Solução Cap (15) Cálculo 1 e 2 George B. Thomas 11 Edição

Prévia do material em texto

CHAPTER 15 MULTIPLE INTEGRALS
15.1 DOUBLE INTEGRALS
 1. 4 y dy dx 4y dx dx 16' ' ' '
0 0 0 0
3 2 3 3a b ’ “� œ � œ œ# #
!
y
3 3
16$
 
 2. (x y 2xy dy dx xy dx' ' '
0 2 0
3 0 3
�
a b ’ “# # !
�#
� œ �
x y
2
# #
 4x 2x dx 2x 0œ � œ � œ'
0
3a b ’ “# # $
!
2x
3
$
 
 3. (x y 1) dx dy yx x dy' ' '
� � �1 1 1
0 1 0
� � œ � �’ “x2#
"
�"
 (2y 2) dy y 2y 1œ � œ � œ'
�1
0 c d# !�"
 
 4. (sin x cos y) dx dy ( cos x) (cos y)x dy' ' '
1 1
1 1 12 2
0
� œ � �c d 1!
 ( cos y 2) dy sin y 2y 2œ � œ � œ'
1
12
1 1 1c d #1
1
 
 5. (x sin y) dy dx x cos y dx' ' '
0 0 0
x
x
1 1
œ �c d !
 (x x cos x) dx (cos x x sin x)œ � œ � �'
0
1 ’ “x2#
1
!
 2œ �1##
 
 6. y dy dx dx sin x dx' ' ' '
0 0 0 0
sin x sin x1 1 1
œ œ’ “y2# ! "# #
 (1 cos 2x) dx x sin 2xœ � œ � œ" " "
!4 4 2 4
'
0
1 � ‘1 1
 
942 Chapter 15 Multiple Integrals
 7. e dx dy e dy ye e dy' ' ' '
1 0 1 1
ln 8 ln y ln 8 ln 8
x y x y y yln y� �œ œ �c d a b!
 (y 1)e e 8(ln 8 1) 8 eœ � � œ � � �c dy y ln 81
 8 ln 8 16 eœ � �
 
 8. dx dy y y dy' ' '
1 y 1
2 y 2#
œ � œ �a b ’ “# #
#
"
y y
3
$ #
 2œ � � � œ � œˆ ‰ ˆ ‰8 7 3 53 3 3 6" "# #
 
 9. 3y e dx dy 3y e dy' ' '
0 0 0
1 1
0
y y
#
#
$ #xy xyœ c d
 3y e 3y dy e y e 2œ � œ � œ �'
0
1Š ‹ ’ “# # $ "
!
y y$ $
 
10. e dy dx x e dx' ' '
1 0 1
4 x 4
y x y x x
0
È
È È3 3
2#
Î Îœ � ‘È È
 (e 1) x dx (e 1) x 7(e 1)œ � œ � œ �3 3 22 3# $Î#
%
"
'
1
4È � ‘ˆ ‰
 
11. dy dx x ln y dx (ln 2) x dx ln 2' ' ' '
1 x 1 1
2 2x 2 2
2x
x
x 3
y œ œ œc d #
12. dy dx (ln 2 ln 1) dx (ln 2) dx (ln 2)' ' ' '
1 1 1 1
2 2 2 2
1
xy xœ � œ œ
" #
13. x y dy dx x y dx x (1 x) dx x x dx' ' ' ' '
0 0 0 0 0
1 1 x 1 1 1x
0
�
"�a b ’ “ ’ “ ’ “# # # # # $� �� œ � œ � � œ � �y (1 x) (1 x)3 3 3$ $ $
 0 0 0œ � � œ � � � � � œ’ “ ˆ ‰ ˆ ‰x x3 4 1 3 4 1 6(1 x)$ % %�# #
"
!
" " " "
14. y cos xy dx dy sin xy dy sin y dy cos y ( 1 1)' ' ' '
0 0 0 0
1 1 11
œ œ œ � œ � � � œc d � ‘1
1 1 1!
" ""
!
1 1
2
15. v u dv du v u du u(1 u) du' ' ' '
0 0 0 0
1 1 u 1 1u
0
�
"�ˆ ‰È È È’ “ ’ “� œ � œ � �v 1 2u u2# #� �#
 u u u du u uœ � � � � œ � � � � œ � � � � œ � � œ �'
0
1 Š ‹ ’ “" " " " " "# # # # # #"Î# $Î# $Î# &Î#
"
!
u u u u 2 2 2 2 2
2 6 3 5 6 3 5 5 10
# # $
 Section 15.1 Double Integrals 943
16. e ln t ds dt e ln t dt (t ln t ln t) dt ln t t ln t t' ' ' '
1 0 1 1
2 ln t 2 2
ln t
0
s s t t
2 4œ œ � œ � � �c d ’ “# #
#
"
 (2 ln 2 1 2 ln 2 2) 1œ � � � � � � œˆ ‰" "4 4
17. 2 dp dv 2 p dv 2 2v dv' ' ' '
� � �
�
2 v 2 2
0 v 0 0
v
v
œ œ �c d�
 2 v 8œ � œc d# 0 2�
 
18. 8t dt ds 4t ds' ' '
0 0 0
1 1 s 1 1 s
0
È
È
�
�
#
#
œ c d#
 4 1 s ds 4 sœ � œ � œ'
0
1 a b ’ “# "
!
s 8
3 3
$
 
19. 3 cos t du dt (3 cos t)u' ' '
� Î � Î
Î Î
1 1
1 1
3 0 3
3 sec t 3
sec t
0œ c d
 3 dt 2œ œ'
� Î
Î
1
1
3
3
1
 
20. dv du du' ' '
0 1 0
3 4 2u 3 4 2u
1
 
�
�4 2u 2u 4
v v
� �
#
œ � ‘
 (3 u) du 3u uœ � # œ � œ !'
0
3 c d# $!
 
21. dx dy' '
2 0
4 4 y) 2Ð � Î
 
22. dy dx' '
�
�
2 0
0 x 2
 
944 Chapter 15 Multiple Integrals
23. dy dx' '
0 x
1 x
#
 
24. dx dy' '
0 1 y
1 1 y
�
�
È
 
25. dx dy' '
1 ln y
e 1
 
26. dy dx' '
1 0
2 ln x
 
27. 16x dx dy' '
0 0
9 9 y12È �
 
28. y dy dx' '
0 0
4 4 xÈ �
 
 
 Section 15.1 Double Integrals 945
29. 3y dy dx' '
�
�
1 0
1 1 xÈ #
 
30. 6x dx dy' '
�
�
2 0
2 4 yÈ #
 
31. dy dx dx dy sin y dy 2' ' ' ' '
0 x 0 0 0
y1 1 1 1
sin y sin y
y yœ œ œ
 
32. 2y sin xy dy dx 2y sin xy dx dy' ' ' '
0 x 0 0
2 2 2 y
# #œ
 2y cos xy dy 2y cos y 2y dyœ � œ � �' '
0 0
2 2
y
0c d a b#
 sin y y 4 sin 4œ � � œ �c d# # #!
 
