964 |||| CHAPTER 15 MULTIPLE INTEGRALSmtm.ufsc.br/~saeger/st_15_2-8.pdf · 964 |||| CHAPTER 15 MULTIPLE INTEGRALS N The function in Example 5 is positive on , so the integral repre-sents

EXAMPLE 5 If , then, by Equation 5,

M

FIGURE 6

y

x

z

0

� [�cos x]0

� 2 [sin y]0

� 2� 1 � 1 � 1

yyR

sin x cos y dA � y� 2

0 sin x dx y

� 2

0 cos y dy

R � �0, � 2 � �0, � 2

964 | | | | CHAPTER 15 MULTIPLE INTEGRALS

N The function in Example 5 is positive on , so the integral repre-sents the volume of the solid that lies above and below the graph of shown in Figure 6.f

RR

f �x, y� � sin x cos y

18. ,

,

20. ,

21. ,

22. ,

23–24 Sketch the solid whose volume is given by the iterated integral.

24.

25. Find the volume of the solid that lies under the planeand above the rectangle

.

26. Find the volume of the solid that lies under the hyperbolicparaboloid and above the square

.R � ��1, 1 � �0, 2z � 4 � x 2 � y 2

R � ��x, y� � 0 � x � 1, �2 � y � 33x � 2y � z � 12

y1

0 y

1

0 �2 � x 2 � y 2 � dy dx

y1

0 y

1

0 �4 � x � 2y� dx dy23.

R � �1, 2 � �0, 1yyR

x

x 2 � y 2 dA

R � �0, 1 � �0, 2yyR

xye x2y dA

R � �0, 1 � �0, 1yyR

x

1 � xy dA

R � �0, � 6 � �0, � 3yyR

x sin�x � y� dA19.

R � ��x, y� � 0 � x � 1, 0 � y � 1yyR

1 � x 2

1 � y 2 dA1–2 Find and .

1. 2.

3–14 Calculate the iterated integral.

4.

5. 6.

7. 8.

10.

11. 12.

13. 14.

15–22 Calculate the double integral.

15. ,

16. ,

, R � ��x, y� � 0 � x � 1, �3 � y � 3yyR

xy 2

x 2 � 1 dA17.

R � ��x, y� � 0 � x � �, 0 � y � � 2yyR

cos�x � 2y� dA

R � ��x, y� � 0 � x � 3, 0 � y � 1yyR

�6x 2y 3 � 5y 4 � dA

y1

0 y

1

0 ss � t ds dty

2

0 y

�

0 r sin2� d� dr

y1

0 y

1

0 xysx 2 � y 2 dy dxy

1

0 y

1

0 �u � v�5 du dv

y1

0 y

3

0 e x�3y dx dyy

4

1 y

2

1 � x

y�

y

x� dy dx9.

y1

0y

2

1 xe x

y dy dxy

2

0 y

1

0 �2x � y�8 dx dy

y� 2

� 6 y

5

�1 cos y dx dyy

2

0 y

� 2

0 x sin y dy dx

y1

0y

2

1 �4x 3 � 9x 2 y2� dy dxy

3

1 y

1

0 �1 � 4xy� dx dy3.

f �x, y� � y � xe yf �x, y� � 12x 2 y3

x10 f �x, y� dyx

50 f �x, y� dx

EXERCISES15.2

34. Graph the solid that lies between the surfacesand for ,

. Use a computer algebra system to approximate thevolume of this solid correct to four decimal places.

35–36 Find the average value of over the given rectangle.

, has vertices , , ,

36. ,

37. Use your CAS to compute the iterated integrals

Do the answers contradict Fubini’s Theorem? Explain what is happening.

38. (a) In what way are the theorems of Fubini and Clairaut similar?

(b) If is continuous on and

for , , show that .txy � tyx � f �x, y�c y da x b

t�x, y� � yx

a y

y

c f �s, t� dt ds

�a, b � �c, d f �x, y�

y1

0 y

1

0

x � y

�x � y�3 dx dyandy1

0 y

1

0

x � y

�x � y�3 dy dx

CAS

R � �0, 4 � �0, 1f �x, y� � e ysx � e y

�1, 0��1, 5��1, 5��1, 0�Rf �x, y� � x 2 y35.

f

� y � � 1� x � � 1z � 2 � x 2 � y 2z � e�x2

cos �x 2 � y 2 �CASFind the volume of the solid lying under the elliptic

paraboloid and above the rectangle.

28. Find the volume of the solid enclosed by the surfaceand the planes , , ,

and .

29. Find the volume of the solid enclosed by the surfaceand the planes , , , ,

and .

30. Find the volume of the solid in the first octant bounded by the cylinder and the plane .

31. Find the volume of the solid enclosed by the paraboloidand the planes , , ,

, and .

; 32. Graph the solid that lies between the surfaceand the plane and is bounded

by the planes , , , and . Then find itsvolume.

33. Use a computer algebra system to find the exact value of theintegral , where . Then usethe CAS to draw the solid whose volume is given by the integral.

R � �0, 1 � �0, 1xxR x 5y 3e x y dA

CAS

y � 4y � 0x � 2x � 0z � x � 2yz � 2xy �x 2 � 1�

y � 4y � 0x � �1x � 1z � 1z � 2 � x 2 � �y � 2�2

y � 5z � 16 � x 2

y � � 4y � 0x � 2x � 0z � 0z � x sec2y

z � 0y � �y � 0x � 1z � 1 � e x sin y

R � ��1, 1 � ��2, 2x 2 4 � y 2 9 � z � 1

27.

SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS | | | | 965

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 not just over rectangles butalso over regions of more general shape, such as the one illustrated in Figure 1. We sup-pose that is a bounded region, which means that can be enclosed in a rectangularregion as in Figure 2. Then we define a new function with domain by

0

y

x

D

y

0 x

D

R

FIGURE 2FIGURE 1

F�x, y� � �0

f �x, y� if

if

�x, y� is in D

�x, y� is in R but not in D1

RFRDD

Df

15.3


is bounded by the circle with center the origin and radius 2

18. is the triangular region with vertices ,

, and

19–28 Find the volume of the given solid.

19. Under the plane and above the regionbounded by and

20. Under the surface and above the region boundedby and

Under the surface and above the triangle with vertices, , and

22. Enclosed by the paraboloid and the planes ,, ,

23. Bounded by the coordinate planes and the plane

24. Bounded by the planes , , , and

25. Enclosed by the cylinders , and the planes ,

26. Bounded by the cylinder and the planes , in the first octant

27. Bounded by the cylinder and the planes ,, in the first octant

28. Bounded by the cylinders and

; 29. Use a graphing calculator or computer to estimate the -coordinates of the points of intersection of the curves

and . If is the region bounded by these curves,estimate .xxD x dA

Dy � 3x � x 2y � x 4x

y 2 � z2 � r 2x 2 � y 2 � r 2

z � 0x � 0y � zx 2 � y 2 � 1

z � 0x � 0x � 2y,y 2 � z2 � 4

y � 4z � 0y � x 2z � x 2

z � 0x � y � 2y � xz � x

3x � 2y � z � 6

z � 0y � xy � 1x � 0z � x 2 � 3y 2

�1, 2��4, 1��1, 1�z � xy21.

x � y 3x � y 2z � 2x � y 2

y � x 4y � xx � 2y � z � 0

�0, 3��1, 2�

�0, 0�yyD

2xy dA, D

D

yyD

�2x � y� dA,17.1–6 Evaluate the iterated integral.

