◆
SECTION 12.1 DOUBLE INTEGRALS OVER RECTANGLES
847
Properties of Double Integrals
We list here three properties of double integrals that can be proved in the same manner as in Section 5.2. We assume that all of the integrals exist. Properties 7 and 8 are
referred to as the linearity of the integral.
yy 关 f 共x, y兲 t共x, y兲兴 dA 苷 yy f 共x, y兲 dA yy t共x, y兲 dA
7
▲ Double integrals behave this way
because the double sums that define
them behave this way.
R
R
R
yy c f 共x, y兲 dA 苷 c yy f 共x, y兲 dA
8
R
where c is a constant
R
If f 共x, y兲 t共x, y兲 for all 共x, y兲 in R, then
yy
9
f 共x, y兲 dA yy t共x, y兲 dA
R
12.1
Exercises
●
●
●
●
●
●
●
●
●
●
●
R
●
●
●
1. Find approximations to xxR 共x 3y 2 兲 dA using the same
subrectangles as in Example 3 but choosing the sample
point to be the (a) upper left corner, (b) upper right corner,
(c) lower left corner, (d) lower right corner of each subrectangle.
2. Find the approximation to the volume in Example 1 if the
Midpoint Rule is used.
●
y
●
●
●
●
●
●
●
●
0
1
2
3
4
1.0
2
0
3
6
5
1.5
3
1
4
8
6
2.0
4
3
0
5
8
2.5
5
5
3
1
4
3.0
7
8
6
3
0
x
3. (a) Estimate the volume of the solid that lies below
the surface z 苷 xy and above the rectangle
R 苷 兵共x, y兲 ⱍ 0 x 6, 0 y 4其. Use a Riemann
sum with m 苷 3, n 苷 2, and take the sample point to
be the upper right corner of each subrectangle.
(b) Use the Midpoint Rule to estimate the volume of the
solid in part (a).
●
●
●
6. A 20-ft-by-30-ft swimming pool is filled with water. The
depth is measured at 5-ft intervals, starting at one corner of
the pool, and the values are recorded in the table. Estimate
the volume of water in the pool.
4. If R 苷 关1, 3兴 关0, 2兴, use a Riemann sum with m 苷 4,
n 苷 2 to estimate the value of xxR 共y 2 2x 2兲 dA. Take
the sample points to be the upper left corners of the
subrectangles.
5. A table of values is given for a function f 共x, y兲 defined on
R 苷 关1, 3兴 关0, 4兴.
(a) Estimate xxR f 共x, y兲 dA using the Midpoint Rule with
m 苷 n 苷 2.
(b) Estimate the double integral with m 苷 n 苷 4 by choosing the sample points to be the points farthest from the
origin.
0
5
10
15
20
0
5
10
15
20
25
30
2
2
2
2
2
3
3
4
3
2
4
4
6
4
2
6
7
8
5
2
7
8
10
6
3
8
10
12
8
4
8
8
10
7
4
7. Let V be the volume of the solid that lies under the graph of
f 共x, y兲 苷 s52 x 2 y 2 and above the rectangle given by
2 x 4, 2 y 6. We use the lines x 苷 3 and y 苷 4
848
■
CHAPTER 12 MULTIPLE INTEGRALS
to divide R into subrectangles. Let L and U be the Riemann
sums computed using lower left corners and upper right
corners, respectively. Without calculating the numbers V , L,
and U , arrange them in increasing order and explain your
reasoning.
54
58
62
8. The figure shows level curves of a function f in the square
R 苷 关0, 1兴 关0, 1兴. Use them to estimate xxR f 共x, y兲 dA to
the nearest integer.
66
y
68
1
14
13
74
12
76
70
70
11
68
10
74
9
0
1
x
9. A contour map is shown for a function f on the square
R 苷 关0, 4兴 关0, 4兴.
(a) Use the Midpoint Rule with m 苷 n 苷 2 to estimate the
value of xxR f 共x, y兲 dA.
(b) Estimate the average value of f .
y
11–13 ■ Evaluate the double integral by first identifying it as
the volume of a solid.
11.
12.
13.
4
■
10
0
0
10 20 30
xxR 3 dA, R 苷 兵共x, y兲 ⱍ 2 x 2, 1 y 6其
xxR 共5 x兲 dA, R 苷 兵共x, y兲 ⱍ 0 x 5, 0 y 3其
xxR 共4 2y兲 dA, R 苷 关0, 1兴 关0, 1兴
■
■
■
■
■
■
■
■
■
■
■
■
14. The integral xxR s9 y dA, where R 苷 关0, 4兴 关0, 2兴,
2
represents the volume of a solid. Sketch the solid.
15. Use a programmable calculator or computer (or the sum
2
0
command on a CAS) to estimate
10
2
x 2y 2
20
yy e
30
R
4
x
10. The contour map shows the temperature, in degrees Fahren-
heit, at 3:00 P.M. on May 1, 1996, in Colorado. (The state
measures 388 mi east to west and 276 mi north to south.)
Use the Midpoint Rule to estimate the average temperature
in Colorado at that time.
dA
where R 苷 关0, 1兴 关0, 1兴. Use the Midpoint Rule with the
following numbers of squares of equal size: 1, 4, 16, 64,
256, and 1024.
16. Repeat Exercise 15 for the integral xxR cos共x 4 y 4 兲 dA.
17. If f is a constant function, f 共x, y兲 苷 k, and
R 苷 关a, b兴 关c, d兴, show that xxR k dA 苷 k共b a兲共d c兲.
18. If R 苷 关0, 1兴 关0, 1兴, show that 0 xxR sin共x y兲 dA 1.
◆
SECTION 12.2 ITERATED INTEGRALS
853
since xab t共x兲 dx is a constant. Therefore, in this case, the double integral of f can be
written as the product of two single integrals:
yy t共x兲h共y兲 dA 苷 y
b
a
d
where R 苷 关a, b兴 关c, d兴
t共x兲 dx y h共y兲 dy
c
R
EXAMPLE 5 If R 苷 关0, 兾2兴 关0, 兾2兴, then
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.
0
y
x
FIGURE 6
12.2
1–2
Exercises
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●
Find x03 f 共x, y兲 dx and x04 f 共x, y兲 dy.
■
1. f 共x, y兲 苷 2x 3x y
■
■
■
■
■
■
●
●
13.
y
2. f 共x, y兲 苷
x2
2
■
●
■
■
■
■
●
yy
R
■
■
14.
yy
R
3–10
3.
■
Calculate the iterated integral.
3
1
yy
1
0
共1 4xy兲 dx dy
15.
4
4.
yy
2
1
1
共x y 兲 dy dx
2
5.
yy
0
1
0
4
2
sx y dx dy
冉 冊
x
y
y
x
7.
yy
9.
y y
10.
1
ln 2
0
1
0
ln 5
0
yy
■
1
0
■
yy 共6x y
2 3
yy xye
R
y y
0
0
■
■
■
■
■
■
■
■
■
y
ⱍ
ⱍ
●
●
●
●
●
●
●
R 苷 兵共x, y兲 0 x 2, 0 y 1其
●
xy 2
dA,
x2 1
R 苷 兵共x, y兲 0 x 1, 3 y 3其
1 x2
dA,
1 y2
R 苷 兵共x, y兲 0 x 1, 0 y 1其
●
ⱍ
ⱍ
R 苷 关0, 兾6兴 关0, 兾3兴
yy xe
xy
R 苷 关0, 1兴 关0, 1兴
dA,
■
■
■
■
■
■
■
■
■
■
■
■
■
Sketch the solid whose volume is given by the iterated
integral.
■
1
17.
yy
18.
yy
■
0
1
0
■
1
0
1
0
共4 x 2y兲 dx dy
共2 x 2 y 2 兲 dy dx
■
■
■
■
■
■
■
■
■
19. Find the volume of the solid lying under the plane
5y 4 兲 dA, R 苷 兵共x, y兲 0 x 3, 0 y 1其
dA,
●
yy x sin共x y兲 dA,
17–18
sin共x y兲 dy dx
Calculate the double integral.
R
12.
兾2 兾2
8.
●
R
e 2xy dx dy
■
■
y1 y0 (x sy ) dx dy
6.
dy dx
2
xy
dy dx
sx 2 y 2 1
■
11–16
11.
1
4
●
R
2
16.
3
●
z 苷 2x 5y 1 and above the rectangle
R 苷 兵共x, y兲 1 x 0, 1 y 4其.
ⱍ
20. Find the volume of the solid lying under the circular
paraboloid z 苷 x 2 y 2 and above the rectangle
R 苷 关2, 2兴 关3, 3兴.
■
■
854
■
CHAPTER 12 MULTIPLE INTEGRALS
21. 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 square
R 苷 关1, 1兴 关2, 2兴.