33. x e dx dy x e dy dx xe dx' ' ' ' '
0 y 0 0 0
1 1 1 x 1
xy xy xy x
0
# #œ œ c d
 xe x dx eœ � œ � œ'
0
1
x xa b ’ “# " �# #
"
!2
x e 2# #
 
34. dy dx dx dy' ' ' '
0 0 0 0
2 4 x 4 4 y� �#
xe xe
4 y 4 y
2y 2y
� �œ
È
 dy dyœ œ œ œ' '
0 0
4 44 y
0
’ “ ’ “x e e e e(4 y) 4 4# )2y 2y 2y# � #
%
!
�"
È
�
 
35. e dx dy e dy dx' ' ' '
0 y/2 0 0
2 ln 3 ln 3 ln 3 2x
x x
È È È
#
œ
#
 2xe dx e e 1 2œ œ œ � œ'
0
ln 3
x x ln 3ln 3
0
È
È
# #c d
 
946 Chapter 15 Multiple Integrals
36. e dy dx e dx dy' ' ' '
0 x 3 0 0
3 1 1 3y
y y
È
Î
$ $
#
œ
 3y e dy e e 1œ œ œ �'
0
1
y y# "
!
$ $c d
 
37. cos 16 x dx dy cos 16 x dy dx' ' ' '
0 y 0 0
1 16 1 2 1 2 xÎ Î Î
"Î%
%a b a b1 1& &œ
 x cos 16 x dxœ œ œ'
0
1 2Î
% &
"Î#
!
"a b ’ “1 sin 16 x80 80a b11 1&
 
38. dy dx dx dy' ' ' '
0 x 0 0
8 2 2 y
$
$
È
" "
� �y 1 y 1% %œ
 dy ln y 1œ œ � œ'
0
2 y
y 1 4 4
ln 17$
%�
" % #
!c da b
 
39. y 2x dA' '
R
a b� #
 y 2x dy dx y 2x dy dxœ � � �' ' ' '
� � � �
� �
1 x 1 0 x 1
0 x 1 1 1 xa b a b# #
 y 2x y dx y 2x y dxœ � � �' '
�
�" �
� � �1 0
0 1
x 1 x
x 1 x 1
� ‘ � ‘" "# # # #
2 2 
 (x 1) 2x (x 1) ( x 1) 2x ( x 1) dxœ � � � � � � � � �'
�1
0 � ‘" "
# #
# # # #
 (1 x) 2x (1 x) (x 1) 2x (x 1) dx� � � � � � � �'
0
1� ‘" "
# #
# # # #
 4 x x dx 4 x x dxœ � � � �' '
�1 0
0 1a b a b$ # $ #
 
 4 4 4 4 8œ � � � � œ � � � œ � œ � œ �’ “ ’ “ ’ “ ˆ ‰ ˆ ‰x x x x 3 4 8 24 3 4 3 4 3 4 3 12 12 12 3( 1) ( 1)% $ % $ % $
0
1�
"
!
� � " "
40. xy dA xy dy dx xy dy dx' '
R
œ �' ' ' '
0 x 2 3 x
2 3 2x 1 2 xÎ �
Î
 xy dx xy dxœ �' '
0 2 3
2 3 12x 2 x
x x
Î
Î
�� ‘ � ‘" "# #
2 2 
 2x x dx x(2 x) x dxœ � � � �' '
0 2 3
2 3 1Î
Î
ˆ ‰ � ‘$ $ # $" " "
# # #
 x dx 2x x dxœ � �' '
0 2 3
2 3 1Î
Î
3
#
$ #a b
 
 x x x 1œ � � œ � � � � œ � � � œ� ‘ � ‘ ˆ ‰ ˆ ‰ ˆ ‰ � ‘ ˆ ‰ˆ ‰ ˆ ‰3 2 3 16 2 4 2 8 6 27 36 16 138 3 8 81 3 9 3 27 81 81 81 81 81% # $ "#Î$2 30Î
41. V x y dy dx x y dx 2x dxœ � œ � œ � � œ � �' ' ' '
0 x 0 0
1 2 x 1 12 x
x
�
�a b ’ “ ’ “ ’ “# # # # � � "
!
y (2 x) (2 x)
3 3 3 3 12 12
7x 2x 7x$ $ %$ $ %
 0 0œ � � � � � œˆ ‰ ˆ ‰2 7 16 43 12 12 12 3"
42. V x dy dx x y dx 2x x x dx x x xœ œ œ � � œ � �' ' ' '
� � �
�
�
2 x 2 2
1 2 x 1 12 x
x
#
#
# # # % $ $ & % "
�#
c d a b � ‘2 1 13 5 4
 œ � � � � � � œ � � � � � � œ œˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰2 16 32 16 40 12 15 320 384 240 189 633 5 4 3 5 4 60 60 60 60 60 60 60 20" "
 Section 15.1 Double Integrals 947
43. V (x 4) dy dx xy 4y dx x 4 x 4 4 x 3x 12x dxœ � œ � œ � � � � �' ' ' '
� � �
�
�
4 3x 4 4
1 4 x 1 1
4 x
3x
#
#c d c da b a b# # #
 x 7x 8x 16 dx x x 4x 16x 12 64œ � � � � œ � � � � œ � � � � �'
�4
1 a b � ‘ ˆ ‰ ˆ ‰$ # % $ # "
�%
"1 7 7 64
4 3 4 3 3
 œ � œ157 6253 4 12
"
44. V (3 y) dy dx 3y dx 3 4 x dxœ � œ � œ � �' ' ' '
0 0 0 0
2 4 x 2 24 x
0
È
È
�
�
#
#’ “ ’ “È Š ‹y2 4 x# ## �#
 x 4 x 6 sin 2x 6 4 3œ � � � � œ � � œ � œ’ “È ˆ ‰ ˆ ‰3 x x 8 16 9 82 6 6 6 3# �" # #
#
!
�$ 1 1
1
45. V 4 y dx dy 4x y x dy 12 3y dy 12y y 24 8 16œ � œ � œ � œ � œ � œ' ' ' '
0 0 0 0
2 3 2 2a b c d a b c d# # # $$! !#
46. V 4 x y dy dx 4 x y dx 4 x dx 8 4x dxœ � � œ � � œ � œ � �' ' ' ' '
0 0 0 0 0
2 4 x 2 2 24 x� �
!
#
#a b a b a b’ “ Š ‹# # # #"# ##y2 x# %
 8x x x 16œ � � œ � � œ œ� ‘4 32 32 480 320 96 1283 10 3 10 30 15$ &" � �#!
47. V 12 3y dy dx 12y y dx 24 12x (2 x) dxœ � œ � œ � � �' ' ' '
0 0 0 0
2 2 x 2 2� a b c d c d# $ $#�! x
 24x 6x20œ � � œ’ “# � #
!
(2 x)
4
%
48. V (3 3x) dy dx (3 3x) dy dx 6 1 x dx 6 (1 x) dx 4 2 6œ � � � œ � � � œ � œ' ' ' ' ' '
� � � � �
� �
1 x 1 0 x 1 1 0
0 x 1 1 1 x 0 1a b# #
49. V (x 1) dy dx xy y dx 1 1 2 1 dxœ � œ � œ � � � � œ �' ' ' ' '
1 1 x 1 1 1
2 1 x 2 2 2
1 x
1 x
� Î
Î
Î
� Î
c d � ‘ ˆ ‰ˆ ‰" " "
x x x
 2 x ln x 2(1 ln 2)œ � œ �c d #"
50. V 4 1 y dy dx 4 y dx 4 sec x dxœ � œ � œ �' ' ' '
0 0 0 0
3 sec x 3 3sec x
0
1 1 1Î Î Îa b ’ “ Š ‹# y3 3sec x$ $
 7 ln sec x tan x sec x tan x 7 ln 2 3 2 3œ � � œ � �2 23 3c dk k ’ “Š ‹È È1Î$!
51. dy dx dx dx lim lim 1 1' ' ' '
1 e 1 1
1
e
b
1
_ _ _
�
�
x
x
" � " "
"
x y x x x b
ln y x
$ $ $
œ œ � œ � œ � � œ’ “ ˆ ‰ � ‘ ˆ ‰
b bÄ_ Ä_
52. (2y 1) dy dx y y dx dx 4 lim sin x' ' ' '
� � � � �
�
�
� �
�1 1/ 1 x 1 1
1 1/ 1 x 1 11/ 1 x
1/ 1 x
b
È
È
ˆ ‰
a b
#
#
#
Î#
#
Î#
� œ � œ œc d c dº# �"
� !
1
1
2
1 xÈ # b 1Ä
 4 lim sin b 0 2œ � œ
b 1Ä �
c d�" 1
53. -dx dy 2 lim tan b tan 0 dy 2 lim dy' ' ' '
�_ �_
_ _ _
" "
� � � �
�" �"
a b a bx 1 y 1 y 1 y 1
2
# # # #
œ � œ
0 0
bŠ ‹Š ‹
b bÄ_ Ä_
1
 2 lim tan b tan 0 (2 )œ � œ œ1 1 1Š ‹ ˆ ‰
bÄ_
�" �" #
#
1
54. xe dx dy e lim xe e dy e lim be e 1 dy' ' ' '
0 0 0 0
x 2y 2y x x 2y b bb
0
_ _ _ _
�Ð � Ñ � � � � � �œ � � œ � � �
b bÄ_ Ä_
c d a b
 e dy lim e 1œ œ � � œ'
0
2y 2b
_
� �
" "
# #bÄ_
a b
55. f(x y) dA f 0 f(0 0) f 0 f 0 f f f' '
R
ß ¸ � ß � ß � ß � ß � � ß � !ß � ß" " " " " " " " " " " " " " "# # # # # #4 8 8 4 4 4 8 8 4
ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰
 0 0œ � � � � � � � œ" " " " " "# # #4 8 4 4 16
3 3ˆ ‰ ˆ ‰
948 Chapter 15 Multiple Integrals
56. f(x y) dA f f f f f f f f' '
R
ß ¸ ß � ß � ß � ß � ß � ß � ß � ß"4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
7 9 9 9 5 11 7 11 9 11 11 11 5 13 7 13� ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰
 f f f f ‘ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰� ß � ß � ß � ß9 13 11 13 7 15 9 154 4 4 4 4 4 4 4
 (25 27 27 29 31 33 31 33 35 37 37 39) 24œ � � � � � � � � � � � œ œ"16 16384
57. The ray meets the circle x y 4 at the point 3 1 the ray is represented by the line y .) œ � œ ß Ê œ16
x
3
# # Š ‹È È
 Thus, f(x y) dA 4 x dy dx 4 x 4 x dx 4x' '
R
ß œ � œ � � � œ � �' ' '
0 x 3 0
3 4 x 3
3
3
0
È È
È
È
È
È
Î
�
#È È’ “a b ” •# ## �x x3 4 x3 3$
#
$Î#a b
È
 œ
20 3
9
È
58. dy dx dx dx 6 ' ' ' ' '
2 0 2 2 2
2 2
0
_ _ _ _
"
� �
�
� � � �a b a bx x (y 1)
3(y 1)
x x x x x x x(x 1)
3 3 dx
# #Î$
"Î$
# # #
œ œ � œ’ “ ˆ ‰
 6 lim dx 6 lim ln (x 1) ln x 6 lim [ln (b 1) ln b ln 1 ln 2]œ � œ � � œ � � � �
b b bÄ_ Ä_ Ä_
'
2
b b
2
ˆ ‰ c d" "�x 1 x
 6 lim ln 1 ln 2 6 ln 2œ � � œ’ “ˆ ‰
bÄ_
"
b
59. V x y dy dx x y dxœ � œ �' ' '
0 x 0
1 2 x 1 2 x
x
�
�a b ’ “# # # y3$
 2x dxœ � � œ � �'
0
1’ “ ’ “# � � "
!
7x 2x 7x
3 3 3 12 12
(2 x) (2 x)$ $ %$ %
 0 0œ � � � � � œˆ ‰ ˆ ‰2 7 16 43 12 1 12 3"#
 
