兲
6
6
19. 7 ka 兾180 , 7 ka 兾180 , 7 ka 6兾90 if vertex is 共0, 0兲 and sides are
along positive axes
2
2
x01 xy1 x0y f 共x, y, z兲 dz dx dy 苷 x01 x0x x0y f 共x, y, z兲 dz dy dx
苷 x01 xz1 xy1 f 共x, y, z兲 dx dy dz 苷 x01 x0y xy1 f 共x, y, z兲 dx dz dy
苷 x01 x0x xzx f 共x, y, z兲 dy dz dx 苷 x01 xz1 xzx f 共x, y, z兲 dy dx dz
35.
37. 30 , ( 553 , 79 , 553 )
39. a 5, 共7a兾12, 7a兾12, 7a兾12兲
2
1
5
41. Ix 苷 Iy 苷 Iz 苷 3 kL
43. 2 kha 4
79
1.
27
4
5. 3 共e 3 1兲
1
3. 1
13. 8兾共3e兲
23. (a)
x01 x0x x0s1y
2
1
17. 16兾3
1
60
15.
7. 3
dz dy dx
(b) 14 13
16
3
11.
65
28
21. 36
1
2
0
y
2
s4x y兾2
x2
x04x xs4x
f 共x, y, z兲 dz dy dx
y兾2
4 s4y
s4x y兾2
苷 x0 xs4y xs4x y兾2 f 共x, y, z兲 dz dx dy
1
s4y4z
x044z xs4y4z
苷 x1
f 共x, y, z兲 dx dy dz
s4y兾2
s4y4z
xs4y4z
苷 x04 xs4y兾2
f 共x, y, z兲 dx dz dy
2
4x 4z
s4x 兾2
f 共x, y, z兲 dy dz dx
苷 x2 xs4x 兾2 x0
1
4x 4z
s44z
苷 x1 xs44z x0
f 共x, y, z兲 dy dx dz
2
4 2y兾2
31. x2 xx x0
f 共x, y, z兲 dz dy dx
sy
x02y兾2 f 共x, y, z兲 dz dx dy
苷 x04 xsy
sy
苷 x02 x042z xsy
f 共x, y, z兲 dx dy dz
4 2y兾2 sy
xsy f 共x, y, z兲 dx dz dy
苷 x0 x0
2
2x 兾2 42z
苷 x2 x0
xx f 共x, y, z兲 dy dz dx
s42z
苷 x02 xs42z
xx42z f 共x, y, z兲 dy dx dz
2
29.
2
3
s9x
y 苷 共1兾m兲 x3
xs9x
2
2
3
s9x
z 苷 共1兾m兲 x3
xs9x
2
2
2
2
3
32
11
24
冉
冊
28
30 128 45 208
,
,
9 44 45 220 135 660
1
240 共68 15兲
1
1
1
(a) 8
(b) 64
(c) 5760
3
L 兾8
The region bounded by the ellipsoid x 2 2y 2 3z 2 苷 1
(b) 共 x, y, z 兲 苷
(c)
2
x15y xsx 2 y 2 dz dy dx
x15y ysx 2 y 2 dz dy dx
x15y zsx 2 y 2 dz dy dx
s9x
x33 xs9x
x15y 共x 2 y 2 兲3兾2 dz dy dx
EXERCISES 15.7
1. (a)
x
2
2
49.
51.
53.
25. 60.533
z
27.
1
19.
9. 4
s9x
x33 xs9x
x15y sx 2 y 2 dz dy dx
(b) 共 x, y, z 兲, where
3
s9x
xs9x
x 苷 共1兾m兲 x3
47. (a)
PAGE 998
N
358 33 571
45. (a) m 苷
(c)
EXERCISES 15.6
N
PAGE 1004
z
z
(b)
π
”4, _ 3 , 5’
2
2
π
” 2, 4 , 1 ’
2
2
2
A123
33.