28. Graph the solid that lies between the surfaces
2
ⱍ ⱍ
22. Find the volume of the solid lying under the hyper-
bolic paraboloid z 苷 y 2 x 2 and above the square
R 苷 关1, 1兴 关1, 3兴.
29–30
30. f 共x, y兲 苷 x sin xy,
■
24. Find the volume of the solid bounded by the elliptic parabo-
loid z 苷 1 共x 1兲 4y , the planes x 苷 3 and y 苷 2,
and the coordinate planes.
2
CAS
■
■
■
R 苷 关0, 兾2兴 关0, 1兴
■
■
■
1
yy
0
the cylinder z 苷 9 y 2 and the plane x 苷 2.
■
■
■
■
■
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.
26. (a) Find the volume of the solid bounded by the surface
z 苷 6 xy and the planes x 苷 2, x 苷 2, y 苷 0,
y 苷 3, and z 苷 0.
(b) Use a computer to draw the solid.
32. (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
27. Use a computer algebra system to find the exact value of the
t共x, y兲 苷 y
integral xxR x 5y 3e xy 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.
12.3
■
31. Use your CAS to compute the iterated integrals
25. Find the volume of the solid in the first octant bounded by
CAS
Find the average value of f over the given rectangle.
R has vertices 共1, 0兲, 共1, 5兲, 共1, 5兲, 共1, 0兲
z 苷 xsx 2 y and the planes x 苷 0, x 苷 1, y 苷 0, y 苷 1,
and z 苷 0.
;
■
29. f 共x, y兲 苷 x 2 y,
23. Find the volume of the solid bounded by the surface
2
ⱍ ⱍ
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.
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
1
F共x, y兲 苷
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
◆
SECTION 12.3 DOUBLE INTEGRALS OVER GENERAL REGIONS
861
If m f 共x, y兲 M for all 共x, y兲 in D, then
11
mA共D兲 yy f 共x, y兲 dA MA共D兲
D
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 4e
e
D
12.3
1–6
Exercises
■
x2
1.
yy
3.
yy
5.
y y
0
0
1
10.
yy
6.
yy
cos 0
e sin dr d
■
■
2
4.
■
■
●
●
●
3 2
yy x y
■
■
1
y
1
0
1
0
■
2
2x
x
v
0
■
15.
■
■
●
●
●
●
●
●
●
●
●
●
●
dA,
yy 共x y兲 dA,
D is bounded by y 苷 sx, y 苷 x 2
yy 共2x y兲 dA,
16.
yy ye
x
dA,
D is the triangular region with vertices 共0, 0兲,
D
ⱍ
D 苷 {共x, y兲 0 x 1, 0 y sx }
D
yy
e y dA,
ⱍ
共2, 4兲, and 共6, 0兲
■
17–24
■
■
■
■
■
■
■
■
■
■
■
Find the volume of the given solid.
17. Under the paraboloid z 苷 x 2 y 2 and above the region
ⱍ
ⱍ
D 苷 兵共x, y兲 0 y 1, 0 x y其
bounded by y 苷 x 2 and x 苷 y 2
18. Under the paraboloid z 苷 3x 2 y 2 and above the region
bounded by y 苷 x and x 苷 y 2 y
19. Under the surface z 苷 xy and above the triangle with
vertices 共1, 1兲, 共4, 1兲, and 共1, 2兲
yy x cos y dA,
yy x sy
3
●
D is bounded by the circle with center the origin and radius 2
■
■
2
yy y
●
D
■
D 苷 兵共x, y兲 0 x 2, x y x其
dA,
●
D
s1 v 2 du dv
2y
dA,
x2 1
D
14.
共x 2 y兲 dy dx
D is bounded by y 苷 0, y 苷 x 2, x 苷 1
20. Bounded by the paraboloid z 苷 x 2 y 2 4 and the planes
x 苷 0, y 苷 0, z 苷 0, x y 苷 1
D
12.
●
D is the triangular region with vertices (0, 2), (1, 1), and 共3, 2兲
D 苷 兵共x, y兲 1 x 2, 0 y 2x其
yy
●
xy dx dy
4y
dA,
3
x 2
yy
●
D
Evaluate the double integral.
D
11.
●
13.
sx dx dy
ey
兾2
0
D
9.
●
yy
D
8.
●
2.
■
7.
●
共x 2y兲 dy dx
y
0
7–16
●
Evaluate the iterated integral.
1
■
●
2
x dA,
2
ⱍ
D 苷 兵共x, y兲 0 y 1, 0 x y其
21. Bounded by the planes x 苷 0, y 苷 0, z 苷 0, and
xyz苷1
■
■
862
CHAPTER 12 MULTIPLE INTEGRALS
22. Bounded by the cylinder y 2 z 2 苷 4 and the planes x 苷 2y,
41–42 ■ Express D as a union of regions of type I or type II
and evaluate the integral.
x 苷 0, z 苷 0 in the first octant
23. Bounded by the cylinder x 2 y 2 苷 1 and the planes y 苷 z,
41.
x 苷 0, z 苷 0 in the first octant
■
■
■
■
■
1
D
24. Bounded by the cylinders x y 苷 r and y z 苷 r
■
y
yy x 2 dA
2
2
■
■
2
2
■
■
2
■
■
(1, 1)
D
2
■
_1
1
0
x
; 25. 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.
_1
42.
; 26. Find the approximate volume of the solid in the first octant
27–28
x=1
x=_1
D
■
Use a computer algebra system to find the exact
volume of the solid.
■
z 苷 8 x 2y and inside the cylinder x y 苷 1
■
■
■
■
■
■
■
■
■
■
43–44
28. Between the paraboloids z 苷 2x 2 y 2 and
2
■
x
y=_1
bounded by the curves y 苷 x 3 x and y 苷 x 2 x
for x 0
2
x=¥
0
27. Under the surface z 苷 x 3 y 4 xy 2 and above the region
2
y=1+≈
D
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.)
CAS
y
yy xy dA
■
■
■
■
■
■
■
■
■
■
■
■
Use Property 11 to estimate the value of the integral.
2
■
■
43.
yy sx
3
D 苷 关0, 1兴 关0, 1兴
y 3 dA,
■
D
29–34
■ Sketch the region of integration and change the order
of integration.
29.
31.
33.
1
x
y0 y0
2
f 共x, y兲 dy dx
ln x
f 共x, y兲 dy dx
y1 y0
4
2
sin x
y0 y0
1
32.
f 共x, y兲 dx dy
y0 yy兾2
兾2
30.
2y
y0 yy
1
2
44.
x 2y 2
dA,
D
D is the disk with center the origin and radius 12
f 共x, y兲 dy dx
■
■
■
■
■
■
■
35–40
■
1
3
1
1
3
9
■
■
■
■
■
■
■
■
y0 y3y e
36.
y0 ysy sx 3 1 dx dy
■
1
yy f 共x, y兲 dA 苷 y y
■
y0 yarcsin y cos x s1 cos 2x dx dy
■
8
2
■
■
4
3y
0
f 共x, y兲 dx dy
x 2 y 2 1, by first identifying the integral as the volume
of a solid.
CAS
dx dy
■
y
49. Compute xxD s1 x 2 y 2 dA, where D is the disk
兾2
3
3
1
is the region bounded by the square with vertices 共5, 0兲
and 共0, 5兲.
x 3 sin共 y 3 兲 dy dx
y0 ysy e x
f 共x, y兲 dx dy y
47. Evaluate xxD 共x 2 tan x y 3 4兲 dA, where
y cos共x 兲 dx dy
39.
40.
2y
0
48. Use symmetry to evaluate xxD 共2 3x 4y兲 dA, where D
y0 yx
1
■
Sketch the region D and express the double integral as an
iterated integral with reversed order of integration.
2
38.
2
■
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.]
y0 yy
1
■
D
dx dy
37.
1
■
ⱍ
35.
2
■
iterated integrals was obtained as follows:
Evaluate the integral by reversing the order of
integration.
x2
■
46. In evaluating a double integral over a region D, a sum of
兾4
0
■
■
45. Prove Property 11.
f 共x, y兲 dx dy
y0 yarctan x f 共x, y兲 dy dx
34.
yy e
■
■
■
■
■
■
■
■
■
50. 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.)
■
◆
SECTION 12.4 DOUBLE INTEGRALS IN POLAR COORDINATES
Now we use Formula 64 in the Table of Integrals:
▲ Instead of using tables, we
could have used the identity
1
cos2 苷 2 共1 cos 2 兲 twice.
V苷6y
兾2
0
Exercises
●
●
●
●
●
●
●
●
●
●
y
0
2 x
y
2 x
y
3
4.
2
x
2
●
●
●
●
●
●
●
●
●
●
●
xxD xy dA,
xxR sx 2 y 2 dA,
2
ⱍ
2
xxD ex y
12.
xxR ye x dA,
13.
xxR arctan共 y兾x兲 dA,
where R 苷 兵共x, y兲 1 x 2 y 2 4, x y x其
14.
xxD x dA,
dA, where D is the region bounded by the
semicircle x 苷 s4 y 2 and the y-axis
where R is the region in the first quadrant
enclosed by the circle x 2 y 2 苷 25
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
■
15–21
R
1
0
●
11.