60. tan x tan x dx dy dx dx dy dx dy' ' ' ' ' ' '
0 0 x 0 y 2 y
2 2 x 2 y 2 2a b�" �" " " "� � �1 � œ œ �
1 1
1 1
1 y 1 y 1 y# # #
Î Î
 dy dy ln 1 y 2 tan y ln 1 yœ � œ � � � �' '
0 2
2 2 2
2
ˆ ‰ ˆ ‰1 y 2
1 y 1 y 2
� � #
� � #
�" "# �" #
!
"
# #
1 1
1
1
y ˆ ‰ � ‘c d a ba b1
1 1
 ln 5 2 tan 2 ln 1 4 2 tan 2 ln 5œ � � � � �ˆ ‰ a b1
1 1 1
� " "�" # �"
#
1
2 21 1
 2 tan 2 2 tan 2 ln 1 4œ � � � ��" �" #" #1 12
ln 5
1
a b
61. To maximize the integral, we want the domain to include all points where the integrand is positive and to
 exclude all points where the integrand is negative. These criteria are met by the points (x y) such thatß
 4 x 2y 0 or x 2y 4, which is the ellipse x 2y 4 together with its interior.� �   � Ÿ � œ# # # # # #
62. To minimize the integral, we want the domain to include all points where the integrand is negative and to
 exclude all points where the integrand is positive. These criteria are met by the points (x y) such thatß
 x y 9 0 or x y 9, which is the closed disk of radius 3 centered at the origin.# # # #� � Ÿ � Ÿ
63. No, it is not possible By Fubini's theorem, the two orders of integration must give the same result.
 Section 15.1 Double Integrals 949
64. One way would be to partition R into two triangles with the
 line y 1. The integral of f over R could then be writtenœ
 as a sum of integrals that could be evaluated by integrating
 first with respect to x and then with respect to y:
 f(x y) dA' '
R
ß
 f(x y) dx dy f(x y) dx dy.œ ß � ß' ' ' '
0 2 2y 1 y 1
1 2 y 2 2 2 y 2
� �
�Ð Î Ñ �Ð Î Ñ
 Partitioning R with the line x 1 would let us write theœ
 integral of f over R as a sum of iterated integrals with
 order dy dx. 
65. e dx dy e e dx dy e e dx dy e dx e dy' ' ' ' ' ' ' '
� � � � � � � �
� � � � � � � �
b b b b b b b b
b b b b b b b b
x y y x y x x y# # # # # # # #œ œ œŒ � Œ �Œ �
 e dx 2 e dx 4 e dx ; taking limits as b gives the stated result.œ œ œ Ä _Œ � Œ � Œ �' ' '
�
� � �
b 0 0
b b b
x x x# # #
# # #
66. dy dx dx dy dy ' ' ' ' ' '
0 0 0 0 0 0
1 3 3 1 3 3
x x x
(y 1) (y 1) (y 1) (y 1)3 3
dy# # $
#Î$ #Î$ #Î$ #Î$� � � �
" "
"
!
œ œ œ’ “
 lim lim lim (y 1) lim (y 1)œ � œ � � �" "
� �
"Î$ "Î$
3 3
dy dy
(y 1) (y 1)b 1 b 1b 1 b 1Ä ÄÄ Ä� �� �
' '
0 b
b 3 b 3
0 b#Î$ #Î$
� ‘ � ‘
 lim (b 1) ( 1) lim (b 1) (2) (0 1) 0 2 1 2œ � � � � � � œ � � � œ �’ “ ’ “ Š ‹È È
b 1 b 1Ä Ä� �
"Î$ "Î$ "Î$ "Î$ $ $
67-70. Example CAS commands:
 :Maple
 f := (x,y) -> 1/x/y;
 q1 := Int( Int( f(x,y), y=1..x ), x=1..3 );
 evalf( q1 );
 value( q1 );
 evalf( value(q1) );
71-76. Example CAS commands:
 :Maple
 f := (x,y) -> exp(x^2);
 c,d := 0,1;
 g1 := y ->2*y;
 g2 := y -> 4;
 q5 := Int( Int( f(x,y), x=g1(y)..g2(y) ), y=c..d );
 value( q5 );
 plot3d( 0, x=g1(y)..g2(y), y=c..d, color=pink, style=patchnogrid, axes=boxed, orientation=[-90,0],
 scaling=constrained, title="#71 (Section 15.1)" );
 r5 := Int( Int( f(x,y), y=0..x/2 ), x=0..2 ) + Int( Int( f(x,y), y=0..1 ), x=2..4 );
 value( r5);
 value( q5-r5 );
67-76. Example CAS commands:
 : (functions and bounds will vary)Mathematica
 You can integrate using the built-in integral signs or with the command . In the command, theIntegrate Integrate
 integration begins with the variable on the right. (In this case, y going from 1 to x).
950 Chapter 15 Multiple Integrals
 Clear[x, y, f]
 f[x_, y_]:= 1 / (x y)
 Integrate[f[x, y], {x, 1, 3}, {y, 1, x}]
 To reverse the order of integration, it is best to first plot the region over which the integration extends. This can be done
 with ImplicitPlot and all bounds involving both x and y can be plotted. A graphics package must be loaded. Remember to
 use the double equal sign for the equations of the bounding curves.
 Clear[x, y, f]
 <<Graphics`ImplicitPlot`
 ImplicitPlot[{x==2y, x==4, y==0, y==1},{x, 0, 4.1}, {y, 0, 1.1}];
 f[x_, y_]:=Exp[x ]2
 Integrate[f[x, y], {x, 0, 2}, {y, 0, x/2}] Integrate[f[x, y], {x, 2, 4}, {y, 0, 1}]�
 To get a numerical value for the result, use the numerical integrator, . Verify that this equals the original.NIntegrate
 Integrate[f[x, y], {x, 0, 2}, {y, 0, x/2}] NIntegrate[f[x, y], {x, 2, 4}, {y, 0, 1}]�
 NIntegrate[f[x, y], {y, 0, 1},{x, 2y, 4}]
 Another way to show a region is with the FilledPlot command. This assumes that functions are given as y = f(x).
 Clear[x, y, f]
 <<Graphics`FilledPlot`
 FilledPlot[{x , 9},{x, 0,3}, AxesLabels {x, y}];2 Ä
 f[x_, y_]:= x Cos[y ]2
 Integrate[f[x, y], {y, 0, 9}, {x, 0, Sqrt[y]}]
67. dy dx 0.603 68. e dy dx 0.558' ' ' '
1 1 0 0
3 x 1 1
x y"
xy ¸ ¸
� �
ˆ ‰
# #
69. tan xy dy dx 0.233 70. 31 x y dy dx 3.142' ' ' '
0 0 1 0
1 1 1 1 x
�" # #¸ � � ¸
�
�
È # È
71. Evaluate the integrals:
 e dx dy' '
0 2y
1 4
x#
 e dy dx e dy dxœ �' ' ' '
0 0 2 0
2 x/2 4 1
x x# #
 e 2 erfi 2 2 erfi 4œ � � � �" "4 4
4ˆ ‰È Èa b a b1 1
 1.1494 10¸ ‚ 6
 The following graphs was generated using
 Mathematica.
 