1. 2.

3. 4.

6.

7–18 Evaluate the double integral.

7.

8.

9.

10.

11.

12. ,

, is bounded by , ,

14. , is bounded by

15. ,

is the triangular region with vertices (0, 2), (1, 1),

16. yyD

xy 2 dA, D is enclosed by x � 0 and x � s1 � y 2

�3, 2�D

yyD

y 3 dA

y � sx and y � x 2DyyD

�x � y� dA

x � 1y � x 2y � 0DyyD

x cos y dA13.

D � ��x, y� � 0 � y � 1, 0 � x � y�yyD

xsy 2 � x 2 dA

yyD

y 2e xy dA, D � ��x, y� � 0 � y � 4, 0 � x � y�

yyD

x 3 dA, D � ��x, y� � 1 � x � e, 0 � y � ln x�

yyD

x dA, D � ��x, y� � 0 � x � �, 0 � y � sin x�

yyD

y

x 5 � 1 dA, D � ��x, y� � 0 � x � 1, 0 � y � x 2�

yyD

y 2 dA, D � ��x, y� � �1 � y � 1, �y � 2 � x � y�

y1

0 y

v

0 s1 � v 2 du dvy

� 2

0 y

cos �

0 e sin � dr d�5.

y2

0 y

2y

y xy dx dyy

1

0 y

x

x 2 �1 � 2y� dy dx

y1

0 y

2

2x �x � y� dy dxy

4

0 y

sy

0 xy 2 dx dy

EXERCISES15.3

EXAMPLE 6 Use Property 11 to estimate the integral , where is the diskwith center the origin and radius 2.

SOLUTION Since and , we have andtherefore

Thus, using , , and in Property 11, we obtain

M

4�

e� yy

D

e sin x cos y dA � 4�e

A�D� � � �2�2M � em � e�1 � 1 e

e�1 � e sin x cos y � e 1 � e

�1 � sin x cos y � 1�1 � cos y � 1�1 � sin x � 1

DxxD e sin x cos y dA

SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS | | | | 973

51–52 Express as a union of regions of type I or type II andevaluate the integral.

52.

53–54 Use Property 11 to estimate the value of the integral.

53. , is the quarter-circle with center the origin

and radius in the first quadrant

54. , is the triangle enclosed by the lines

, , and

55–56 Find the average value of over region .

55. , is the triangle with vertices , and

56. , is enclosed by the curves , , and

57. Prove Property 11.

In evaluating a double integral over a region , a sum of iterated integrals was obtained as follows:

Sketch the region and express the double integral as an iterated integral with reversed order of integration.

59. Evaluate , where[Hint: Exploit the fact that

is symmetric with respect to both axes.]

60. Use symmetry to evaluate , where is the region bounded by the square with vertices and .

61. Compute , where is the disk, by first identifying the integral as the volume

of a solid.

62. Graph the solid bounded by the plane and the paraboloid and find its exact volume.(Use your CAS to do the graphing, to find the equations ofthe boundary curves of the region of integration, and to eval-uate the double integral.)

z � 4 � x 2 � y 2x � y � z � 1CAS

x 2 � y 2 � 1Dxx

D s1 � x 2 � y 2 dA

�0, �5��5, 0�

DxxD �2 � 3x � 4y� dA

DD � ��x, y� � x 2 � y 2 � 2�.

xxD �x 2 tan x � y 3 � 4� dA

D

yyD

f �x, y� dA � y1

0 y

2y

0 f �x, y� dx dy � y

3

1 y

3�y

0 f �x, y� dx dy

D58.

x � 1y � x 2y � 0Df �x, y� � x sin y

�1, 3��0, 0�, �1, 0�Df �x, y� � xy

Df

x � 1y � 2xy � 0

TyyT

sin4�x � y� dA

12

QyyQ

e��x2�y2�2

dA

0

_1

1

_1

x=y-Á

y=(x+1)@

y

x0

1

_1

_1 1

D(1, 1)

x

y

yyD

y dAyyD

x 2 dA51.

D; 30. Find the approximate volume of the solid in the first octant that is bounded by the planes , , and andthe cylinder . (Use a graphing device to estimatethe points of intersection.)

31–32 Find the volume of the solid by subtracting two volumes.

31. The solid enclosed by the parabolic cylinders , and the planes ,

32. The solid enclosed by the parabolic cylinder and theplanes ,


33. 34.

35–38 Use a computer algebra system to find the exact volumeof the solid.

35. Under the surface and above the regionbounded by the curves and for

36. Between the paraboloids andand inside the cylinder

37. Enclosed by

38. Enclosed by

39–44 Sketch the region of integration and change the order ofintegration.

39. 40.

41. 42.

44.

45–50 Evaluate the integral by reversing the order of integration.

46.

47. 48.

49.

50. y8

0 y

2

sy3 ex4

dx dy

y1

0 y

� 2

arcsin y cos x s1 � cos2x dx dy

y1

0 y

1

x e x y dy dxy

4

0 y

2

sx

1

y3 � 1 dy dx

ys�

0 y

s�

y cos�x 2� dx dyy

1

0 y

3

3y e x2

dx dy45.

y1

0 y

� 4

arctan x f �x, y� dy dxy

2

1 y

ln x

0 f �x, y� dy dx43.

y3

0 y

s9�y

0 f �x, y� dx dyy

3

0 y

s9�y 2

�s9�y 2 f �x, y� dx dy

y1

0 y

4

4x f �x, y� dy dxy

4

0 y

sx

0 f �x, y� dy dx

z � x 2 � y 2 and z � 2y

z � 1 � x 2 � y 2 and z � 0

x 2 � y 2 � 1z � 8 � x 2 � 2y 2z � 2x 2 � y 2

x � 0y � x 2 � xy � x 3 � xz � x 3y 4 � xy 2

CAS

y1

0 y

1�x 2

0 �1 � x� dy dxy

1

0 y

1�x

0 �1 � x � y� dy dx

z � 2 � yz � 3yy � x 2

2x � 2y � z � 10 � 0x � y � z � 2y � x 2 � 1y � 1 � x 2

y � cos xz � xz � 0y � x

EXAMPLE 4 Find the volume of the solid that lies under the paraboloid ,above the -plane, and inside the cylinder .

SOLUTION The solid lies above the disk whose boundary circle has equationor, after completing the square,

(See Figures 9 and 10.) In polar coordinates we have and , sothe boundary circle becomes , or . Thus the disk is given by

and, by Formula 3, we have

M � 2[32 � � sin 2� �

18 sin 4�]0

� 2� 2�3

2��

2 � �3�

2

� 2 y� 2

0 [1 � 2 cos 2� �

12 �1 � cos 4��] d�

� 8 y� 2

0 cos4� d� � 8 y

� 2

0 �1 � cos 2�

2 �2

d�� 4 y� 2

�� 2 cos4� d�

� y� 2

�� 2 r 4

4 0

2 cos �

d� V � yyD

�x 2 � y 2 � dA � y� 2

�� 2 y

2 cos �

0 r 2 r dr d�

D � ��r, � � � �� 2 � � � � 2, 0 � r � 2 cos � �

Dr � 2 cos �r 2 � 2r cos �x � r cos �x 2 � y 2 � r 2

�x � 1�2 � y 2 � 1

x 2 � y 2 � 2xD

x 2 � y 2 � 2xxyz � x 2 � y 2V


FIGURE 9

0

y

x1 2

D

(x-1)@+¥=1

(or r=2 cos ¨)

FIGURE 10

y

x

z

5–6 Sketch the region whose area is given by the integral and eval-uate the integral.