■
R
●
where R 苷 兵共x, y兲 1 x 2 y 2 9, y 0其
R
0
●
where D is the disk with center the origin and radius 3
10.
R
3.
9.
2
2
●
■ Evaluate the given integral by changing to polar
coordinates.
y
2.
兾2
0
]
9–14
1–6 ■ A region R is shown. Decide whether to use polar coordinates 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.
1.
●
[
cos2 d 苷 6 12 14 sin 2
1 3
ⴢ
苷
2 2
2
苷6ⴢ
12.4
867
■
■
■
■
■
■
■
■
■
■
■
■
Use polar coordinates to find the volume of the given
solid.
0
1
3 x
15. Under the paraboloid z 苷 x 2 y 2 and above the
disk x 2 y 2 9
16. Inside the sphere x 2 y 2 z 2 苷 16 and outside the
cylinder x 2 y 2 苷 4
y
5.
6.
5
17. A sphere of radius a
y
18. Bounded by the paraboloid z 苷 10 3x 2 3y 2 and the
plane z 苷 4
2
R
2
19. Above the cone z 苷 sx 2 y 2 and below the sphere
R
2
x 2 y 2 z2 苷 1
5
0
x
0
x
2
20. Bounded by the paraboloids z 苷 3x 2 3y 2 and
z 苷 4 x2 y2
21. Inside both the cylinder x 2 y 2 苷 4 and the ellipsoid
4x 2 4y 2 z 2 苷 64
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
■
22. (a) A cylindrical drill with radius r1 is used to bore a hole
7–8 ■ Sketch the region whose area is given by the integral and
evaluate the integral.
7.
■
2
y y
7
4
■
r dr d
■
■
8.
■
■
■
兾2
y y
0
■
4 cos 0
■
r dr d
■
■
■
■
through the center of a sphere of radius r2 . 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 r1 or r2 .
■
■
868
23–24
■
CHAPTER 12 MULTIPLE INTEGRALS
32. (a) We define the improper integral (over the entire plane ⺢ 2 兲
Use a double integral to find the area of the region.
23. One loop of the rose r 苷 cos 3
I 苷 yy e共x
24. The region enclosed by the cardioid r 苷 1 sin ■
■
25–28
■
■
■
■
■
■
■
■
■
⺢
■
al
■
Evaluate the iterated integral by converting to polar
coordinates.
25.
26.
27.
■
1
yy
0
s1x
2
ex
2
y 2
a
y y
2
sa y
0
yy
s4y 2
s4y
0
■
2
2
28.
■
e共x
2
y 2 兲
dy dx
dA
e共x
2
y 2 兲
dA 苷 (b) An equivalent definition of the improper integral in
part (a) is
x 2 y 2 dx dy
■
Da
dy dx
yy
0
2
■
y
where Da is the disk with radius a and center the origin.
Show that
共x 2 y 2 兲3兾2 dx dy
■
y y
2
dA 苷 y
yy e
0
a
y 2 兲
共x 2y 2 兲
苷 lim
■
2
2
■
■
s2xx 2
0
■
sx 2 y 2 dy dx
■
■
■
共x 2y 2 兲
yy e
dA 苷 lim
al
⺢2
■
共x 2y 2 兲
yy e
dA
Sa
where Sa is the square with vertices 共a, a兲. Use this
to show that
29. A swimming pool is circular with a 40-ft diameter. The
depth 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.
y
30. An agricultural sprinkler distributes water in a circular pat-
2
ex dx y
2
ey dy 苷 (c) Deduce that
tern 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) 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.
y
2
ex dx 苷 s
(d) By making the change of variable t 苷 s2 x, show that
y
ex
2
兾2
dx 苷 s2
(This is a fundamental result for probability and
statistics.)
31. Use polar coordinates to combine the sum
33. Use the result of Exercise 32 part (c) to evaluate the follow1
y y
1兾s2
x
s1x 2
xy dy dx y
s2
1
y
x
0
xy dy dx y
2
s2
y
s4x 2
xy dy dx
into one double integral. Then evaluate the double integral.
12.5
ing integrals.
0
Applications of Double Integrals
(a)
●
y
0
2
x 2ex dx
●
●
(b)
●
●
y
0
●
sx ex dx
●
●
●
●
●
We have already seen one application of double integrals: computing volumes.
Another geometric application is finding areas of surfaces and this will be done in the
next section. In this section we explore physical applications such as computing mass,
electric charge, center of mass, and moment of inertia. We will see that these physical
ideas are also important when applied to probability density functions of two random
variables.
Density and Mass
In Chapter 6 we were able to use single integrals to compute moments and the center
of mass of a thin plate or lamina with constant density. But now, equipped with the
double integral, we can consider a lamina with variable density. Suppose the lamina
occupies a region D of the xy-plane and its density (in units of mass per unit area) at
◆
SECTION 12.5 APPLICATIONS OF DOUBLE INTEGRALS
12.5
Exercises
●
●
●
●
●
●
●
●
●
●
1. Electric charge is distributed over the rectangle 1 x 3,
●
CAS
0 y 2 so that the charge density at 共x, y兲 is
共x, y兲 苷 2xy y 2 (measured in coulombs per square
meter). Find the total charge on the rectangle.
共x, y兲 苷 cxy
■
■
■
■
■
■
■
■
9. A lamina occupies the part of the disk x 2 y 2 1 in the
first quadrant. Find its center of mass if the density at any
point is proportional to its distance from the x-axis.
f 共x, y兲 苷
11. Find the center of mass of a lamina in the shape of an
12. A lamina occupies the region inside the circle x 2 y 2 苷 2y
●
共x, y兲 苷 xy
■
■
■
■
■
■
■
再
4xy if 0 x 1, 0 y 1
0
otherwise
再
0.1e共0.5x0.2y兲 if x 0, y 0
0
otherwise
22. (a) A lamp has two bulbs of a type with an average lifetime
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.
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.
13. Find the moments of inertia Ix, Iy, I0 for the lamina of
Exercise 3.
14. Find the moments of inertia Ix, Iy, I0 for the lamina of
Exercise 10.
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?
●
(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.
isosceles 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.
16. Consider a square fan blade with sides of length 2 and the
●
Cx共1 y兲 if 0 x 1, 0 y 2
0
otherwise
function
density at any point is proportional to the square of its distance from the origin.
Exercise 7.
●
21. Suppose X and Y are random variables with joint density
10. Find the center of mass of the lamina in Exercise 9 if the
15. Find the moments of inertia Ix, Iy, I0 for the lamina of
●
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.
共x, y兲 苷 x
■
●
再
f 共x, y兲 苷
line y 苷 x 2; 共x, y兲 苷 3
■
■
20. (a) Verify that
7. D is bounded by the parabola x 苷 y 2 and the
■
●
(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.
共x, y兲 苷 y
■
■
f 共x, y兲 苷
6. D is bounded by the parabola y 苷 9 x 2 and the x-axis;
■
●
■ Use a computer algebra system to find the mass, center of mass, and moments of inertia of the lamina that occupies
the region D and has the given density function.
and Y is
共x, y兲 苷 x y
■
●
19. The joint density function for a pair of random variables X
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 sx 2 y 2
3. D 苷 兵共x, y兲 0 x 2, 1 y 1其; 共x, y兲 苷 xy 2
ⱍ
●
17–18
■
■
8. D 苷 兵共x, y兲 0 y cos x, 0 x 兾2其;
●
18. D is enclosed by the cardioid r 苷 1 cos ;
Find the mass and center of mass of the lamina that
occupies the region D and has the given density function .
4.
●
ⱍ
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.
ⱍ
D 苷 兵共x, y兲 ⱍ 0 x a, 0 y b其;
●
17. D 苷 兵共x, y兲 0 y sin x, 0 x 其 ;
2. Electric charge is distributed over the disk
3–8
●
877
CAS
23. 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.
■
■
878
CHAPTER 12 MULTIPLE INTEGRALS
24. Xavier and Yolanda both have classes that end at noon and
tance between them. Consider a circular city of radius 10 mi
in which the population is uniformly distributed. For an
uninfected individual at a fixed point A共x 0 , y0 兲, assume that
the probability function is given by
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
再
ex if x 0
f1共x兲 苷
0
if x 0
再
f 共P兲 苷 201 关20 d共P, A兲兴
y if 0 y 10
f2共 y兲 苷
0
otherwise
1
50
where d共P, A兲 denotes the distance between P and A.
(a) Suppose the exposure of a person to the disease is the
sum of the probabilities of catching the disease from all
members of the population. Assume that the infected
people are uniformly distributed throughout the city,
with k infected individuals per square mile. Find a
double integral that represents the exposure of a person
residing at A.