5. 6, ( 4 , 2 )
4
3
||||
5
0
2
2
2
2
2
2
π
4
2
4
1
y
0
π
_3
x
y
x
2
2
2
2
2
2
2
(s2, s2, 1)
3. (a) (s2, 7兾4 , 4)
(b)
(2, 2 s3, 5)
(2, 4兾3 , 2)
5. Vertical half-plane through the z-axis
(b) r 苷 2 sin 9. (a) z 苷 r 2
z
11.
1
z=1
2
2
2
2
x
2
y
7. Circular paraboloid
A124
||||
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
13. Cylindrical coordinates: 6 r 7, 0 2, 0 z 20
15.
17.
共9兾4兲 (2 s3 )
z
3
z
64 兾3
4
π
6
4
4
x
y
x
19. 共e 6 e 5兲
21. 2兾5
384
(a) 162
(b) 共0, 0, 15兲
27. 0
Ka 2兾8, 共0, 0, 2a兾3兲
(a) xxxC h共P兲t共P兲 dV , where C is the cone
(b) ⬇3.1 1019 ft-lb
17.
23.
25.
29.
EXERCISES 15.8
N
PAGE 1010
z
1. (a)
y
兾2 3 2
0
0 0
f 共r cos , r sin , z兲 r dz dr d
23. 15兾16
25. 1562兾15
312,500 兾7
(s3 1) a 3兾3 29. (a) 10 (b) (0, 0, 2.1)
(0, 525
296 , 0)
(a) (0, 0, 38 a) (b) 4Ka 5兾15
35. 共2兾3兲[1 (1兾s2 )], (0, 0, 3兾[8(2 s2 )])
37. 5兾6
39. (4 s2 5 )兾15
41.
43. 136兾99
19.
21.
27.
31.
33.
x x x
(0, 0, 1)
(1, 0, 0)
1
0
y
x
EXERCISES 15.9
z
(b)
(
1
2
s2, s6, s2 )
1
2
π π
”2, 3 , 4 ’
π
4
2
0
π
3
y
x
3.
5.
7.
9.
(a) 共4, 兾3, 兾6兲 (b) (s2, 3兾2, 3兾4)
Half-cone
Sphere, radius 12 , center (0, 12, 0)
(a) cos2 苷 sin 2
(b) 2 共sin2 cos2 cos2兲 苷 9
11.
N
PAGE 1020
1. 16
3. sin cos2
5. 0
7. The parallelogram with vertices (0, 0), (6, 3), (12, 1), (6, 2)
9. The region bounded by the line y 苷 1, the y-axis, and y 苷 sx
11. 3
13. 6
15. 2 ln 3
4
(b) 1.083 10 12 km 3
17. (a) 3 abc
3
8
19. 5 ln 8
21. 2 sin 1
23. e e1
2
CHAPTER 15 REVIEW
N
PAGE 1021
True-False Quiz
1. True
3. True
5. True
7. False
z
Exercises
2
1. ⬇64.0
3. 4e 2 4e 3
5. 2 sin 1
7. 3
9. x0 x24 f 共r cos , r sin 兲 r dr d
11. The region inside the loop of the four-leaved rose r 苷 sin 2 in
1
∏=2
2
2
13.
y
x
z
13.
x
the first quadrant
1
1
7
1
15. 2 e 6 2
17. 4 ln 2
19. 8
2 sin 1
64
21. 81兾5
23. 40.5
25. 兾96
27. 15
2
3
31. 3
33. 2ma 兾9
1
(b) ( 13 , 158 )
35. (a) 4
1
(c) Ix 苷 12 , Iy 苷 241 ; y 苷 1兾s3, x 苷 1兾s6
37. 共0, 0, h兾4兲
41. 0.0512
39. 97.2
1
(b) 13
(c) 451
43. (a) 15
2
y
3π
˙=
4
∏=1
15. 0 兾4, 0 cos 45.
sy
x01 x01z xsy
f 共x, y, z兲 dx dy dz
PROBLEMS PLUS
1. 30
3.
1
2
N
sin 1
PAGE 1024
7. (b) 0.90
47. ln 2
29. 176
49. 0