 Section 15.2 Areas, Moments, and Centers of Mass 951
72. Evaluate the integrals:
 x cos y dy dx x cos y dx dy' ' ' '
0 x 0 0
3 9 9 y
2 2
2
a b a bœ È
 0.157472œ ¸ �sin 814
a b
 The following graphs was generated using
 Mathematica.
 
73. Evaluate the integrals:
 x y xy dx dy x y xy dy dx' ' ' '
0 y 0 x /32
2 4 2y 8 x
2 2 2 2
3 2
3
È È
a b a b� œ �
 97.4315œ ¸67,520693
 The following graphs was generated using
 Mathematica.
 
74. Evaluate the integrals:
 e dx dy e dy dx' ' ' '
0 0 0 0
2 4 y 4 4 x� �#
xy xyœ
È
 20.5648¸
 The following graphs was generated using
 Mathematica.
 
952 Chapter 15 Multiple Integrals
75. Evaluate the integrals:
 dy dx' '
1 0
2 x#
1
x y�
 dx dy dx dyœ �' ' ' '
0 1 1 y
1 2 4 2
1 1
x y x y� �
È
 1 ln 0.909543� � ¸ˆ ‰274
 The following graphs was generated using
 Mathematica.
 
76. Evaluate the integrals:
 dx dy dy dx' ' ' '
1 y 1 1
2 8 8 x
3
3
1 1
x y x yÈ È2 2 2 2� �œ
È
 0.866649¸
 The following graphs was generated using
 Mathematica.
 
15.2 AREAS, MOMENTS, AND CENTERS OF MASS
 1. dy dx (2 x) dx 2x 2,' ' '
0 0 0
2 2 x 2�
œ � œ � œ’ “x2
#
#
!
 or dx dy (2 y) dy 2' ' '
0 0 0
2 2 y 2�
œ � œ
 
 2. dy dx (4 2x) dx 4x x 4,' ' '
0 2x 0
2 4 2 2
0œ � œ � œc d#
 or dx dy dy 4' ' '
0 0 0
4 y 2 4Î
œ œ
y
#
 
 3. dx dy y y 2 dy' ' '
� � �
�
2 y 2 2
1 y 1#
œ � � �a b#
 2yœ � � �’ “y y3
$ #
#
"
�#
 2 2 4œ � � � � � � œˆ ‰ ˆ ‰" "# #3 38 9
 
 Section 15.2 Areas, Moments, and Centers of Mass 953
 4. dx dy 2y y dy y' ' '
0 y 0
2 y y 2
�
�
#
œ � œ �a b ’ “# #
#
!
y
3
$
 4œ � œ8 43 3
 
 5. dy dx e dx e 2 1 1' ' '
0 0 0
ln 2 e ln 2
x x ln 2
0
x
œ œ œ � œc d
 
 6. dy dx ln x dx x ln x x' ' '
1 ln x 1
e 2 ln x e
e
1œ œ �c d
 (e e) (0 1) 1œ � � � œ
 
 7. dx dy 2y 2y dy y y' ' '
0 y 0
1 2y y 1
#
#
�
œ � œ �a b � ‘# # $ "
!
2
3
 œ
"
3
 
 8. dx dy y 1 2y 2 dy' ' '
� � �
�
1 2y 2 1
1 y 1 1
#
#
œ � � �a b# #
 1 y dy yœ � œ � œ'
�1
1 a b ’ “#
"
�"
y
3 3
4$
 
 9. dx dy 2y dy y' ' '
0 y 3 0
6 2y 6
#
Î
œ � œ �Š ‹ ’ “y y3 9
# $
#
'
!
 36 12œ � œ2169
 
10. dy dx 3x x dx x x' ' '
0 x 0
3 2x x 3
�
�
#
œ � œ �a b � ‘# # $" $
!
3
2 3
 9œ � œ27 9# #
 
954 Chapter 15 Multiple Integrals
11. dy dx' '
0 sin x
4 cos x1Î
 (cos x sin x) dx sin x cos xœ � œ �'
0
4
4
0
1
1
Î
Îc d
 (0 1) 2 1œ � � � œ �Š ‹ ÈÈ È2 2# #
 
12. dx dy y 2 y dy 2y' ' '
� �
�
�
1 y 1
2 y 2 2 2
1
#
œ � � œ � �a b ’ “# y y2 3
# $
 2 4 2 5œ � � � � � œ � œˆ ‰ ˆ ‰8 93 3" " "# # #
 
13. dy dx dy dx' ' ' '
� � � Î
� �
1 2x 0 x 2
0 1 x 2 1 x
�
 (1 x) dx 1 dxœ � � �' '
�1 0
0 2ˆ ‰x
#
 x x 1 (2 1)œ � � � œ � � � � � œ’ “ ’ “ ˆ ‰x x 32 4
# #
0 2
1 0�
"
# #
 
14. dy dx dy dx' ' ' '
0 x 4 0 0
2 0 4 x
#
�
�
È
 4 x dx x dxœ � �' '
0 0
2 4a b# "Î#
 4x x 8œ � � œ � � œ’ “ � ‘ ˆ ‰x 2 8 16 323 3 3 3 3
$
2
0
4
0
$Î#
 