(b) Evaluate the integral for the case in which A is the center of the city and for the case in which A is located on
the edge of the city. Where would you prefer to live?
(Xavier arrives sometime after noon and is more likely to
arrive promptly than late. Yolanda always arrives by
12:10 P.M. 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.
25. When studying the spread of an epidemic, we assume that
the probability that an infected individual will spread the
disease to an uninfected individual is a function of the dis-
12.6
Surface Area
●
1
(u¸, √¸)
√=√¸
●
●
●
●
●
●
●
●
●
●
●
●
●
r共u, v兲 苷 x共u, v兲 i y共u, v兲 j z共u, v兲 k
x 苷 x共u, v兲
0
y 苷 y共u, v兲
z 苷 z共u, v兲
u
where 共u, v兲 varies throughout a region D in the uv-plane.
We will find the area of S by dividing S into patches and approximating the area of
each patch by the area of a piece of a tangent plane. So first let’s recall from Section 11.4 how to find tangent planes to parametric surfaces.
Let P0 be a point on S with position vector r共u0 , v0 兲. If we keep u constant by putting u 苷 u0 , then r共u0 , v兲 becomes a vector function of the single parameter v and
defines a grid curve C1 lying on S. (See Figure 1.) The tangent vector to C1 at P0 is
obtained by taking the partial derivative of r with respect to v :
r
z
P¸
ru
r√
C¡
FIGURE 1
●
or, equivalently, by parametric equations
u=u¸
D
x
●
In this section we apply double integrals to the problem of computing the area of a
surface. We start by finding a formula for the area of a parametric surface and then, as
a special case, we deduce a formula for the surface area of the graph of a function of
two variables.
We recall from Section 10.5 that a parametric surface S is defined by a vectorvalued function of two parameters
√
0
●
2
rv 苷
x
y
z
共u0 , v0兲 i 共u0 , v0兲 j 共u0 , v0兲 k
v
v
v
Similarly, if we keep v constant by putting v 苷 v0 , we get a grid curve C2 given by
r共u, v0 兲 that lies on S, and its tangent vector at P0 is
C™
y
3
ru 苷
x
y
z
共u0 , v0兲 i 共u0 , v0兲 j 共u0 , v0兲 k
u
u
u
SECTION 12.6 SURFACE AREA
surface area formula in Equation 6
and the arc length formula
L苷
y
a
冑 冉 冊
1
2
dy
dx
881
Thus, the surface area formula in Definition 4 becomes
▲ Notice the similarity between the
b
◆
A共S兲 苷
6
dx
yy
D
from Section 6.3.
冑 冉 冊 冉 冊
z
x
1
z
y
2
2
dA
EXAMPLE 2 Find the area of the part of the paraboloid z 苷 x 2 y 2 that lies under the
plane z 苷 9.
z
SOLUTION The plane intersects the paraboloid in the circle x 2 y 2 苷 9, z 苷 9. There-
9
fore, the given surface lies above the disk D with center the origin and radius 3. (See
Figure 4.) Using Formula 6, we have
A苷
yy
D
冑 冉 冊 冉 冊
z
x
1
z
y
2
2
dA 苷 yy s1 共2x兲 2 共2y兲 2 dA
D
苷 yy s1 4共x 2 y 2 兲 dA
D
D
y
3
x
Converting to polar coordinates, we obtain
FIGURE 4
A苷y
2
0
y
3
0
s1 4r 2 r dr d 苷 y
2
0
3
苷 2 ( 18 ) 23 共1 4r 2 兲3兾2 ]0 苷
3
d y rs1 4r 2 dr
0
(37s37 1)
6
A common type of surface is a surface of revolution S obtained by rotating the
curve y 苷 f 共x兲, a x b, about the x-axis, where f 共x兲 0 and f is continuous. In
Exercise 23 you are asked to use a parametric representation of S and Definition 4 to
prove the following formula for the area of a surface of revolution:
A 苷 2
7
y
b
f 共x兲s1 关 f 共x兲兴 2 dx
a
12.6
1–12
■
Exercises
●
●
●
●
●
●
●
●
●
Find the area of the surface.
1. The part of the plane z 苷 2 3x 4y that lies above the
rectangle 关0, 5兴 关1, 4兴
2. The part of the plane 2x 5y z 苷 10 that lies inside the
cylinder x 2 y 2 苷 9
3. The part of the plane 3x 2y z 苷 6 that lies in the
first octant
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
6. The part of the surface z 苷 x y 2 that lies above the tri-
angle with vertices 共0, 0兲, 共1, 1兲, and 共0, 1兲
7. The surface with parametric equations x 苷 uv, y 苷 u v,
z 苷 u v, u 2 v 2 1
8. The helicoid (or spiral ramp) with vector equation
r共u, v兲 苷 u cos v i u sin v j v k, 0 u 1, 0 v 9. The part of the surface y 苷 4x z 2 that lies between the
planes x 苷 0, x 苷 1, z 苷 0, and z 苷 1
4. The part of the plane with vector equation
r共u, v兲 苷 具1 v, u 2v, 3 5u v 典 that is given by
0 u 1, 0 v 1
10. The part of the paraboloid x 苷 y 2 z 2 that lies inside the
5. The part of the hyperbolic paraboloid z 苷 y 2 x 2 that lies
11. The part of the surface z 苷 xy that lies within the cylinder
between the cylinders x 2 y 2 苷 1 and x 2 y 2 苷 4
●
cylinder y 2 z 2 苷 9
x2 y2 苷 1
■
882
CHAPTER 12 MULTIPLE INTEGRALS
12. The surface z 苷 3 共x 3兾2 y 3兾2 兲, 0 x 1, 0 y 1
2
■
■
■
■
■
■
■
■
■
■
■
■
(b) Use the parametric equations in part (a) to graph the
hyperboloid for the case a 苷 1, b 苷 2, c 苷 3.
(c) Set up, but do not evaluate, a double integral for the surface area of the part of the hyperboloid in part (b) that
lies between the planes z 苷 3 and z 苷 3.
;
■
13. (a) Use the Midpoint Rule for double integrals (see Sec-
CAS
tion 12.1) with four squares to estimate the surface area
of the portion of the paraboloid z 苷 x 2 y 2 that lies
above the square 关0, 1兴 关0, 1兴.
(b) Use a computer algebra system to approximate the surface area in part (a) to four decimal places. Compare
with the answer to part (a).
21. Find the area of the part of the sphere x 2 y 2 z 2 苷 4z
that lies inside the paraboloid z 苷 x 2 y 2.
22. The figure shows the surface created when the cylinder
y 2 z 2 苷 1 intersects the cylinder x 2 z 2 苷 1. Find the
area of this surface.
14. (a) Use the Midpoint Rule for double integrals with
CAS
CAS
m 苷 n 苷 2 to estimate the area of the surface
z 苷 xy x 2 y 2, 0 x 2, 0 y 2.
(b) Use a computer algebra system to approximate the surface area in part (a) to four decimal places. Compare
with the answer to part (a).
surface z 苷 共1 x 兲兾共1 y 兲 that lies above the square
2
surface.
17. Find the exact area of the surface z 苷 1 2x 3y 4y 2,
1 x 4, 0 y 1.
18. (a) Set up, but do not evaluate, a double integral for the area
of the surface with parametric equations x 苷 au cos v,
y 苷 bu sin v, z 苷 u 2, 0 u 2, 0 v 2.
CAS
of revolution (see Equations 10.5.3) to derive Formula 7.
16. Find, to four decimal places, the area of the part of the
ⱍ x ⱍ ⱍ y ⱍ 1. Illustrate by graphing this part of the
;
(b) Eliminate the parameters to show that the surface is an
elliptic paraboloid and set up another double integral for
the surface area.
(c) Use the parametric equations in part (a) with a 苷 2 and
b 苷 3 to graph the surface.
(d) For the case a 苷 2, b 苷 3, use a computer algebra
system to find the surface area correct to four decimal
places.
24–25
■ Use Formula 7 to find the area of the surface obtained
by rotating the given curve about the x-axis.
24. y 苷 x 3,
■
■
25. y 苷 sx,
0x2
■
■
■
■
■
hyperboloid of one sheet.
■
4x9
■
■
■
z-axis the circle in the xz-plane with center 共b, 0, 0兲 and
radius a b. Parametric equations for the torus are
x 苷 b cos a cos cos y 苷 b sin a cos sin z 苷 a sin where and are the angles shown in the figure. Find the
surface area of the torus.
z
(x, y, z)
represent an ellipsoid.
(b) Use the parametric equations in part (a) to graph the
ellipsoid for the case a 苷 1, b 苷 2, c 苷 3.
(c) Set up, but do not evaluate, a double integral for the surface area of the ellipsoid in part (b).