5. 6.

7–14 Evaluate the given integral by changing to polar coordinates.

7. ,where is the disk with center the origin and radius 3

8. , where is the region that lies to the left of the-axis between the circles and

9. , where is the region that lies above the -axis within the circle

10. ,where

, where D is the region bounded by thesemicircle and the y-axis

12. , where is the region in the first quadrant enclosedby the circle x 2 � y 2 � 25

RxxR yex dA

x � s4 � y 2

xxD e�x2�y2

dA11.

R � ��x, y� � x 2 � y 2 � 4, x � 0�xx

R s4 � x 2 � y 2 dA

x 2 � y 2 � 9xRxx

R cos�x 2 � y 2� dA

x 2 � y 2 � 4x 2 � y 2 � 1yRxx

R �x � y� dA

DxxD xy dA

y� 2

0 y

4 cos �

0 r dr d�y

2�

� y

7

4 r dr d�

1–4 A region is shown. Decide whether to use polar coordinatesor rectangular coordinates and write as an iteratedintegral, where is an arbitrary continuous function on .

2.

3. 4.

0

y

x

6

3

0

y

x_1 1

1

0

y

x_1 1

1 y=1-≈

0 4

4

y

x

1.

RfxxR f �x, y� dA

R

EXERCISES15.4

SECTION 15.4 DOUBLE INTEGRALS IN POLAR COORDINATES | | | | 979

33. A swimming pool is circular with a 40-ft diameter. The depthis constant along east-west lines and increases linearly from2 ft at the south end to 7 ft at the north end. Find the volume ofwater in the pool.

34. An agricultural sprinkler distributes water in a circular patternof radius 100 ft. It supplies water to a depth of feet per hourat a distance of feet from the sprinkler.(a) If , what is the total amount of water supplied

per hour to the region inside the circle of radius centeredat the sprinkler?

(b) Determine an expression for the average amount of waterper hour per square foot supplied to the region inside thecircle of radius .

Use polar coordinates to combine the sum

into one double integral. Then evaluate the double integral.

36. (a) We define the improper integral (over the entire plane

where is the disk with radius and center the origin.Show that

(b) An equivalent definition of the improper integral in part (a)is

where is the square with vertices . Use this toshow that

(c) Deduce that

(d) By making the change of variable , show that

(This is a fundamental result for probability and statistics.)

37. Use the result of Exercise 36 part (c) to evaluate the followingintegrals.

(a) (b) y�

0 sx e�x dxy

�

0 x 2e�x2 dx

y�

�� e�x2 2 dx � s2�

t � s2 x

y�

�� e�x2 dx � s�

y�

�� e�x2 dx y

�

�� e�y2 dy � �

��a, �a�Sa

yy� 2

e��x2�y2 � dA � lima l �

yySa

e��x2�y2 � dA

y�

�� y

�

�� e��x2�y2 � dA � �

aDa

� lima l �

yyDa

e��x2�y2 � dA

I � yy� 2

e��x2�y2 � dA � y�

�� y

�

�� e��x2�y2 � dy dx

�2�

y1

1 s2 y

x

s1�x 2 xy dy dx � y

s2

1 y

x

0 xy dy dx � y

2

s2 y

s4�x 2

0 xy dy dx

35.

R

R0 R � 100

re�r

,where

14. , where is the region in the first quadrant that liesbetween the circles and

15–18 Use a double integral to find the area of the region.

One loop of the rose

16. The region enclosed by the curve

17. The region within both of the circles and

18. The region inside the cardioid and outside thecircle

19–27 Use polar coordinates to find the volume of the given solid.

19. Under the cone and above the disk

20. Below the paraboloid and above the -plane

21. Enclosed by the hyperboloid and the plane

22. Inside the sphere and outside the cylinder

23. A sphere of radius

24. Bounded by the paraboloid and the plane in the first octant

Above the cone and below the sphere

26. Bounded by the paraboloids and

27. Inside both the cylinder and the ellipsoid

28. (a) A cylindrical drill with radius is used to bore a holethrough the center of a sphere of radius . Find the volumeof the ring-shaped solid that remains.

(b) Express the volume in part (a) in terms of the height ofthe ring. Notice that the volume depends only on , not on or .

29–32 Evaluate the iterated integral by converting to polar coordinates.

29. 30.

31. 32. y2

0 y

s2x�x 2

0 sx 2 � y 2

dy dxy1

0y

s2�y 2

y �x � y� dx dy

ya

0 y

0

�sa 2 �y 2 x 2 y dx dyy

3

�3 y

s9�x 2

0 sin�x 2 � y2� dy dx

r2r1

hh

r2

r1

4x 2 � 4y 2 � z2 � 64x 2 � y 2 � 4

z � 4 � x 2 � y 2z � 3x 2 � 3y 2

x 2 � y 2 � z2 � 1z � sx 2 � y 2 25.

z � 7z � 1 � 2x 2 � 2y 2

a

x 2 � y 2 � 4x 2 � y 2 � z 2 � 16

z � 2�x 2 � y 2 � z2 � 1

xyz � 18 � 2x 2 � 2y 2

x 2 � y 2 � 4z � sx 2 � y 2

r � 3 cos �r � 1 � cos �

r � sin �r � cos �

r � 4 � 3 cos �

r � cos 3�15.

x 2 � y 2 � 2xx 2 � y 2 � 4Dxx

D x dA

R � ��x, y� � 1 � x 2 � y 2 � 4, 0 � y � x�xx

R arctan� y x� dA13.

Saeger

EXAMPLE 8 A factory produces (cylindrically shaped) roller bearings that are sold ashaving diameter 4.0 cm and length 6.0 cm. In fact, the diameters X are normally distrib-uted with mean 4.0 cm and standard deviation 0.01 cm while the lengths Y are normallydistributed with mean 6.0 cm and standard deviation 0.01 cm. Assuming that X and Y areindependent, write the joint density function and graph it. Find the probability that abearing randomly chosen from the production line has either length or diameter thatdiffers from the mean by more than 0.02 cm.

SOLUTION We are given that X and Y are normally distributed with , and. So the individual density functions for X and Y are

Since X and Y are independent, the joint density function is the product:

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

Then the probability that either X or Y differs from its mean by more than 0.02 cm isapproximately

M1 � 0.91 � 0.09

� 0.91

�5000

� y

4.02

3.98 y

6.02

5.98 e�5000��x�4�2� y�6�2� dy dx

P�3.98 � X � 4.02, 5.98 � Y � 6.02� � y4.02

3.98 y

6.02

5.98 f �x, y� dy dx

�5000

� e�5000��x�4�2� y�6�2�

f �x, y� � f1�x� f2�y� �1

0.0002� e��x�4�2�0.0002e��y�6�2�0.0002

f2�y� �1

0.01s2� e�� y�6�2�0.0002f1�x� �

1

0.01s2� e��x�4�2�0.0002

�1 � �2 � 0.01�2 � 6.0,�1 � 4.0


FIGURE 9Graph of the bivariate normal jointdensity function in Example 8

1500

1000

500

0

y6.05

65.95

x

4.05

4

3.95

4. ;

is the triangular region with vertices , , ;

6. is the triangular region enclosed by the lines , ,and ;

7. is bounded by , , , and ;

8. is bounded by , , and ;

9. ;

10. is bounded by the parabolas and ;��x, y� � sx

x � y 2y � x 2D

��x, y� � yD � ��x, y� 0 y sin��x�L�, 0 x L�

��x, y� � xx � 1y � 0y � sx D

��x, y� � yx � 1x � 0y � 0y � e xD

��x, y� � x 22x y � 6y � xx � 0D

��x, y� � x y�0, 3��2, 1��0, 0�D5.