15. (a) average sin (x y) dy dx cos (x y) dx [ cos (x ) cos x] dxœ � œ � � œ � � �" " "
1 1 1# # #
' ' ' '
0 0 0 00
1 1 1 1
1c d 1
 sin (x ) sin x [( sin 2 sin ) ( sin sin 0)] 0œ � � � œ � � � � � œ" "
1 1# #
c d1 1 1 110
 (b) average sin (x y) dy dx cos (x y) dx cos x cos x dxœ � œ � � œ � � �" Î#! #Š ‹1#
#
# #
' ' ' '
0 0 0 0
21 1 1 1Î
2 2
1 1
1 1c d � ‘ˆ ‰
 sin x sin x sin sin sin sin 0œ � � � œ � � � � � œ2 2 3 4
1 1 1
1 1 1
# # #
� ‘ � ‘ˆ ‰ ˆ ‰ ˆ ‰
# # #
1
0 1
16. average value over the square xy dy dx dx dx 0.25;œ œ œ œ œ' ' ' '
0 0 0 0
1 1 1 1’ “xy2 4x
#
"
! #
"
 average value over the quarter circle xy dy dx dxœ œ"ˆ ‰14 ' ' '0 0 0
1 1 x 1 1 x
0
È
È
�
�
#
#
 
4 xy
21 ’ “
#
 x x dx 0.159. The average value over the square is larger.œ � œ � œ ¸2 2 x x2 41 1 1'0
1a b ’ “$
"
!
"
#
# 4
17. average height x y dy dx x y dx 2x dxœ � œ � œ � œ � œ" " " "# # # #
# #
! !#4 4 3 4 3 3 3 3
y 8 x 4x 8' ' ' '
0 0 0 0
2 2 2 2a b ’ “ ’ “ˆ ‰$ $
 Section 15.2 Areas, Moments, and Centers of Mass 955
18. average dy dx dxœ œ" " "(ln 2) xy (ln 2) x
ln y
# #
' ' '
ln 2 ln 2 ln 2
2 ln 2 2 ln 2 2 ln 2 2 ln 2
ln 2
 ’ “
 (ln 2 ln ln 2 ln ln 2) dx ln xœ � � œ œ" " " "#(ln 2) x ln 2 x ln dx# ' 'ln 2 ln 2
2 ln 2 2 ln 2 2 ln 2
ln 2
ˆ ‰ ˆ ‰ c d
 (ln 2 ln ln 2 ln ln 2) 1œ � � œˆ ‰"#ln 
19. M 3 dy dx 3 2 x x dx ; M 3x dy dx 3 xy dxœ œ � � œ œ œ' ' ' ' ' '
0 x 0 0 x 0
1 2 x 1 1 2 x 1
y
2 x
x
� �
�
# #
#a b c d# #7
 3 2x x x dx ; M 3y dy dx y dx 4 5x x dxœ � � œ œ œ œ � � œ' ' ' ' '
0 0 x 0 0
1 1 2 x 1 1
x
2 x
x
a b c d a b$ # # # %# #5 3 3 194 5
�
�
#
#
 x and yÊ œ œ5 3814 35
20. M dy dx 3 dx 9 ; I y dy dx dx 27 ; R 3;œ œ œ œ œ œ œ œ$ $ $ $ $ $' ' ' ' ' '
0 0 0 0 0 0
3 3 3 3 3 3
x x
3
0
# ’ “ É Èy3 MI
$
x
 I x dy dx x y dx 3x dx 27 ; R 3y y0 0 0 0
3 3 3 3
œ œ œ œ œ œ$ $ $ $' ' ' '# # #$!c d É ÈIMy
21. M dx dy 4 y dy ; M x dx dy x dyœ œ � � œ œ œ' ' ' ' ' '
0 y 2 0 0 y 2 0
2 4 y 2 2 4 y 2
y
4 y
y 2
# #
#
Î Î
� �
�
Î
Š ‹ c dy 143
#
# #
" #
 16 8y y dy ; M y dx dy 4y y dyœ � � � œ œ œ � � œ"# #
# #' ' ' '
0 0 y 2 0
2 2 4 y 2
xŠ ‹ Š ‹y y4 15 3128 10
% $
 