20. (a) Show that the parametric equations x 苷 a cosh u cos v,
y 苷 b cosh u sin v, z 苷 c sinh u, represent a
■
26. The figure shows the torus obtained by rotating about the
19. (a) Show that the parametric equations x 苷 a sin u cos v,
y 苷 b sin u sin v, z 苷 c cos u, 0 u , 0 v 2,
;
y
23. Use Definition 4 and the parametric equations for a surface
2
CAS
x
15. Find the area of the surface with vector equation
r共u, v兲 苷 具cos 3u cos 3v, sin 3u cos 3v, sin 3v 典 , 0 u ,
0 v 2. State your answer correct to four decimal
places.
CAS
z
0
¨
x
(b, 0, 0)
å
y
■
■
890
CHAPTER 12 MULTIPLE INTEGRALS
Then, if the density is 共x, y, z兲 苷 , the mass is
m 苷 yyy dV 苷 y
E
苷y
1
y
1
2
苷
1
1
y y
1
y
1
x
dz dx dy
0
x dx dy 苷 y2
1
y2
y 冋 册
x苷1
x2
2
1
1
dy
x苷y 2
1
共1 y 4 兲 dy 苷 y 共1 y 4 兲 dy
1
0
冋 册
1
y5
苷 y
5
苷
0
4
5
Because of the symmetry of E and about the xz-plane, we can immediately say
that Mxz 苷 0 and, therefore, y 苷 0. The other moments are
Myz 苷 yyy x dV 苷 y
E
苷y
1
y
1
苷
2
3
1
y2
y
1
0
1
1
1
2
y y
y
1
0
y
共x, y, z兲 苷
Exercises
●
●
●
●
●
●
●
●
1. Evaluate the integral in Example 1, integrating first with
●
●
●
●
3–6
respect to z , then x, and then y.
2. Evaluate the integral xxxE 共xz y 3兲 dV , where
ⱍ
E 苷 兵共x, y, z兲 1 x 1, 0 y 2, 0 z 1其
using three different orders of integration.
■
x
0
1
y y
冋册
y2
冉
y
1
2
3
1
z
xz
yyy
■
1
y7
7
苷
0
4
7
z dz dx dy
2
1
1
1
y2
y y
x 2 dx dy
2
7
●
冊
●
苷 ( 57 , 0, 145 )
●
●
●
●
●
●
●
●
●
Evaluate the iterated integral.
5.
0
●
dy
x苷y 2
y
dx dy 苷
●
yyy
3
x
0
冋册
冋 册
x苷1
x3
3
Myz Mxz Mxy
,
,
m
m
m
3.
0
1
z苷0
●
■
x dz dx dy
z苷x
z2
2
共1 y 6 兲 dy 苷
Therefore, the center of mass is
12.7
1
1
3
y2
共1 y 6 兲 dy 苷
E
苷
1
y y
x 2 dx dy 苷 Mxy 苷 yyy z dV 苷 y
苷
1
1
0
1
0
0
6xz dy dx dz
s1z 2
0
■
ze y dx dz dy
■
■
■
■
1
2x
4.
yy y
6.
yyy
0
1
0
■
x
z
y
0
0
■
y
0
2xyz dz dy dx
2
zey dx dy dz
■
■
■
■
◆
SECTION 12.7 TRIPLE INTEGRALS
7–14
7.
■
891
21–22
■ Use the Midpoint Rule for triple integrals (Exercise 20) to estimate the value of the integral. Divide B into
eight sub-boxes of equal size.
Evaluate the triple integral.
xxxE 2x dV , where
ⱍ
E 苷 {共x, y, z兲 0 y 2, 0 x s4 y 2, 0 z y}
21.
兲 dV , where
E 苷 兵共x, y, z兲 0 x 1, 0 y x, x z 2x其
5
8.
xxxE yz cos共x
9.
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
ⱍ
ⱍ
xxxE 6xy dV ,
10.
xxxE xz dV ,
11.
xxxE z dV ,
12.
xxxE 共x 2y兲 dV , where E is bounded by the parabolic
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
22.
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
where E is the solid tetrahedron with vertices
共0, 0, 0兲, 共0, 1, 0兲, 共1, 1, 0兲, and 共0, 1, 1兲
■
■
23–24
where E is bounded by the planes x 苷 0, y 苷 0,
z 苷 0, y z 苷 1, and x z 苷 1
■
■
■
■
■
■
■
■
■
■
■
■
Sketch the solid whose volume is given by the iterated
integral.
1
1x
23.
yy y
24.
yy y
cylinder y 苷 x and the planes x 苷 z, x 苷 y, and z 苷 0
0
0
22z
0
dy dz dx
2
13.
xxxE x dV , where E is bounded by the paraboloid
x 苷 4y 4z and the plane x 苷 4
2
14.
■
■
xxxE z dV , where E is bounded by the cylinder y z 苷 9
and the planes x 苷 0, y 苷 3x, and z 苷 0 in the first octant
15–18
■
■
2y
4y 2
0
0
■
■
■
■
■
■
■
2
■
■
■
■
■
■
15. The tetrahedron enclosed by the coordinate planes and the
plane 2x y z 苷 4
16. The solid bounded by the elliptic cylinder 4x 2 z 2 苷 4 and
the planes y 苷 0 and y 苷 z 2
17. The solid bounded by the cylinder x 苷 y and the planes
■
■
■
■
■
■
■
19. (a) Express the volume of the wedge in the first octant that
CAS
is 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 the back Reference
Pages) or a computer algebra system to find the exact
value of the triple integral in part (a).
■
■
CAS
■
z 苷 y 2x
27. z 苷 0,
z 苷 y,
x 苷1y
28. 9x 2 4y 2 z 2 苷 1
■
■
■
■
■
■
■
■
■
■
1
1
yy y
sx
1y
0
■
f 共x, y, z兲 dz dy dx
■
Rewrite this integral as an equivalent iterated integral in the
five other orders.
z
1
20. (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 xi, yj, zk 兲 of the box Bijk. Use the Midpoint Rule to
2
2
2
estimate xxxB ex y z dV , where B is the cube defined
by 0 x 1, 0 y 1, 0 z 1. Divide B into
eight cubes of equal size.
(b) Use a computer algebra system to approximate the integral in part (a) correct to two decimal places. Compare
with the answer to part (a).
■
2
0
■
■
y苷6
y 苷 2,
2
■
y 苷 0,
x 苷 0,
18. The solid enclosed by the paraboloids z 苷 x 2 y 2 and
■
■
29. The figure shows the region of integration for the integral
z 苷 0 and x z 苷 1
■
■
26. z 苷 0,
■
2
z 苷 18 x y
■
■ Express the integral xxx f 共x, y, z兲 dV as an iterated
E
integral in six different ways, where E is the solid bounded by
the given surfaces.
25. x 2 z 2 苷 4,
Use a triple integral to find the volume of the given
2
■
25–28
solid.
■
dx dz dy
2
2
■
2
0
z=1-y
0
1
y=œ„
x
x
y
■
■
892
CHAPTER 12 MULTIPLE INTEGRALS
30. The figure shows the region of integration for the integral
1
yy
1x 2
0
0
y
1x
0
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
f 共x, y, z兲 dy dz dx
Rewrite this integral as an equivalent iterated integral in the
five other orders.
CAS
40. If E is the solid of Exercise 14 with density function
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
z
1
z=1-≈
41. Find the moments of inertia for a cube of constant density k
and side length L if one vertex is located at the origin and
three edges lie along the coordinate axes.
0
1
1
x
31–32
42. Find the moments of inertia for a rectangular brick with
y
dimensions a, b, and c, mass M , and constant density if the
center of the brick is situated at the origin and the edges are
parallel to the coordinate axes.
y=1-x
43. The joint density function for random variables X , Y , and Z
■
Write five other iterated integrals that are equal to the
given iterated integral.
1
1
y
31.
yyy
32.
yy y
■
0
y
0
0
■
f 共x, y, z兲 dz dx dy
0
x2
1
is 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.
y
0
■
f 共x, y, z兲 dz dy dx
■
■
44. Suppose X , Y , and Z are random variables with joint density
■
■
■
■
■
■
■
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.
■
33–36
■ Find the mass and center of mass of the given solid E
with the given density function .
共x, y, z兲 苷 2
33. E is the solid of Exercise 9;
45–46 ■ The average value of a function f 共x, y, z兲 over a solid
region E is defined to be
34. E is bounded by the parabolic cylinder z 苷 1 y 2 and the
planes x z 苷 1, x 苷 0, and z 苷 0; 共x, y, z兲 苷 4
fave 苷
35. E is the cube given by 0 x a, 0 y a, 0 z a;
共x, y, z兲 苷 x y z
2
2
2
z 苷 0, x y z 苷 1; 共x, y, z兲 苷 y
■
37–38
■
■
■
■
■
■
■
■
■
■
■
Set up, but do not evaluate, integral expressions for
(a) the mass, (b) the center of mass, and (c) the moment of inertia about the z -axis.