��x, y� � cxyD � ��x, y� 0 x a, 0 y b�Electric charge is distributed over the rectangle ,so that the charge density at is

(measured in coulombs per square meter).Find the total charge on the rectangle.

2. Electric charge is distributed over the disk so that the charge density at is (measured in coulombs per square meter). Find the total chargeon the disk.

3–10 Find the mass and center of mass of the lamina that occupiesthe region and has the given density function .

3. ; ��x, y� � xy 2D � ��x, y� 0 x 2, �1 y 1�

�D

� �x, y� � x y x 2 y 2�x, y�x 2 y 2 4

� �x, y� � 2xy y 2�x, y�0 y 2

1 x 31.

EXERCISES15.5

The joint density function for a pair of random variables and is

(a) Find the value of the constant .(b) Find .(c) Find .

28. (a) Verify that

is a joint density function.(b) If and are random variables whose joint density func-

tion is the function in part (a), find

(i) (ii)(c) Find the expected values of and .

Suppose and are random variables with joint density function

(a) Verify that is indeed a joint density function.(b) Find the following probabilities.

(i) (ii)(c) Find the expected values of and .

30. (a) A lamp has two bulbs of a type with an average lifetimeof 1000 hours. Assuming that we can model the proba-bility of failure of these bulbs by an exponential densityfunction with mean , find the probability thatboth of the lamp’s bulbs fail within 1000 hours.

(b) Another lamp has just one bulb of the same type as inpart (a). If one bulb burns out and is replaced by a bulb of the same type, find the probability that the two bulbsfail within a total of 1000 hours.

31. Suppose that and are independent random variables,where is normally distributed with mean 45 and standarddeviation 0.5 and is normally distributed with mean 20 andstandard deviation 0.1.(a) Find .(b) Find .

32. Xavier and Yolanda both have classes that end at noon andthey agree to meet every day after class. They arrive at thecoffee shop independently. Xavier’s arrival time is andYolanda’s arrival time is , where and are measured inminutes after noon. The individual density functions are

(Xavier arrives sometime after noon and is more likely toarrive promptly than late. Yolanda always arrives by 12:10 PM

and is more likely to arrive late than promptly.) After Yolandaarrives, she’ll wait for up to half an hour for Xavier, but hewon’t wait for her. Find the probability that they meet.

f2�y� � � 150 y

0

if 0 y 10

otherwisef1�x� � �e�x

0

if x 0

if x � 0

YXYX

P�4�X � 45�2 100�Y � 20�2 2�P�40 X 50, 20 Y 25�

YX

YXCAS

� � 1000

YXP�X 2, Y 4�P�Y 1�

f

f �x, y� � �0.1e��0.5x0.2y�

0

if x 0, y 0

otherwise

YX29.

YXP(X

12 , Y

12 )P(X

12 )

fYX

f �x, y� � �4xy

0

if 0 x 1, 0 y 1

otherwise

P�X Y 1�P�X 1, Y 1�

C

f �x, y� � �Cx�1 y�0

if 0 x 1, 0 y 2

otherwise

YX27.11. A lamina occupies the part of the disk in the

first quadrant. Find its center of mass if the density at anypoint is proportional to its distance from the -axis.

12. Find the center of mass of the lamina in Exercise 11 if the density at any point is proportional to the square of itsdistance from the origin.

13. The boundary of a lamina consists of the semicirclesand together with the portions

of the -axis that join them. Find the center of mass of thelamina if the density at any point is proportional to its dis-tance from the origin.

14. Find the center of mass of the lamina in Exercise 13 if thedensity at any point is inversely proportional to its distancefrom the origin.

Find the center of mass of a lamina in the shape of an isos-celes right triangle with equal sides of length if the densityat any point is proportional to the square of the distance fromthe vertex opposite the hypotenuse.

16. A lamina occupies the region inside the circle but outside the circle . Find the center of mass if the density at any point is inversely proportional to its dis-tance from the origin.

17. Find the moments of inertia , , for the lamina of Exercise 7.



20. Consider a square fan blade with sides of length 2 and thelower left corner placed at the origin. If the density of theblade is , is it more difficult to rotate theblade about the -axis or the -axis?

21–22 Use a computer algebra system to find the mass, center of mass, and moments of inertia of the lamina that occupies theregion and has the given density function.

21. ;

22. is enclosed by the cardioid ;

23–26 A lamina with constant density occupies thegiven region. Find the moments of inertia and and the radiiof gyration and .

23. The rectangle

24. The triangle with vertices , , and

25. The part of the disk in the first quadrant

26. The region under the curve from to x � �x � 0y � sin x

x 2 y 2 a2

�0, h��b, 0��0, 0�

0 x b, 0 y h

yxIyIx

��x, y� � �CAS

��x, y� � sx 2 y 2

r � 1 cos �D

��x, y� � xyD � ��x, y� 0 y sin x, 0 x � �

D

CAS

yx��x, y� � 1 0.1x

I0IyIx

I0IyIx

I0IyIx

x 2 y 2 � 1x 2 y 2 � 2y

a15.

xy � s4 � x 2 y � s1 � x 2

x

x 2 y 2 1

SECTION 15.5 APPLICATIONS OF DOUBLE INTEGRALS | | | | 989


20. The solid bounded by the cylinder and the planes, and

21. The solid enclosed by the cylinder and the planes and

22. The solid enclosed by the paraboloid and theplane

(a) Express the volume of the wedge in the first octant that iscut from the cylinder by the planes and

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 thetriple integral in part (a).

24. (a) In the Midpoint Rule for triple integrals we use a tripleRiemann sum to approximate a triple integral over a box

, where is evaluated at the center of the box . Use the Midpoint Rule to estimate

, where is the cube defined by, , . Divide 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 theanswer to part (a).

25–26 Use the Midpoint Rule for triple integrals (Exercise 24) toestimate the value of the integral. Divide into eight sub-boxes ofequal size.

25. , where

26. , where


28.

29–32 Express the integral as an iterated integralin six different ways, where is the solid bounded by the givensurfaces.

29. ,

30. , ,

31. , ,

32. , , , x � y � 2z � 2z � 0y � 2x � 2

y � 2z � 4z � 0y � x 2

x � 2x � �2y 2 � z2 � 9

y � 0y � 4 � x 2 � 4z2

ExxxE

f �x, y, z� dV

y2

0 y

2�y

0 y

4�y 2

0 dx dz dyy

1

0 y

1�x

0 y

2�2z

0 dy dz dx27.

B � ��x, y, z� � 0 � x � 4, 0 � y � 2, 0 � z � 1xxxB

sin�xy 2z 3� dV

B � ��x, y, z� � 0 � x � 4, 0 � y � 8, 0 � z � 4

xxxB 1

ln�1 � x � y � z� dV

B

CAS

B0 � z � 40 � y � 40 � x � 4Bxxx

B sx 2 � y 2 � z 2 dVBijk

�xi, yj, zk �f �x, y, z�B

CAS

x � 1y � xy 2 � z2 � 1

23.

x � 16x � y 2 � z 2

z � 1y � z � 5x 2 � y 2 � 9

y � 9z � 4z � 0,y � x 21. Evaluate the integral in Example 1, integrating first with

respect to , then , and then .