#
Î
�
 x and yÊ œ œ64 535 7
22. M dy dx (3 x) dx ; M x dy dx xy dx 3x x dxœ œ � œ œ œ œ � œ' ' ' ' ' ' '
0 0 0 0 0 0 0
3 3 x 3 3 3 x 3 3
y
3 x
0
� �
�9 9
# #
#
 c d a b
 x 1 and y 1, by symmetryÊ œ œ
23. M 2 dy dx 2 1 x dx 2 ; M 2 y dy dx y dxœ œ � œ œ œ œ' ' ' ' ' '
0 0 0 0 0 0
1 1 x 1 1 1 x 1
x
1 x
0
È È
È
� �
�
# #
#È ˆ ‰ c d# # #1 14
 1 x dx x y and x 0, by symmetryœ � œ � œ Ê œ œ'
0
1a b ’ “#
"
!
x 2 4
3 3 3
$
1
24. M ; M x dy dx xy dx 5x x dx ;œ œ œ œ � œ125 6256 1
$ $
y 0 x 0 0
5 6x x 5 56x x
x
$ $ $' ' ' '
�
�
#
#c d a b# $ #
 M y dy dx y dx 35x 12x x dx x and y 5x 0 x 0 0
5 6x x 5 56x x
x
œ œ œ � � œ Ê œ œ$ ' ' ' '
�
�
#
#
$ $ $
# # #
# # $ %c d a b 625 56
25. M dy dx ; M x dy dx xy dx x a x dxœ œ œ œ œ � œ' ' ' ' ' '
0 0 0 0 0 0
a a x a a x a a
y
a x
0
È È
È
# # # #
# #
� �
�1a a
4 3
# $c d È # #
 x y , by symmetryÊ œ œ 4a31
26. M dy dx sin x dx 2; M y dy dx y dx sin x dxœ œ œ œ œ œ' ' ' ' ' ' '
0 0 0 0 0 0 0
sin x sin x
x
sin x
0
1 1 1 1 1
" "
# #
# #c d
 (1 cos 2x) dx x and yœ � œ Ê œ œ" #4 4 8'0
1
1 1 1
27. I y dy dx dx 4 x dx 4 ; I 4 , by symmetry;x y2 4 x 2 2
2 4 x 2 24 x
4 x
œ œ œ � œ œ' ' ' '
� � � � �
�
�
� �
È
È
È
È
#
#
#
#
# # $Î#’ “ a by3 32
$
1 1
 I I I 8o x yœ � œ 1
28. I x dy dx sin x 0 dx (1 cos 2x) dxy
2 sin x x 2 2
0
œ œ � œ � œ' ' ' '
1 1 1
1 1 1
ˆ ‰
# #
Î
# # "
# #a b 1
29. M dy dx e dx lim e dx 1 lim e 1; M x dy dx xe dxœ œ œ œ � œ œ œ' ' ' ' ' ' '
�_ �_ �_ �_
0 e 0 0 0 e 0
0 b 0
x x b x
y
x x
b bÄ �_ Ä �_
 lim xe dx lim xe e 1 lim be e 1; M y dy dxœ œ � œ � � � œ � œ
b b bÄ �_ Ä �_ Ä �_
' ' '
b 0
0 0 e
x x x b b0
b xc d a b
�_x
956 Chapter 15 Multiple Integrals
 e dx lim e dx x 1 and yœ œ œ Ê œ � œ" " " "# #' '
�_
0 0
2x 2x
bbÄ �_ 4 4
30. M x dy dx lim xe dx lim 1 1y 0 0 0
e b
x 2 b
0
œ œ œ � � œ' ' '
_
� Î
� Î
#
#
#
Î
x 2
x 2b bÄ_ Ä_
� ‘"
e
31. M (x y) dx dy xy dy 2y 2y dy ;œ � œ � œ � � œ � � œ' ' ' '
0 y 0 0
2 y y 2 2y y
y�
�
�
�
#
#
’ “ Š ‹ ’ “x 82 2 10 3 15y y y 2y
#
% & % $
$ #
#
#
!
 I y (x y) dx dy xy dy 2y 2y dy ;x 0 y 0 0
2 y y 2 2y y
y
œ � œ � œ � � œ' ' ' '
�
�
�
�
#
#
# $ & %’ “ Š ‹x y y2 2 10564
# # '
 R 2x œ œ œÉ É ÉIM 7 78 2x
32. M 5x dx dy 5 dy 12 4y 16y dy 23 3œ œ œ � � œ' ' ' '
� Î � Î � Î
Î � Î Î
�
È È È
È È È
È
È
3 2 4y 3 2 3 2
3 2 12 4y 3 2 3 212 4y
4y
#
#
#
#
’ “ a b Èx 52
#
#
# %
33. M (6x 3y 3) dy dx 6xy y 3y dx 12 12x dx 8;œ � � œ � � œ � œ' ' ' '
0 x 0 0
1 2 x 1 12 x
x
�
�� ‘ a b3# # #
 M x(6x 3y 3) dy dx 12x 12x dx 3; M y(6x 3y 3) dy dxy x0 x 0 0 x
1 2 x 1 1 2 x
œ � � œ � œ œ � �' ' ' ' '
� �a b$
 14 6x 6x 2x dx x and yœ � � � œ Ê œ œ'
0
1a b# $ #17 3 178 16
34. M (y 1) dx dy 2y 2y dy ; M y(y 1) dx dy 2y 2y dy ;œ � œ � œ œ � œ � œ' ' ' ' ' '
0 y 0 0 y 0
1 2y y 1 1 2y y 1
x
# #
# #
� �a b a b$ # %"# 415
 M x(y 1) dx dy 2y 2y dy x and y ; I y (y 1) dx dyy 0 y 0 0 y
1 2y y 1 1 2y y
œ � œ � œ Ê œ œ œ �' ' ' ' '
# #
# #
� �a b# % #4 8 815 15 15 x
 2 y y dyœ � œ'
0
1a b$ & "6
35. M (x y 1) dx dy (6y 24) dy 27; M y(x y 1) dx dy y(6y 24) dy 14;œ � � œ � œ œ � � œ � œ' ' ' ' ' '
0 0 0 0 0 0
1 6 1 1 6 1
x
 M x(x y 1) dx dy (18y 90) dy 99 x and y ; I x (x y 1) dx dyy y0 0 0 0 0
1 6 1 1 6
œ � � œ � œ Ê œ œ œ � �' ' ' ' '11 143 27 #
 216 dy 432; R 4œ � œ œ œ'
0
1
yˆ ‰ Éy3 6 M11 Iy
36. M (y 1) dy dx x dx ; M y(y 1) dy dx dxœ � œ � � � œ œ � œ � �' ' ' ' ' '
� � � �1 x 1 1 x 1
1 1 1 1 1 1
x
# #
Š ‹ Š ‹x 3 32 5 x x15 6 3
% ' %
# # #
#
 ; M x(y 1) dy dx x dx 0 x 0 and y ; I x (y 1) dy dxœ œ � œ � � œ Ê œ œ œ �48 3x x 935 14y y1 x 1 1 x
1 1 1 1 1' ' ' ' '
� � �
# #
Š ‹# # $ #
&
 x dx ; Rœ � � œ œ œ'
�1
1
yŠ ‹ É É3x x 16 32 2 35 M 14I
# ' % y
37. M (7y 1) dy dx x dx ; M y(7y 1) dy dx dx ;œ � œ � œ œ � œ � œ' ' ' ' ' '
� � � �1 0 1 1 0 1
1 x 1 1 x 1
x
# #
Š ‹ Š ‹7x 31 7x x 1315 3 2 15
% ' %
#
#
 M x(7y 1) dy dx x dx 0 x 0 and y ; I x (7y 1) dy dxy y1 0 1 1 0
1 x 1 1 x
œ � œ � œ Ê œ œ œ �' ' ' ' '
� � �
# #
Š ‹7x 1331
&
#
$ #
 x dx ; Rœ � œ œ œ'
�1
1
yŠ ‹ É É7x 7 215 M 31I
'
#
% y
38. M 1 dy dx 2 dx 60; M y 1 dy dx 1 dx 0;œ � œ � œ œ � œ � œ' ' ' ' ' '
0 1 0 0 1 0
20 1 20 20 1 20
x
� �
ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰’ “Š ‹x x x x20 10 20 0 y# #
"
�"
#
 M x 1 dy dx 2x dx x and y 0; I y 1 dy dxy x0 1 0 0 1
20 1 20 20 1
œ � œ � œ Ê œ œ œ �' ' ' ' '
� �
ˆ ‰ ˆ ‰Š ‹x x 2000 100 x20 10 3 9 20
# #
 1 dx 20; Rœ � œ œ œ2 x3 20 M 3
I'
0
20
x
ˆ ‰ É Éx "
 Section 15.2 Areas, Moments, and Centers of Mass 957
39. M (y 1) dx dy 2y 2y dy ; M y(y 1) dx dy 2 y y dy ;œ � œ � œ œ � œ � œ' ' ' ' '
0 y 0 0 y
1 y 1 1 y
x
1
0
� �
a b a b'# $ #5 73 6
 M x(y 1) dx dy 0 dy 0 x 0 and y ; I y (y 1) dx dy 2y 2y dyy x0 y 0 0 y 0
1 y 1 1 y 1
œ � œ œ Ê œ œ œ � œ �' ' ' ' ' '
� �
7
10
# % $a b
 R ; I x (y 1) dx dy 2y 2y dy R ;œ Ê œ œ œ � œ � œ Ê œ œ9 310 M 10 3 10 M 10I 3 6
I 3 2
x y y0 y 0
1 y 1É a b Éx È È' ' '
�
# % $" y
 I I I Ro x yœ � œ Ê œ œ65 M 5
I 3 2
! É o È
40. M 3x 1 dx dy 2y 2y dy ; M y 3x 1 dx dy 2y 2y dy ;œ � œ � œ œ � œ � œ' ' ' ' ' '
0 y 0 0 y 0
1 y 1 1 y 1
x
� �
a b a b a b a b# $ # % ##3 1615
 M x 3x 1 dx dy 0 x 0 and y ; I y 3x 1 dx dy 2y 2y dyy x0 y 0 y 0
1 y 1 y 1
œ � œ Ê œ œ œ � œ � œ' ' ' ' '
� �
a b a b a b# # # & $32 545 6
 R ; I x 3x 1 dx dy 2 y y dy R ;Ê œ œ œ � œ � œ Ê œ œx y y0 y 0
1 y 1É a b ˆ ‰ É ÉIM 3 5 3 30 M 455 3 11 11Ix
È ' ' '
�
# # & $" y
 I I I Ro x y oœ � œ Ê œ œ6 25 M
I
5
É o È
41. dy dx 10,000 1 e 10,000 1 e ' ' ' ' '
� � � � !
!
5 2 5 5
5 0 5 510,000e
1 1
dx dx dx
1 1
y
� �
�# �#
� �k k k kx x x x
# #
# #
œ � œ � �a b a b ’ “
 10,000 1 e 2 ln 1 10,000 1 e 2 ln 1œ � � � � � �a b a b� ‘ � ‘ˆ ‰ ˆ ‰� �2 2x x# #
! &
�& !
 10,000 1 e 2 ln 1 10,000 1 e 2 ln 1 40,000 1 e ln 43,329œ � � � � � œ � ¸a b a b a b� ‘ � ‘ ˆ ‰ˆ ‰ ˆ ‰�# �# �## #5 5 72
42. 100(y 1) dx dy 100(y 1)x dy 100(y 1) 2y 2y dy 200 y y dy' ' ' ' '
0 y 0 0 0
1 2y y 1 1 1
2y y
y
#
#
#
#
�
�
� œ � œ � � œ �c d a b a b# $
 200 (200) 50œ � œ œ’ “ ˆ ‰y y2 4 4
# %
"
!
"
43. M dy dx 2a 1 x dx 2a x ; M y dy dxœ œ � œ � œ œ' ' ' ' '
� �
� �
1 0 0 1 0
1 a 1 x 1 1 a 1 x
x
ˆ ‰ ˆ ‰
# #
a b ’ “#
"
!
x 4a
3 3
$
 1 2x x dx a x y . The angle between theœ � � œ � � œ Ê œ œ œ2a 2x x 8a 2a3 5 15 M 5