CAS
■
■
■
■
■
■
■
■
■
■
x y 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
2
■
■
■
■
■
■
■
■
■
47. Find the region E for which the triple integral
■
■
39. Let E be the solid in the first octant bounded by the cylinder
2
46. Find the average value of the function
f 共x, y, z兲 苷 x y z over the tetrahedron with vertices
共0, 0, 0兲, 共1, 0, 0兲, 共0, 1, 0兲, and 共0, 0, 1兲.
共x, y, z兲 苷 sx 2 y 2 z 2
■
the cube with side length L that lies in the first octant with
one vertex at the origin and edges parallel to the coordinate
axes.
共x, y, z兲 苷 x 2 y 2 z 2
38. The hemisphere x 2 y 2 z 2 1, z 0;
■
E
45. Find the average value of the function f 共x, y, z兲 苷 xyz over
■
37. The solid of Exercise 13;
yyy f 共x, y, z兲 dV
where V共E 兲 is the volume of E. For instance, if is a density
function, then ave is the average density of E.
36. E is the tetrahedron bounded by the planes x 苷 0, y 苷 0,
■
1
V共E兲
yyy 共1 x
E
is a maximum.
2
2y 2 3z 2 兲 dV
■
■
■
■
898
CHAPTER 12 MULTIPLE INTEGRALS
Figure 11 shows how E is swept out if we integrate first with respect to , then ,
and then . The volume of E is
V共E兲 苷 yyy dV 苷 y
2
0
兾4
y y
0
0
E
苷y
2
0
2
苷
3
d
y
y
兾4
0
兾4
0
sin 1–4
●
●
●
●
●
●
●
●
●
●
2
4
4
r dz d dr
3.
y y y
4.
■
5–6
0
0
r
兾6
兾2
0
0
2
3
0
y y y
0
兾2
■
■
2
1
2.
■
■
兾2
2
y yy
0
9r 2
0
0
●
●
●
●
●
¨ varies from 0 to 2π.
●
●
●
●
●
■
■
z
●
●
●
●
●
z
■
■
y
r dz dr d
■
■
Set up the triple integral of an arbitrary continuous function f 共x, y, z兲 in cylindrical or spherical coordinates over the
solid shown.
■
■
7–14
■
■
■
x
2
■
■
y
■
■
■
■
■
■
■
Use cylindrical coordinates.
7. 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.
8. Evaluate xxxE 共x 3 xy 2 兲 dV , where E is the solid in the first
3
octant that lies beneath the paraboloid z 苷 1 x 2 y 2.
9. Evaluate xxxE y dV , where E is the solid that lies between
2
x
0
1
■
5.
8
苷
x
6.
2 sin d d d
■
册
兾4
z
2 sin d d d
■
冋
cos 4
4
y
●
Sketch the solid whose volume is given by the integral
and evaluate the integral.
yy y
d
苷0
˙ varies from 0 to π/4 while
¨ is constant.
■
1.
苷cos 2
sin cos d 苷
3
x
y
∏ varies from 0 to cos ˙ while
˙ and ¨ are constant.
Exercises
冋册
3
3
z
x
12.8
2 sin d d d
3
z
FIGURE 11
cos y
the cylinders x 2 y 2 苷 1 and x 2 y 2 苷 4, above the
xy-plane, and below the plane z 苷 x 2.
◆
SECTION 12.8 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
24. Find the mass and center of mass of a solid hemisphere of
10. Evaluate xxxE xz dV , where E is bounded by the planes
z 苷 0, z 苷 y, and the cylinder x 2 y 2 苷 1 in the halfspace y 0.
radius a if the density at any point is proportional to its
distance from the base.
■
11. Evaluate xxxE x 2 dV , where E is the solid that lies within
■
■
■
■
■
■
■
■
■
■
■
■
1
s1x 2
1
s1x 2
y y
16. Evaluate xxxH 共x y 兲 dV , where H is the hemispherical
2
3
y y
17. 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.
2 2
18. Evaluate xxxE xe 共x y z 兲 dV , where E is the solid that lies
between the spheres x y z 苷 1 and
x 2 y 2 z 2 苷 4 in the first octant.
2
2
19. Evaluate xxxE sx 2 y 2 z 2 dV , where E is bounded below
■
■
■
■
■
■
■
y
2x 2y 2
x 2y 2
共x 2 y 2 兲3兾2 dz dy dx
30. Evaluate the integral by changing to spherical coordinates:
region that lies above the xy-plane and below the sphere
x 2 y 2 z 2 苷 1.
2
■
■
0
CAS
s9y 2
0
y
s18x 2y 2
sx 2y 2
共x 2 y 2 z 2 兲 dz dx dy
31. In the Laboratory Project on page 699 we investigated the
family of surfaces 苷 1 15 sin m sin n that have been
used as models for tumors. The “bumpy sphere” with
m 苷 6 and n 苷 5 is shown. Use a computer algebra system
to find its volume.
by the cone 苷 兾6 and above by the sphere 苷 2.
20. 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.
21. (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).
22. 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.
23. (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.
■
29. Evaluate the integral by changing to cylindrical coordinates:
x 2 y 2 z 2 1.
2
■
(b) Use a computer to draw the torus.
;
15. Evaluate xxxB 共x 2 y 2 z 2 兲 dV , where B is the unit ball
2
■
28. (a) Find the volume enclosed by the torus 苷 sin .
■
Use spherical coordinates.
2
■
27. Evaluate xxxE z dV , where E lies above the paraboloid
■
15–24
■
z 苷 x 2 y 2 and below the plane z 苷 2y. Use either the
Table of Integrals (on the back Reference Pages) or a
computer algebra system to evaluate the integral.
loids z 苷 x 2 y 2 and z 苷 36 3x 2 3y 2.
(b) Find the centroid of the region E in part (a).
■
■
radius a by two planes that intersect along a diameter at an
angle of 兾6.
14. (a) Find the volume of the region E bounded by the parabo-
■
■
26. Find the volume of the smaller wedge cut from a sphere of
CAS
the paraboloid z 苷 4x 2 4y 2 and the plane z 苷 a 共a 0兲
if S has constant density K .
■
■
above the cone z 苷 sx 2 y 2 and below the sphere
x 2 y 2 z 2 苷 1.
13. Find the mass and center of mass of the solid S bounded by
■
■
25. Find the volume and centroid of the solid E that lies
r 苷 a cos 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.
■
■
■
Use cylindrical or spherical coordinates, whichever
seems more appropriate.
2
12. (a) Find the volume of the solid that the cylinder
;
■
25–28
the cylinder x y 苷 1, above the plane z 苷 0, and below
the cone z 2 苷 4x 2 4y 2.
2
899
32. Show that
y y y
sx 2 y 2 z 2 e共x
2
y2z2 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.)
900
■
CHAPTER 12 MULTIPLE INTEGRALS
much work was done in forming Mount Fuji if the
land was initially at sea level?
33. When studying the formation of mountain ranges, geolo-
gists 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
Applied
Project
Roller Derby
h
å
P
Suppose that a solid ball (a marble), a hollow ball (a squash ball), a solid cylinder (a steel
bar), and a hollow cylinder (a lead pipe) roll down a slope. Which of these objects reaches
the bottom first? (Make a guess before proceeding.)
To answer this question we consider a ball or cylinder with mass m, radius r , and moment
of inertia I (about the axis of rotation). If the vertical drop is h, then the potential energy at
the top is mth. Suppose the object reaches the bottom with velocity v and angular velocity
, so v 苷 r. The kinetic energy at the bottom consists of two parts: 21 mv 2 from translation
(moving down the slope) and 21 I 2 from rotation. If we assume that energy loss from rolling
friction is negligible, then conservation of energy gives
mth 苷 12 mv2 12 I 2
1. Show that
v2 苷
2th
1 I*
where I* 苷
I
mr 2
2. If y共t兲 is the vertical distance traveled at time t, then the same reasoning as used in
Problem 1 shows that v 2 苷 2ty兾共1 I*兲 at any time t. Use this result to show that y
satisfies the differential equation
dy
苷
dt
冑
2t
共sin 兲 sy
1 I*
where is the angle of inclination of the plane.
3. By solving the differential equation in Problem 2, show that the total travel time is
T苷
冑
2h共1 I*兲
t sin 2
This shows that the object with the smallest value of I* wins the race.
4. Show that I* 苷 2 for a solid cylinder and I* 苷 1 for a hollow cylinder.
1
5. Calculate I* for a partly hollow ball with inner radius a and outer radius r . Express your
answer in terms of b 苷 a兾r. What happens as a l 0 and as a l r ?
6. Show that I* 苷 5 for a solid ball and I* 苷 3 for a hollow ball. Thus, the objects finish in
2
2
the following order: solid ball, solid cylinder, hollow ball, hollow cylinder.