2. Evaluate the integral , where

using three different orders of integration.

3–8 Evaluate the iterated integral.

3. 4.

5. 6.

7.

8.

9–18 Evaluate the triple integral.

9. , where

10. , where

, where lies under the plane and above the region in the -plane bounded by the curves

, , and

12. , where is bounded by the planes , ,, and

13. , where is bounded by the parabolic cylinderand the planes , , and

14. , where is bounded by the parabolic cylindersand and the planes and

15. , where is the solid tetrahedron with vertices, , , and

16. , where is the solid tetrahedron with vertices, , , and

17. , where is bounded by the paraboloid and the plane

18. , where is bounded by the cylinder and the planes , , and in the first octant

19–22 Use a triple integral to find the volume of the given solid.

The tetrahedron enclosed by the coordinate planes and theplane 2x � y � z � 4

19.

z � 0y � 3xx � 0y 2 � z2 � 9ExxxE z dV

x � 4x � 4y2 � 4z2ExxxE x dV

�1, 0, 1��1, 1, 0��1, 0, 0��0, 0, 0�TxxxT xyz dV

�0, 0, 1��0, 1, 0��1, 0, 0��0, 0, 0�TxxxT x

2 dV

z � x � yz � 0x � y 2y � x 2ExxxE xy dV

x � �1x � 1z � 0z � 1 � y2ExxxE x 2e y dV

2x � 2y � z � 4z � 0y � 0x � 0ExxxE y dV

x � 1y � 0y � sx xy

z � 1 � x � yExxxE 6xy dV11.

E � ��x, y, z� � 0 � x � 1, 0 � y � x, x � z � 2xxxx

E yz cos�x 5 � dV

E � {�x, y, z� � 0 � y � 2, 0 � x � s4 � y 2 , 0 � z � y}xxxE 2x dV

ys�

0 y

x

0 y

xz

0 x 2 sin y dy dz dx

y��2

0 y

y

0 y

x

0 cos�x � y � z� dz dx dy

y1

0 y

z

0 y

y

0 ze�y2

dx dy dzy3

0 y

1

0 y

s1�z 2

0 ze y dx dz dy

y1

0 y

2x

x y

y

0 2xyz dz dy dxy

1

0 y

z

0 y

x�z

0 6xz dy dx dz

E � ��x, y, z� � �1 � x � 1, 0 � y � 2, 0 � z � 1

xxxE �xz � y 3� dV

xzy

EXERCISES15.6

SECTION 15.6 TRIPLE INTEGRALS | | | | 999

40. is the tetrahedron bounded by the planes , , , ;

41–44 Assume that the solid has constant density .

41. Find the moments of inertia for a cube with side length if one vertex is located at the origin and three edges lie along thecoordinate axes.

42. Find the moments of inertia for a rectangular brick with dimen-sions , , and and mass if the center of the brick is situ-ated at the origin and the edges are parallel to the coordinateaxes.

43. Find the moment of inertia about the -axis of the solid cylin-der , .

44. Find the moment of inertia about the -axis of the solid cone.

45–46 Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertiaabout the -axis.

45. The solid of Exercise 21;

46. The hemisphere , ;

47. Let be the solid in the first octant bounded by the cylinderand the planes , , and with the

density function . Use a computeralgebra system to find the exact values of the following quan-tities for .(a) The mass(b) The center of mass(c) The moment of inertia about the -axis

48. If is the solid of Exercise 18 with density function, find the following quantities, correct

to three decimal places.(a) The mass(b) The center of mass(c) The moment of inertia about the -axis

49. The joint density function for random variables , , and isif , and

otherwise.(a) Find the value of the constant .(b) Find .(c) Find .

50. Suppose , , and are random variables with joint densityfunction if , , ,and otherwise.(a) Find the value of the constant .(b) Find .(c) Find .P�X � 1, Y � 1, Z � 1�

P�X � 1, Y � 1�C

f �x, y, z� � 0z � 0y � 0x � 0f �x, y, z� � Ce��0.5x�0.2y�0.1z�

ZYX

P�X � Y � Z � 1�P�X � 1, Y � 1, Z � 1�

Cf �x, y, z� � 0

0 � x � 2, 0 � y � 2, 0 � z � 2f �x, y, z� � CxyzZYX

z

�x, y, z� � x 2 � y 2ECAS

z

E

�x, y, z� � 1 � x � y � zz � 0x � 0y � zx 2 � y 2 � 1

ECAS

�x, y, z� � sx 2 � y 2 � z 2

z � 0x 2 � y 2 � z2 � 1

�x, y, z� � sx 2 � y 2

z

sx 2 � y 2 � z � hz

0 � z � hx 2 � y 2 � a 2z

Mcba

L

k

�x, y, z� � yx � y � z � 1z � 0y � 0x � 0E33. The figure shows the region of integration for the integral

Rewrite this integral as an equivalent iterated integral in thefive other orders.

34. The figure shows the region of integration for the integral

Rewrite this integral as an equivalent iterated integral in thefive other orders.

35–36 Write five other iterated integrals that are equal to thegiven iterated integral.

36.

37–40 Find the mass and center of mass of the solid with thegiven density function .

37. is the solid of Exercise 11;

38. is bounded by the parabolic cylinder and theplanes , , and ;

is the cube given by , , ; �x, y, z� � x 2 � y 2 � z2

0 � z � a0 � y � a0 � x � aE39.

�x, y, z� � 4z � 0x � 0x � z � 1z � 1 � y 2E

�x, y, z� � 2E

E

y1

0 y

x2

0 y

y

0 f �x, y, z� dz dy dx

y1

0 y

1

y y

y

0 f �x, y, z� dz dx dy35.

1

1

1

z=1-≈

y=1-x

0

y

x

z

y1

0 y

1�x2

0 y

1�x

0 f �x, y, z� dy dz dx

0

z

1

x

1 y

z=1-y

y=œ„x

y1

0 y

1

sx y

1�y



20. Evaluate , where is enclosed by the planes and and by the cylinders and

.

Evaluate , where is the solid that lies within thecylinder , above the plane , and below thecone .

22. Find the volume of the solid that lies within both the cylinderand the sphere .

23. (a) Find the volume of the region bounded by the parabo-loids and .

(b) Find the centroid of (the center of mass in the casewhere the density is constant).

24. (a) Find the volume of the solid that the cylinder cuts out of the sphere of radius centered at the origin.

; (b) Illustrate the solid of part (a) by graphing the sphere andthe cylinder on the same screen.

25. Find the mass and center of mass of the solid bounded bythe paraboloid and the plane if

has constant density .

26. Find the mass of a ball given by if thedensity at any point is proportional to its distance from the -axis.

27–28 Evaluate the integral by changing to cylindrical coordinates.

27.

28.

29. When studying the formation of mountain ranges, geologistsestimate the amount of work required to lift a mountain fromsea level. Consider a mountain that is essentially in the shapeof a right circular cone. Suppose that the weight density of the material in the vicinity of a point is and the height is .(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, anddensity a constant 200 lb�ft . How much work was donein forming Mount Fuji if the land was initially at sea level?