M# $ & #
#
# % #
"
!
'
0
1 a b ’ “ x Š ‹Š ‹
8a
15
4a
3
#
)
 x-axis and the line segment from the fulcrum to the center of mass on the y-axis plus 45° must be no more than
 90° if the center of mass is to lie on the left side of the line x 1 tan a .œ Ê � Ÿ Ê Ÿ Ê Ÿ) 1 1 14 5 4
2a 5
# #
�" ˆ ‰
 Thus, if 0 a , then the appliance will have to be tipped more than 45° to fall over.� Ÿ 5#
44. f(a) I (y a) dy dx dx (2 a) a ; thus f (a) 0 4(2 a) 4aœ œ � œ � œ � � œ Ê � � �a 0 0 0
4 2 4' ' '# $ $ w # #�’ “ c d(2 a)3 3 3a 4
$
$
 0 a (2 a) 0 4 4a 0 a 1. Since f (a) 8(2 a) 8a 16 0, a 1 gives aœ Ê � � œ Ê � � œ Ê œ œ � � œ � œ# # ww
 minimum value of I .a
45. M dy dx dx 2 sin x 2 0 ; M x dy dxœ œ œ œ � œ œ' ' ' ' '
0 1 1 x 0 0 1 1 x
1 1 1 x 1 1 1 1 x
y
� Î � � Î �
Î � Î �
È È
È È
# #
# #
2
1 xÈ � �"
"
! ## c d ˆ ‰1 1
 dx 2 1 x 2 x and y 0 by symmetryœ œ � � œ Ê œ œ'
0
1
2x 2
1 xÈ � #
"Î# "
!
#
’ “a b
1
46. (a) I x dx Rœ œ Ê œ œ'
� Î
Î
L 2
L 2
$
# "$ $
$
L L L
12 12 L 2 3
$
$É † È
 (b) I x dx Rœ œ Ê œ œ'
0
L
$
# "$ $
$
L L L
3 3 L 3
$
$É † È
47. (a) M dx dy 2 y y dy 2 2 " "# ##
"
!
œ œ œ � œ � œ œ Ê œ' ' '
0 y 0
1 2y y 1
#
#
�
$ $ $ $ $a b ’ “ ˆ ‰y y2 3 6 3 3
# $
$
958 Chapter 15 Multiple Integrals
 (b) average value , so the values are the sameœ œ œ œ
' '
' '
0 y
1 2y y
0 y
1 2y y
#
�
#
#
�
#
(y 1) dx dy
 dx dy
� Š ‹
Š ‹
"
#
"
3
3
# $
48. Let (x y ) be the location of the weather station in county i for i 1 254. The average temperaturei iß œ ßá ß
 in Texas at time t is approximately , where T(x y ) is the temperature at time t at the! !
ß!
254
i 1œ
 T(x y ) A
A i i
i i i?
ß
 weather station in county i, A is the area of county i, and A is the area of Texas.?i
49. (a) x 0 M x (x y) dy dx 0œ œ Ê œ ß œMMy y ' '
R
$
 (b) I (x h) (x y) dA x (x y) dA 2hx (x y) dA h (x y) dAL œ � ß œ ß � ß � ß' ' ' ' ' ' ' '
R R R R
# # #
$ $ $ $
 I 0 h (x y) dA I mhœ � � ß œ �y c m# #' '
R
$
Þ Þ
50. (a) I I mh I I mh 14 ; I I mh 12 14c m L x 5/7 y y 11 14 xÞ Þ œ Îœ � Ê œ � œ � œ œ � œ � œ# # #œ # #39 5 23 11 475 7 35 14 14ˆ ‰ ˆ ‰
 (b) I I mh 14 ; I I mh 14 24x 1 x 5 7 y 2œ œ Î œœ � œ � œ œ � œ � œ# ## #23 2 9 47 1735 7 5 14 14ˆ ‰ ˆ ‰y 11 14œ Î
51. M y dA y dA M M x ; likewise, y ;xp p
" #
�
œ
' ' ' '
R R
"
#
" #
� �
� �� œ � Ê œ œx x
M M M M
m m m m" #
" # " #
" # " #
x x y y
 thus x y M M M M m x m x m y m yc i j i j i jœ � œ � � � œ � � �" "� � " " # # " #" #m m m mx x y y
" # " #
" # " #
c d c da b a b a b a b
 m x y m x yœ � � � œ"� �" " # #" #
�
m m m m
m m
" # " #
" " # #c da b abi j i j c c
52. From Exercise 51 we have that Pappus's formula is true for n 2. Assume that Pappus's formula is true forœ
 n k 1, i.e., that (k 1) . The first moment about x of k nonoverlapping plates isœ � � œc
!
!
k 1
i 1
k 1
i 1
�
œ
�
œ
 m
 m
i i
i
c
 y dA y dA M M x ; similarly, y ;! � �k 1
i 1
�
œ
' ' ' '
R Ri k
i k x x
M M M M
 m m m m
� œ � Ê œ œ
c
c c
Ð � Ñ
Ð � Ñ Ð � Ñ
k 1 k
x x y yk 1 k 1k k
i k i k
� �
� �� � � �! !k 1 k 1
i 1 i 1
� �
œ œ
 thus (k) x y M M M Mc i j i jœ � œ � � �"!k
i 1
i
k 1 k k 1 k
œ
Ð � Ñ Ð � Ñ
 m
� ‘ˆ ‰ ˆ ‰
x x y yc c
 m x m x m y m yœ � � �"!k
i 1
i
œ
 m
” •Œ � Œ �Œ � Œ �! !k 1 k 1
i 1 i 1
� �
œ œ
i k k i k kc ci j
 m x y m x yœ � � � œ"
� �
! !
!
k k 1
i 1 i 1
k 1
i 1
œ œ
�
œ
�
 m m
i k k k
 m (k 1) m
i i
i k k– —� �a b a b!k 1
i 1
�
œ
c c
c c
i j i j
� �
 , and by mathematical induction the statement follows.œ m m m m
m m m m
" " # # � �
" # �
c c c c� �á� �
� �á� �
k 1 k 1 k k
k 1 k
53. (a) x and yc œ œ Ê œ œ8( 3 ) 2(3 3.5 ) 14 318 2 10 5 107 31i j i j i j� � � ��
 (b) x and yc œ œ Ê œ œ8( 3 ) 6(5 2 ) 38 3614 14 7 719 18i j i j i j� � � �
 (c) x and yc œ œ Ê œ œ2(3 3.5 ) 6(5 2 ) 36 198 8 2 89 19i j i j i j� � � �
 (d) x and yc œ œ Ê œ œ8( 3 ) 2(3 3.5 ) 6(5 2 ) 44 4316 16 4 1611 43i j i j i j i j� � � � � �
54. c œ œ œ œ15 7 48(12 )15 48 4 63 4 63 4 7
15(3 28 ) 48(48 4 ) 2349 612 261 68ˆ ‰34 i j i j i j i j i j i j� � �
�
� � � � �
† † †
 x and yÊ œ œ261 178 7#
 Section 15.3 Double Integrals in Polar Form 959
55. Place the midpoint of the triangle's base at the origin and above the semicircle. Then the center of
 mass of the triangle is 0 , and the center of mass of the disk is 0 from Exercise 25. Fromˆ ‰ ˆ ‰ß ß �h 4a3 31
 Pappus's formula, , so the centroid is on the boundaryc œ œ
(ah)
ah ah
ˆ ‰ ˆ ‰Š ‹ Š ‹
Š ‹ Š ‹
h a 4a ah 2a
3 2 3 3
a a
j j j� �
� �
1
1
1 1
# # $
# #
# #
�
 if ah 2a 0 h 2a h a 2 . In order for the center of mass to be inside T we must have# $ # #� œ Ê œ Ê œ È
 ah 2a 0 or h a 2.# $� � � È
56. Place the midpoint of the triangle's base at the origin and above the square. From Pappus's formula,
 , so the centroid is on the boundary if 0 h 3s 0 h s 3.c œ � œ Ê � œ Ê œ
Š ‹Š ‹ ˆ ‰
Š ‹
sh h s
3 2
sh
#
#
#
j j�
� #
# #
s
sh s
6
#
# $
�
s
È
15.3 DOUBLE INTEGRALS IN POLAR FORM
 1. dy dx r dr d d' ' ' ' '
�
�
1 0 0 0 0
1 1 x 1È #
œ œ œ
1 1
) )
"
# #
1
 2. dy dx r dr d d' ' ' ' '
� � �
�
1 1 x 0 0 0
1 1 x 2 1 2
È
È
#
#
œ œ œ
1 1
) ) 1
"
#
 3. x y dx dy r dr d d' ' ' ' '
0 0 0 0 0
1 1 y 2 1 2È � Î Î# a b# # $ "� œ œ œ1 1) )4 81
 4. x y dx dy r dr d d' ' ' ' '
� � �
�
1 1 y 0 0 0
1 1 y 2 1 2
È
È
#
# a b# # $ " #� œ œ œ1 1) )4 1
 5. dy dx r dr d d a' ' ' ' '
� � �
�
a a x 0 0 0
a a x 2 a 2
È
È
# #
# #
œ œ œ
1 1
) ) 1
a
2
# #
 6. x y dx dy r dr d 4 d 2' ' ' ' '
0 0 0 0 0
2 4 y 2 2 2È � Î Î# a b# # $� œ œ œ1 1) ) 1
 7. x dx dy r cos dr d 72 cot csc d 36 cot 36' ' ' ' '
0 0 4 0 4
6 y 2 6 csc 2 2
4œ œ œ � œ
1 1
1 ) 1
1
1
Î Î
Î Î
Î
Î
# # #
) ) ) ) ) )c d
 8. y dy dx r sin dr d tan sec d' ' ' ' '
0 0 0 0 0
2 x 4 2 sec 4
œ œ œ
1 ) 1Î Î
# #
) ) ) ) )
8 4
3 3
 9. dy dx dr d 2 1 dr d 2 (1 ln 2) d' ' ' ' ' ' '
� � �
Î Î Î
1 1 x 0 0
0 0 3 2 1 3 2 1 3 2
È #
2 2r
1 x y 1 r 1 r� � � �
"È # # œ œ � œ �
1 1 1
1 1 1
) ) )ˆ ‰
 (1 ln 2)œ � 1
10. dx dy dr d 4 1 dr d 4 1 d' ' ' ' ' ' '
� � � Î Î Î
Î Î Î
1 1 y 2 0 2 0 2
1 0 3 2 1 3 2 1 3 2
È
#
4 x y
1 x y 1 r 1 r 4
4rÈ # #
# # # #
#�
� � � �
"œ œ � œ �
1 1 1
1 1 1
) ) )ˆ ‰ ˆ ‰1
 4œ �1 1#
11. e dx dy re dr d (2 ln 2 1) d (2 ln 2 1)' ' ' ' '
0 0 0 0 0
ln 2 (ln 2) y 2 ln 2 2
r
È
# #
� Î Î
 