◆
SECTION 12.9 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
12.9
1–6
■
Exercises
●
●
●
●
●
●
●
●
●
Find the Jacobian of the transformation.
1. x 苷 u 4 v,
y 苷 3u 2v
2. x 苷 u 2 v 2,
y 苷 u 2 v2
3. x 苷
u
,
uv
y苷
4. x 苷 sin ,
5. x 苷 uv,
6. x 苷 e
■
●
■
v
■
■
■
■
■
■
■
■
■
■
■
■
7–10
■ Find the image of the set S under the given
transformation.
10. S is the disk given by u 2 v 2 1;
11–16
■
■
■
■
●
●
●
●
●
●
●
●
■
■
■
■
■
■
■
■
■
■
■
19.
■
■
20.
x 苷 au, y 苷 bv
■
■
■
■
■
Use the given transformation to evaluate the integral.
where R is the region bounded by the lines
2x y 苷 1, 2x y 苷 3, 3x y 苷 1, and 3x y 苷 2
x 2y
dA, where R is the parallelogram bounded
cos共x
y兲
R
by the lines y 苷 x, y 苷 x 1, x 2y 苷 0, and
x 2y 苷 2
yy
冉 冊
yx
dA, where R is the trapezoidal region
yx
R
with vertices 共1, 0兲, 共2, 0兲, 共0, 2兲, and 共0, 1兲
xxR 共3x 4y兲 dA,
where R is the region bounded by the
lines y 苷 x, y 苷 x 2, y 苷 2x, and y 苷 3 2x;
x 苷 13 共u v兲, y 苷 13 共v 2u兲
21.
12.
xxR 共x y兲 dA,
where R is the square with vertices 共0, 0兲,
共2, 3兲, 共5, 1兲, and 共3, 2兲; x 苷 2u 3v, y 苷 3u 2v
22.
xxR sin共9x 2 4y 2 兲 dA, where R is the region in the first
quadrant bounded by the ellipse 9x 2 4y 2 苷 1
13.
xxR x 2 dA,
23.
xxR e xy dA,
14.
where R is the region bounded by the ellipse
9x 4y 苷 36; x 苷 2u, y 苷 3v
2
xxR 共x 2 xy y 2 兲 dA,
where R is the region bounded
by the ellipse x xy y 2 苷 2;
x 苷 s2u s2兾3 v, y 苷 s2u s2兾3 v
2
15.
xxR xy dA,
■
xxR xy dA,
11.
2
●
■ Evaluate the integral by making an appropriate change
of variables.
9. S is the triangular region with vertices 共0, 0兲, 共1, 1兲, 共0, 1兲;
x 苷 u2, y 苷 v
■
●
19–23
8. S is the square bounded by the lines u 苷 0, u 苷 1, v 苷 0,
v 苷 1; x 苷 v, y 苷 u共1 v 2 兲
■
●
18. Evaluate xxxE x 2 y dV , where E is the solid of Exercise 17(a).
ⱍ
7. S 苷 兵共u, v兲 0 u 3, 0 v 2其;
x 苷 2u 3v, y 苷 u v
■
●
ellipsoid x 2兾a 2 y 2兾b 2 z 2兾c 2 苷 1. Use the transformation x 苷 au, y 苷 bv, z 苷 cw.
(b) Earth is not a perfect sphere; rotation has resulted in
flattening at the poles. So the shape can be approximated by an ellipsoid with a 苷 b 苷 6378 km and
c 苷 6356 km. Use part (a) to estimate the volume of
Earth.
z 苷 e uvw
,
●
17. (a) Evaluate xxxE dV, where E is the solid enclosed by the
z 苷 uw
uv
●
xy 苷 1, xy 苷 2, xy 2 苷 1, xy 2 苷 2; u 苷 xy, v 苷 xy 2.
Illustrate by using a graphing calculator or computer to
draw R.
y 苷 cos y苷e
,
●
2
; 16. xxR y dA, where R is the region bounded by the curves
uv
y 苷 vw,
uv
●
909
where R is the region in the first quadrant
bounded by the lines y 苷 x and y 苷 3x and the hyperbolas
xy 苷 1, xy 苷 3; x 苷 u兾v, y 苷 v
■
yy cos
where R is given by the inequality
ⱍxⱍ ⱍyⱍ 1
■
■
■
■
■
■
■
■
■
■
■
24. Let f be continuous on 关0, 1兴 and let R be the triangular
region with vertices 共0, 0兲, 共1, 0兲, and 共0, 1兲. Show that
yy f 共x y兲 dA 苷 y
1
0
R
u f 共u兲 du
■
910
12
■
CHAPTER 12 MULTIPLE INTEGRALS
Review
CONCEPT CHECK
1. Suppose f is a continuous function defined on a rectangle
R 苷 关a, b兴 关c, d 兴.
(a) Write an expression for a double Riemann sum of f .
If f 共x, y兲 0, what does the sum represent?
(b) Write the definition of xxR f 共x, y兲 dA as a limit.
(c) What is the geometric interpretation of xxR f 共x, y兲 dA if
f 共x, y兲 0? What if f takes on both positive and negative values?
(d) How do you evaluate xxR f 共x, y兲 dA?
(e) What does the Midpoint Rule for double integrals say?
(f) Write an expression for the average value of f .
2. (a) How do you define xxD f 共x, y兲 dA if D is a bounded
region that is not a rectangle?
(b) What is a type I region? How do you evaluate
xxD f 共x, y兲 dA if D is a type I region?
(c) What is a type II region? How do you evaluate
xxD f 共x, y兲 dA if D is a type II region?
(d) What properties do double integrals have?
3. How do you change from rectangular coordinates to polar
coordinates in a double integral? Why would you want to
do it?
4. If a lamina occupies a plane region D and has density func-
tion 共x, y兲, write expressions for each of the following in
terms of double integrals.
(a) The mass
(b) The moments about the axes
(c) The center of mass
(d) The moments of inertia about the axes and the origin
5. Let f be a joint density function of a pair of continuous ran-
dom variables X and Y .
(a) Write a double integral for the probability that X lies
between a and b and Y lies between c and d.
(b) What properties does f possess?
(c) What are the expected values of X and Y ?
6. Write an expression for the area of a surface S for each of
the following cases.
(a) S is a parametric surface given by a vector function
r共u, v兲, 共u, v兲 僆 D .
(b) S has the equation z 苷 f 共x, y兲, 共x, y兲 僆 D.
(c) S is the surface of revolution obtained by rotating the
curve y 苷 f 共x兲, a x b, about the x-axis.
7. (a) Write the definition of the triple integral of f over a
rectangular box B.
(b) How do you evaluate xxxB f 共x, y, z兲 dV ?
(c) How do you define xxxE f 共x, y, z兲 dV if E is a bounded
solid region that is not a box?
(d) What is a type 1 solid region? How do you evaluate
xxxE f 共x, y, z兲 dV if E is such a region?
(e) What is a type 2 solid region? How do you evaluate
xxxE f 共x, y, z兲 dV if E is such a region?
(f) What is a type 3 solid region? How do you evaluate
xxxE f 共x, y, z兲 dV if E is such a region?
8. Suppose a solid object occupies the region E and has den-
sity function 共x, y, z兲. Write expressions for each of the
following.
(a) The mass
(b) The moments about the coordinate planes
(c) The coordinates of the center of mass
(d) The moments of inertia about the axes
9. (a) How do you change from rectangular coordinates to
cylindrical coordinates in a triple integral?
(b) How do you change from rectangular coordinates to
spherical coordinates in a triple integral?
(c) In what situations would you change to cylindrical or
spherical coordinates?
10. (a) If a transformation T is given by x 苷 t共u, v兲,
y 苷 h共u, v兲, what is the Jacobian of T ?
(b) How do you change variables in a double integral?
(c) How do you change variables in a triple integral?
◆
CHAPTER 12 REVIEW
911
T R U E – FA L S E Q U I Z
Determine whether the statement is true or false. If it is true, explain why.
If it is false, explain why or give an example that disproves the statement.
1.
2.
2
y y
1
6
0
1
y y
1
6
0
1
0
x 2 sin共x y兲 dx dy 苷 y
e
x 2y 2
y
2
1
4.
4
1
y y (x
1
2
0
sy ) sin共x 2 y 2 兲 dx dy 9
5. The integral
x 2 sin共x y兲 dy dx
2
2
y yy
0
sin y dx dy 苷 0
2
16
3
y 2 dA 苷
D
2
r
dz dr d
represents the volume enclosed by the cone z 苷 sx 2 y 2
and the plane z 苷 2.
3. If D is the disk given by x 2 y 2 4, then
yy s4 x
0
6. The integral xxxE kr 3 dz dr d represents the moment of
inertia about the z -axis of a solid E with constant density k.