P

3

h�P�t�P�P

y3

�3 y

s9�x 2

0 y

9�x 2�y 2

0 sx2 � y2 dz dy dx

y2

�2 y

s4�y 2

�s4�y 2 y

2

sx 2�y 2 xz dz dx dy

z

x 2 � y 2 � z2 a 2B

KSz � a �a 0�z � 4x 2 � 4y 2

S

ar � a cos �

Ez � 36 � 3x 2 � 3y 2z � x 2 � y 2

E

x 2 � y 2 � z2 � 4x 2 � y 2 � 1

z2 � 4x 2 � 4y 2z � 0x 2 � y 2 � 1

ExxxE x 2 dV21.

x 2 � y 2 � 9x 2 � y 2 � 4z � x � y � 5

z � 0ExxxE x dV1–2 Plot the point whose cylindrical coordinates are given. Thenfind the rectangular coordinates of the point.

1. (a) (b)

2. (a) (b)

3–4 Change from rectangular to cylindrical coordinates.

(a) (b)

4. (a) (b)

5–6 Describe in words the surface whose equation is given.

5. 6.

7–8 Identify the surface whose equation is given.

7. 8.

9–10 Write the equations in cylindrical coordinates.

(a) (b)

10. (a) (b)

11–12 Sketch the solid described by the given inequalities.

11. , ,

12. ,

13. A cylindrical shell is 20 cm long, with inner radius 6 cm andouter radius 7 cm. Write inequalities that describe the shell in an appropriate coordinate system. Explain how you havepositioned the coordinate system with respect to the shell.

; 14. Use a graphing device to draw the solid enclosed by the paraboloids and .

15–16 Sketch the solid whose volume is given by the integral and evaluate the integral.

15. 16.

17–26 Use cylindrical coordinates.

Evaluate , where is the region that liesinside the cylinder and between the planes

and .

18. Evaluate , where is the solid in the firstoctant that lies beneath the paraboloid .

19. Evaluate , where is enclosed by the paraboloid, the cylinder , and the -plane.xyx 2 � y 2 � 5z � 1 � x 2 � y 2

ExxxE e z dV

z � 1 � x 2 � y 2ExxxE �x 3 � xy 2 � dV

z � 4z � �5x 2 � y 2 � 16

ExxxE sx 2 � y 2 dV17.

y��2

0 y

2

0 y

9�r 2

0 r dz dr d�y

4

0 y

2�

0 y

4

r r dz d� dr

z � 5 � x 2 � y 2z � x 2 � y 2

r z 20 � ��2

0 z 1��2 � ��20 r 2

�x 2 � y 2 � z2 � 13x � 2y � z � 6

x 2 � y2 � 2yz � x 2 � y29.

2r 2 � z2 � 1z � 4 � r 2

r � 5� � ��4

�4, �3, 2�(2s3, 2, �1)(�1, �s3 , 2)�1, �1, 4�3.

�1, 3��2, 2��1, �, e�

�4, ��3, 5��2, ��4, 1�

EXERCISES15.7


19–20 Set up the triple integral of an arbitrary continuous functionin cylindrical or spherical coordinates over the solid

shown.

19. 20.

21–34 Use spherical coordinates.

Evaluate , where is the ball with center the origin and radius 5.

22. Evaluate , where is the solidhemisphere , .

23. Evaluate , where lies between the spheresand in the first octant.

24. Evaluate , where is enclosed by the spherein the first octant.

25. Evaluate , where is bounded by the -plane and the hemispheres and

.

26. Evaluate , where lies between the spheres and and above the cone .

27. Find the volume of the part of the ball that lies betweenthe cones and .

28. Find the average distance from a point in a ball of radius to its center.

29. (a) Find the volume of the solid that lies above the coneand below the sphere .

(b) Find the centroid of the solid in part (a).

Find the volume of the solid that lies within the sphere, above the -plane, and below the cone

.

31. Find the centroid of the solid in Exercise 25.

32. Let be a solid hemisphere of radius whose density at anypoint is proportional to its distance from the center of the base.(a) Find the mass of .(b) Find the center of mass of .(c) Find the moment of inertia of about its axis.

33. (a) Find the centroid of a solid homogeneous hemisphere ofradius .

(b) Find the moment of inertia of the solid in part (a) about adiameter of its base.

a

HH

H

aH

z � sx 2 � y 2

xyx 2 � y 2 � z 2 � 430.

� � 4 cos �� 3

a

� � ��3� � ��6� � a

� � ��3� � 4� � 2ExxxE xyz dV

y � s16 � x 2 � z 2 y � s9 � x 2 � z 2

xzExxxE x 2 dV

x 2 � y 2 � z2 � 9Exxx

E esx2�y2�z2 dV

x 2 � y 2 � z 2 � 4x 2 � y 2 � z 2 � 1ExxxE

z dV

z � 0x 2 � y 2 � z2 � 9Hxxx

H �9 � x 2 � y 2 � dV

BxxxB �x 2 � y 2 � z2 �2 dV21.

z

xy

3

2

z

x y21

f �x, y, z�1–2 Plot the point whose spherical coordinates are given. Thenfind the rectangular coordinates of the point.

(a) (b)

2. (a) (b)

3–4 Change from rectangular to spherical coordinates.

3. (a) (b)

4. (a) (b)

5–6 Describe in words the surface whose equation is given.

6.

7–8 Identify the surface whose equation is given.

7. 8.

9–10 Write the equation in spherical coordinates.

9. (a) (b)

10. (a) (b)

11–14 Sketch the solid described by the given inequalities.

11. , ,

12. ,

13. ,

14. ,

15. A solid lies above the cone and below thesphere . Write a description of the solid interms of inequalities involving spherical coordinates.

16. (a) Find inequalities that describe a hollow ball with diameter30 cm and thickness 0.5 cm. Explain how you havepositioned the coordinate system that you have chosen.

(b) Suppose the ball is cut in half. Write inequalities thatdescribe one of the halves.

17–18 Sketch the solid whose volume is given by the integral andevaluate the integral.

18. y2�

0 y

�

��2 y

2

1 �2 sin � d� d� d

y��6

0 y

��2

0 y

3

0 �2 sin � d� d d�17.

x 2 � y 2 � z2 � zz � sx 2 � y 2

� � csc �� 2

3��4 � � � �� 1

��2 � � � �2 � � � 3

0 � � ��20 � � � ��2� � 2

x � 2y � 3z � 1x 2 � 2x � y 2 � z 2 � 0

x 2 � z2 � 9z2 � x 2 � y 2

� 2 �sin2� sin2 � cos2�� 9� � sin sin �

� � 3� � ��35.

(�1, 1, s6 )(0, s3 , 1)

�0, �1, �1�(1, s3 , 2s3 )

�4, 3��4, ��3��5, �, ��2�

�2, ��3, ��4��1, 0, 0�1.

EXERCISES15.8

SECTION 15.8 TRIPLE INTEGRALS IN SPHERICAL COORDINATES | | | | 1011

43. The surfaces have been used asmodels for tumors. The “bumpy sphere” with and

is shown. Use a computer algebra system to find thevolume it encloses.

44. Show that

(The improper triple integral is defined as the limit of a triple integral over a solid sphere as the radius of the sphereincreases indefinitely.)