EXERCISES
■ Write xx f 共x, y兲 dA as an iterated integral, where R is
R
the region shown and f is an arbitrary continuous function on R.
1. A contour map is shown for a function f on the square
9–10
R 苷 关0, 3兴 关0, 3兴. Use a Riemann sum with nine terms to
estimate the value of xxR f 共x, y兲 dA. Take the sample points
to be the upper right corners of the squares.
y
4
9.
y
3
y
10.
4
R
R
2
10
9
_4
0
_2
2
4 x
4
5
6
■
■
■
■
■
■
11.
Describe the region whose area is given by the integral
3
0
2
兾6
0
0
y y y
0
1
3 x
2
■
1
1
0
x
共 y 2xe y 兲 dx dy
yy
7.
y yy
0
0
■
13.
0
1
0
4.
cos共x 2 兲 dy dx
5.
s1y 2
0
■
y sin x dz dy dx
■
■
■
■
r dr d
■
■
■
1
yy
0
1
0
1
■
ye xy dx dy
ex
yy
8.
yy y
■
x
0
1
0
■
1
y
sy
0
■
2 sin d d d
0
■
1
x
■
e x兾y dy dx
■
■
■
1
14.
■
■
■
yy
0
■
1
y2
■
y sin共x 2 兲 dx dy
■
■
Calculate the value of the multiple integral.
1
dA, where
共x
y兲2
R
R 苷 兵共x, y兲 0 x 1, 2 y 4其
15.
yy
16.
xxD x 3 dA,
ⱍ
xy dz dx dy
■
1
yy
15–28
3xy 2 dy dx
6.
3
1
■ Calculate the iterated integral by first reversing the
order of integration.
Calculate the iterated integral.
2
■
13–14
Exercise 1.
2
■
and evaluate the integral.
2. Use the Midpoint Rule to estimate the integral in
yy
1sin 1
■
12. Describe the solid whose volume is given by the integral
1
3.
■
y y
2
1
■
4 x
7
2
3–8
0
_4
8
■
where
D 苷 兵共x, y兲 1 x 1, x 2 1 y x 1其
ⱍ
■
■
■
912
17.
18.
19.
CHAPTER 12 MULTIPLE INTEGRALS
xxD xy dA, where D is bounded by y 2 苷 x 3 and y 苷 x
xxD xe y dA, where D is bounded by y 苷 0, y 苷 x 2, x 苷 1
xxD 共xy 2x 3y兲 dA, where D is the region in the first
36. A lamina occupies the part of the disk x 2 y 2 a 2 that
xxD y dA,
37. (a) Find the centroid of a right circular cone with height h
lies in the first quadrant.
(a) Find the centroid of the lamina.
(b) Find the center of mass of the lamina if the density
function is 共x, y兲 苷 xy 2.
quadrant bounded by x 苷 1 y 2, y 苷 0, x 苷 0
20.
where D is the region in the first quadrant that lies
above the hyperbola xy 苷 1 and the line y 苷 x and below
the line y 苷 2
21.
xxD 共x 2 y 2 兲3兾2 dA,
22.
xxD sx 2 y 2 dA, where D is the closed disk with radius 1
and base radius a. (Place the cone so that its base is in
the xy-plane with center the origin and its axis along the
positive z -axis.)
(b) Find the moment of inertia of the cone about its axis
(the z -axis).
where D is the region in the first quadrant bounded by the lines y 苷 0 and y 苷 s3x and the circle
x2 y2 苷 9
38. (a) Set up, but don’t evaluate, an integral for the surface
and center 共0, 1兲
23.
24.
25.
xxxE x 2z dV ,
where
E 苷 兵共x, y, z兲 0 x 2, 0 y 2x, 0 z x其
ⱍ
CAS
xxxT y dV ,
where T is the tetrahedron bounded by the planes
x 苷 0, y 苷 0, z 苷 0, and 2x y z 苷 2
39. Find the area of the part of the surface z 苷 x 2 y that lies
above the triangle with vertices (0, 0), (1, 0), and (0, 2).
xxxE y 2z 2 dV ,
where E is bounded by the paraboloid
x 苷 1 y z 2 and the plane x 苷 0
2
xxxE z dV ,
27.
xxxE yz dV ,
28.
■
CAS
40. Graph the surface z 苷 x sin y, 3 x 3, y ,
and find its surface area correct to four decimal places.
where E is bounded by the planes y 苷 0, z 苷 0,
x y 苷 2 and the cylinder y 2 z 2 苷 1 in the first octant
26.
area of the parametric surface given by the vector
function r共u, v兲 苷 v 2 i uv j u 2 k, 0 u 3,
3 v 3.
(b) Use a computer algebra system to approximate the surface area correct to four significant digits.
41. Use polar coordinates to evaluate
where E lies above the plane z 苷 0, below the
plane z 苷 y, and inside the cylinder x 2 y 2 苷 4
s2
y y
0
xxxH z 3sx 2 y 2 z 2 dV ,
where H is the solid hemisphere
with center the origin, radius 1, that lies above the xy-plane
■
■
■
■
■
■
■
■
■
■
■
■
yy
0
29. Under the paraboloid z 苷 x 2 4y 2 and above the rectangle
s1x 2
0
xy-plane with vertices 共1, 0兲, 共2, 1兲, and 共4, 0兲
31. The solid tetrahedron with vertices 共0, 0, 0兲, 共0, 0, 1兲,
CAS
共0, 2, 0兲, and 共2, 2, 0兲
2
and y z 苷 3
44. Find the center of mass of the solid tetrahedron with
45. The joint density function for random variables X and Y is
33. One of the wedges cut from the cylinder x 9y 苷 a by
2
2
2
the planes z 苷 0 and z 苷 mx
f 共x, y兲 苷
34. Above the paraboloid z 苷 x 2 y 2 and below the half-cone
z 苷 sx 2 y 2
■
共x 2 y 2 z 2 兲2 dz dy dx
vertices 共0, 0, 0兲, 共1, 0, 0兲, 共0, 2, 0兲, 共0, 0, 3兲 and density
function 共x, y, z兲 苷 x 2 y 2 z 2.
32. Bounded by the cylinder x y 苷 4 and the planes z 苷 0
■
s1x 2y 2
0
y 苷 e x, find the approximate value of the integral xxD y 2 dA.
(Use a graphing device to estimate the points of intersection
of the curves.)
30. Under the surface z 苷 x 2 y and above the triangle in the
■
y
2
; 43. If D is the region bounded by the curves y 苷 1 x and
R 苷 关0, 2兴 关1, 4兴
■
1
dx dy
1 x2 y2
■
Find the volume of the given solid.
2
y
42. Use spherical coordinates to evaluate
1
29–34
s4y 2
■
■
■
■
■
■
■
■
35. Consider a lamina that occupies the region D bounded by
the parabola x 苷 1 y 2 and the coordinate axes in the first
quadrant with density function 共x, y兲 苷 y.
(a) Find the mass of the lamina.
(b) Find the center of mass.
(c) Find the moments of inertia about the x- and y-axes.
■
再
C共x y兲 if 0 x 3, 0 y 2
0
otherwise
(a) Find the value of the constant C.
(b) Find P共X 2, Y 1兲.
(c) Find P共X Y 1兲.
46. A lamp has three bulbs, each of a type with average lifetime
800 hours. If we model the probability of failure of the
bulbs by an exponential density function with mean 800,
find the probability that all three bulbs fail within a total of
1000 hours.
CHAPTER 12 REVIEW
47. Rewrite the integral
1
1
1y
1
x2
0
f 共x, y, z兲 dz dy dx
as an iterated integral in the order dx dy dz.
48. Give five other iterated integrals that are equal to
2
y3
yy y
0
913
51. Use the change of variables formula and an appropriate
y y y
0
◆
y2
0
f 共x, y, z兲 dz dx dy
49. Use the transformation u 苷 x y, v 苷 x y to evaluate
xxR 共x y兲兾共x y兲 dA, where R is the square with vertices
共0, 2兲, 共1, 1兲, 共2, 2兲, and 共1, 3兲.
50. Use the transformation x 苷 u 2, y 苷 v 2, z 苷 w 2 to
find the volume of the region bounded by the surface
sx sy sz 苷 1 and the coordinate planes.
transformation to evaluate xxR xy dA, where R is the square
with vertices 共0, 0兲, 共1, 1兲, 共2, 0兲, and 共1, 1兲.
1
dA, where n is an integer and D
2
共x
y 2 兲n兾2
D
is the region bounded by the circles with center the origin and radii r and R, 0 r R.
(b) For what values of n does the integral in part (a) have a
limit as r l 0 ?
1
(c) Find yyy 2
dV , where E is the region
2
共x
y
z 2 兲n兾2
E
bounded by the spheres with center the origin and radii
r and R, 0 r R.
(d) For what values of n does the integral in part (c) have a
limit as r l 0 ?
52. (a) Evaluate yy