45. (a) Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere andbelow by the cone (or ), where

, is

(b) Deduce that the volume of the spherical wedge given by, , is

(c) Use the Mean Value Theorem to show that the volume inpart (b) can be written as

where lies between and , lies between and , , , and .� � � 2 � �1 � 2 � 1� � � 2 � �1� 2

�1��

� 2�1��

V � �� 2 sin ��

� �

V �� 2

3 � �13

3 �cos �1 � cos � 2 �� 2 � 1 �

�1 � � � � 21 � � 2�1 � � � � 2

V �2�a 3

3 �1 � cos� 0 �

0 � � 0 � ��2� � � 0z � r cot � 0

r 2 � z2 � a 2

e��x2�y 2�z2� dx dy dz � 2�y�

�� y�

�� y�

�� sx 2 � y 2 � z2

n � 5m � 6

� � 1 �15 sin m sin n�CAS

34. Find the mass and center of mass of a solid hemisphere ofradius if the density at any point is proportional to its distance from the base.

35–38 Use cylindrical or spherical coordinates, whichever seemsmore appropriate.

Find the volume and centroid of the solid that lies above the cone and below the sphere

.

36. Find the volume of the smaller wedge cut from a sphere ofradius by two planes that intersect along a diameter at anangle of .

37. Evaluate , where lies above the paraboloid and below the plane . Use either the

Table of Integrals (on Reference Pages 6–10) or a computeralgebra system to evaluate the integral.

38. (a) Find the volume enclosed by the torus .

; (b) Use a computer to draw the torus.

39–40 Evaluate the integral by changing to spherical coordinates.

39.

40.

; 41. Use a graphing device to draw a silo consisting of a cylinderwith radius 3 and height 10 surmounted by a hemisphere.

42. The latitude and longitude of a point 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 posi-tive -axis to pass through the North Pole. The positive -axispasses through the point where the prime meridian (themeridian through Greenwich, England) intersects the equator.Then the latitude of is and the longitude is

. Find the great-circle distance from LosAngeles (lat. N, long. W) to Montréal (lat.

N, long. W). Take the radius of the earth to be3960 mi. (A great circle is the circle of intersection of asphere and a plane through the center of the sphere.)

73.60 45.50 118.25 34.06

� � 360 � � � 90 � � P

xz

��P

ya

�a y

sa 2�y 2

�sa 2�y 2 y

sa 2�x 2�y 2

�sa 2�x 2�y 2 �x 2z � y 2z � z3� dz dx dy

y1

0 y

s1�x 2

0 y

s2�x 2�y 2

sx 2�y 2 xy dz dy dx

� � sin �

z � 2yz � x 2 � y 2

ExxxE z dVCAS

��6a

x 2 � y 2 � z2 � 1z � sx 2 � y 2

E35.

a

A122 || | | APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES

51. Minimum 53. Maximum ; saddle points (0, 0), (0, 3), (3, 0)55. Maximum , minimum 57. Maximum , minima ,saddle points 59. Maximum ,minimum 61. Maximum 1, minimum 63.65.

PROBLEMS PLUS N PAGE 948

1. 3. (a) (b) Yes7.

CHAPTER 15

EXERCISES 15.1 N PAGE 958

1. (a) 288 (b) 1443. (a) (b) 05. (a) �6 (b) �3.57.9. (a) (b) 15.511. 60 13. 315. 1.141606, 1.143191, 1.143535, 1.143617, 1.143637, 1.143642


1. , 3. 10 5. 2 7. 9.11. 0 13. 15. 17.19. 21.

23.

25. 27. 29. 2 31.33.

35.37. Fubini’s Theorem does not apply. The integrand has an infinitediscontinuity at the origin.

56

2

0

y1

0

x10

z

21e � 57

643

1662747.5

z

yx

0

1

1

4

12�e 2 � 3�1

2 (s3 � 1) �1

12�9 ln 221

2�

212 ln 2261,632�453x 2500y 3

�248U � V � L

� 2�2 � 4.935

s6�2, 3s2�2x � w�3, base � w�3L2W 2, 14 L2W 2

P(2 � s3), P(3 � s3)�6, P(2s3 � 3)�3(�3�1�4, 3�1�4

s2, �31�4 ), (�3�1�4, �3�1�4s2, �31�4 )

�1f (�s2�3, �1�s3) � �2�(3s3)

f (�s2�3, 1�s3) � 2�(3s3)��1, �1�, �1, 0�

f �1, �1� � �3f ��1, 0� � 2f �2, 4� � �64f �1, 2� � 4

f �1, 1� � 1f ��4, 1� � �11 EXERCISES 15.3 N PAGE 972

1. 32 3. 5. 7. 9. 11.

13. 15. 17. 0 19. 21.

23. 6 25. 27. 29. 0, 1.213, 0.713 31.

33.

35. 13,984,735,616�14,549,535 37.

39. 41.

43.

45. 47. 49. 51. 153. 55.

59. 61.


1. 3.

5.

7. 0 9. 11. 13.

15. 17. 19. 21.

23. 25.

27.

29. 31.

33. 35. 37. (a) (b) s� �2s� �415161800� ft3

2s2�312��1 � cos 9��8��3�(64 � 24s3)

�2��3�[1 � (1�s2)]43 �a 3

43�16

3 �18�� 2��12

364 � 2��2��1 � e�4 �1

2� sin 9

33��2y

0 x

4 7

R

x1�1 x

�x�1��20 f �x, y� dy dxx

3��20 x4

0 f �r cos �, r sin ��r dr d�

2��38�

34��16�e�1�16 � xxQ e��x 2�y 2� 2

dA � ��16

13 (2s2 � 1)1

3 ln 916 �e 9 � 1�

y

x0

x=2

y=ln x or x=e†

ln 2

1 2

y=0

xln 2

0 x2e y f �x, y� dx dy

≈+¥=9

y=0

y

x0

3

3–3

x=4

y=0

y=œ„x

y

x0

2

4

x3

�3 xs9�x 2

0 f �x, y� dy dxx

20 x4

y 2 f �x, y� dx dy

��2

0

z

y

x

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

643

13

12815

318

718

14720

12 �1 � cos 1�

12e16 �

172�4

3e � 1310

APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES | | | | A123


1. 3. 5.

7. ,

9. , 11. 13.15. if vertex is (0, 0) and sides are along positive axes17.19. , , if vertex is and sides arealong positive axes

21. , , ,

,23. , ; ,25. , ; ,27. (a) (b) 0.375 (c)29. (b) (i)(ii) (c) 2, 531. (a) (b)33. (a) , where D isthe disk with radius 10 mi centered at the center of the city(b) , on the edge


1. 3. 1 5. 7. 9. 4 11.

13. 15. 17. 19. 21.

23. (a) (b)

25. 60.53327.

29.

31.

� x20 xs4�2z

�s4�2z x4�2z

x 2 f �x, y, z� dy dx dz

� x2�2 x

2�x 2�20 x4�2z

x 2 f �x, y, z� dy dz dx

� x40 x2�y�2

0 xsy�sy f �x, y, z� dx dz dy

� x20 x4�2z

0 xsy�sy f �x, y, z� dx dy dz

� x40 xsy

�sy x2�y�20 f �x, y, z� dz dx dy

x2�2 x

4x 2 x2�y�2


� x1�1 x

s4�4z 2

�s4�4z 2 x4�x 2�4z 2

0 f �x, y, z� dy dx dz

� x2�2 x

s4�x 2�2�s4�x 2�2 x

4�x 2�4z 2

0 f �x, y, z� dy dz dx

� x40 xs4�y�2

�s4�y�2 xs4�y�4z 2

�s4�y�4z 2 f �x, y, z� dx dz dy

� x1�1 x

4�4z 2

0 xs4�y�4z 2

�s4�y�4z 2 f �x, y, z� dx dy dz

� x40 xs4�y

�s4�y xs4�x 2�y�2�s4�x 2�y�2 f �x, y, z� dz dx dy

x2�2 x

4�x 2

0 xs4�x 2�y�2�s4�x 2�y�2 f �x, y, z� dz dy dx

z

y

x

01

2

1

14 � �

13x

10 xx

0 xs1�y 2

0 dz dy dx

36�16316��31

608��3e�

6528�

13

13 �e 3 � 1�27

4

200�k�3 � 209k, 200(��2 �89 )k � 136k

xxD �k�20�[20 � s�x � x0 �2 � �y � y0 �2 ] dA

�0.632�0.5001 � e�1.8 � e�0.8 � e�1 � 0.3481

e�0.2 � 0.8187

548 � 0.10421

2

a�2a�2�a4��16�a4��16h�s3b�s3�b3h�3�bh3�3

I0 � � 4�16 � 9� 2�64Iy � 116 �� 4 � 3� 2 �

Ix � 3� 2�64�x, y � � 2�

3�

1

�,

16

9��m � � 2�8

�0, 0�7ka6�907ka6�1807ka6�180

116�e 4 � 1�, 18�e 2 � 1�, 1

16�e 4 � 2e 2 � 3��2a�5, 2a�5�

�0, 45��14��( 38, 3��16)�L�2, 16��9��L�4

e 2 � 1

2�e 2 � 1�,

4�e 3 � 1�9�e 2 � 1��1

4�e 2 � 1�

6, ( 34 , 32 )4

3 , (43 , 0)64

3 C33.

35.

37. 39.41. 43.

45. (a)

(b) , where

(c)

47. (a)

(b)

(c)

49. (a) (b) (c)51.53. The region bounded by the ellipsoid


1. (a) (b)

3. (a) (b)5. Vertical half-plane through the -axis 7. Circular paraboloid9. (a) (b)11.

x

z

y2

2

z=11

r � 2 sin �z � r 2z

(2, 4��3, 2)(s2, 7��4, 4)(2, �2s3, 5)(s2, s2, 1)

0

z

yx

”4, _ , 5’

5

4

π

3

π3

_

0

z

y

x

π4

2 1

” 2, , 1 ’π

4

x 2 � 2y 2 � 3z2 � 1L3�8

15760

164

18

1240 �68 � 15��

�x, y, z � � 28

9� � 44,

30� � 128

45� � 220,

45� � 208

135� � 660�332 � �

1124

x3

�3 xs9�x 2

�s9�x 2 x5�y1 �x 2 � y 2�3�2 dz dy dx

x3

�3 xs9�x 2

�s9�x 2 x5�y1 zsx 2 � y 2 dz dy dxz � �1�m�

x3

�3 xs9�x 2

�s9�x 2 x5�y1 ysx 2 � y 2 dz dy dxy � �1�m�

x3

�3 xs9�x 2

�s9�x 2 x5�y1 xsx 2 � y 2 dz dy dxx � �1�m�

�x, y, z �m � x

3�3 x

s9�x 2

�s9�x 2 x5�y1 sx 2 � y 2 dz dy dx

12�kha 4Ix � Iy � Iz � 2

3 kL5a 5, �7a�12, 7a�12, 7a�12�79

30 , ( 358553 , 33

79 , 571553 )

x10 x1

z xx

z f �x, y, z� dy dx dz� x

10 xx

0 xxz f �x, y, z� dy dz dx �

x10 xy

0 x1y f �x, y, z� dx dz dy� x

10 x1

z x1

y f �x, y, z� dx dy dz �

x10 xx

0 xy0 f �x, y, z� dz dy dxx

10 x1

y xy

0 f �x, y, z� dz dx dy �

� x10 x�1�z�2

0 x1�zsx

f �x, y, z� dy dx dz

� x10 x1�sx

0 x1�zsx f �x, y, z� dy dz dx

� x10 x1�y

0 xy 2

0 f �x, y, z� dx dz dy

� x10 x1�z

0 xy 2

0 f �x, y, z� dx dy dz

� x10 xy 2

0 x1�y0 f �x, y, z� dz dx dy

x10 x1

sx x1�y


A124 || | | APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES

13. Cylindrical coordinates: , ,

15.

17. 19. 21.23. (a) (b)25. 27. 029. (a) , where is the cone(b) ft-lb


1. (a)(0, 0, 1)

(b)

3. (a) (b)5. Half-cone7. Sphere, radius , center 9. (a) (b)

11.

13.

15. 0 � � � ��4, 0 � � � cos �

x

z

y

˙=3π4

∏=1

∏=2

z

yx

22

2

� 2�sin2� cos2� � cos2�� 9cos2� � sin2�(0, 12, 0)1

2

(s2, 3��2, 3��4)�4, ��3, ��6�

( 12 s2, 1

2 s6, s2)

0

z

yx

”2, , ’

2

π4

π3

π

3

π

4

0

z

yx

(1, 0, 0)

1

�3.1 � 1019Cxxx

C h�P�t�P� dV�Ka 2�8, �0, 0, 2a�3�

�0, 0, 15�162�2��5��e 6 e 5�384�

64��3

x 4 y4

z

4

0 � z � 200 � � � 2�6 � r � 7 17.

19.21. 23. 25.27. 29. (a) (b) (0, 0, 2.1)31.33. (a) (b)

35.37. 39.41. 43.


1. 16 3. 5. 07. The parallelogram with vertices (0, 0), (6, 3), (12, 1), (6, 2)9. The region bounded by the line , the y-axis, and 11. 13. 15. 2 ln 317. (a) (b)19. 21. 23.

CHAPTER 15 REVIEW N PAGE 1021

True-False Quiz

1. True 3. True 5. True 7. False

Exercises

1. 3. 5. 7.9.11. The region inside the loop of the four-leaved rose inthe first quadrant13. 15. 17. 19.

21. 23. 40.5 25. 27. 29. 17631. 33.35. (a) (b)(c)37.39. 97.2 41. 0.051243. (a) (b) (c)

45. 47. 49. 0

PROBLEMS PLUS N PAGE 1024

1. 30 3. 7. (b) 0.9012 sin 1

ln 2x10 x1z

0 xsysy f �x, y, z� dx dy dz

145

13

115

�0, 0, h�4�Ix � 1

12 , Iy � 124; y � 1�s3, x � 1�s6( 1

3, 815 )1

4

2ma 3�923

6415��9681��5

814 ln 21

2 e 6 72

12 sin 1

r � sin 2�x

�

0 x42 f �r cos �, r sin �� r dr d�

23

12 sin 14e 2 4e � 3�64.0

e e132 sin 18

5 ln 81.083 � 1012 km34

3 �abc6�3

y � sxy � 1

sin2� cos2�

136��99(4s2 5 )�155��6

�2��3�[1 (1�s2)], (0, 0, 3�[8(2 s2)])4K�a 5�15(0, 0, 38 a)

(0, 525296 , 0)

10�(s3 1)�a 3�31562��1515��16312,500��7

x��20 x3

0 x20 f �r cos �, r sin �, z� r dz dr d�

�9��4� (2 s3)

x y

z

π

6

3

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964 |||| CHAPTER 15 MULTIPLE INTEGRALSmtm.ufsc.br/~saeger/st_15_2-8.pdf · 964 |||| CHAPTER 15 MULTIPLE INTEGRALS N The function in Example 5 is positive on , so the integral repre-sents