Èx y# #� œ œ � œ �
1 1
) )
1
#
12. e dy dx re dr d 1 d' ' ' ' '
0 0 0 0 0
1 1 x 2 1 2
x y
È
ˆ ‰
� Î Î
� �
#
# #
 œ œ � � œ
1 1
� " "
#
�r
e 4e
(e 1)#
) )ˆ ‰ 1
13. dy dx r dr d 2 cos 2 sin cos d' ' ' ' '
0 0 0 0 0
2 1 (x 1) 2 2 cos 2È � � Î Î#
x y r(cos sin )
x y r
� �
�
#
# # #
œ œ �
1 ) 1
) )
) ) ) ) )a b
 sin 1œ � � œ œ �� ‘) )sin 2 22 ) 1 1# �# #1Î20
960 Chapter 15 Multiple Integrals
14. xy dx dy sin cos r dr d sin cos d sin' ' ' ' '
0 1 (y 1) 2 0 2
2 0 2 sin 
2
� � � Î Î
Î
È
#
 
# # % ( )œ œ œ œ �
1 1
1 ) 1
1
1
) ) ) ) ) ) )
32 4 4
5 5 5c d
15. ln x y 1 dx dy 4 ln r 1 r dr d 2 (ln 4 1) d (ln 4 1)' ' ' ' '
� � �
� Î Î
1 1 y 0 0 0
1 1 y 2 1 2
È
È
#
#
 a b a b# # #� � œ � œ � œ �1 1) ) 1
16. dy dx 4 dr d 4 d 2 d' ' ' ' ' '
� � �
� Î Î Î
1 1 x 0 0 0 0
1 1 x 2 1 2 2
È
È
#
#
2 2r
1 x y 1 r 1 ra b a b� � � "�
"
!# # ## # #
œ œ � œ œ
1 1 1
) ) ) 1� ‘
17. r dr d 2 (2 sin 2 ) d 2( 1)' ' '
0 0 0
2 2 2 sin 2 21 ) 1Î � ÎÈ
 ) ) ) 1œ � œ �
18. A 2 r dr d 2 cos cos dœ œ � œ' ' '
0 1 0
2 1 cos 21 ) 1Î � Î
 ) ) ) )a b# �8 41
19. A 2 r dr d 144 cos 3 d 12œ œ œ' ' '
0 0 0
6 12 cos 3 61 ) 1Î Î
) ) ) 1
#
20. A r dr d dœ œ œ' ' '
0 0 0
2 4 3 21 ) 1Î
 ) ) )
8 64
9 27
# 1$
21. A r dr d 2 sin d 1œ œ � � œ �' ' '
0 0 0
2 1 sin 21 ) 1Î � Î
) ) )
1 3 cos 2 3
2 8
ˆ ‰
# #
) 1
22. A 4 r dr d 2 2 cos d 4œ œ � � œ �' ' '
0 0 0
2 1 cos 21 ) 1Î � Î
 ) ) )ˆ ‰3 cos 2 32# # ) 1
23. M 3r sin dr d (1 cos ) sin d 4x 0 0 0
1 cos 
œ œ � œ' ' '
1 ) 1�
# $
) ) ) ) )
24. I y k x y dy dx k r sin dr d d ;x
a a x 0 0 0
a a x 2 a 2
œ � œ œ œ' ' ' ' '
� � �
�
È
È
# #
# #
# # # & # �
#c da b 1 1) ) )ka 1 cos 2 ka6 6' ') 1
 I k x y dy dx k r dr d do
a a x 0 0 0
a a x 2 a 2
œ � œ œ œ' ' ' ' '
� � �
�
È
È
# #
# # a b# # &# 1 1) )ka ka6 3' '1
25. M 2 dr d 2 (6 sin 3) d 6 2 cos 6 3 2œ œ � œ � � œ �' ' '
1 1
1 ) 1
1
1
Î Î
Î Î
Î
Î6 3 6
2 6 sin 2
2
6) ) ) ) ) 1c d È
26. I r dr d cos 2 cos d 2 sin 2o 2 1 2
3 2 1 cos 3 2 3 2
2
œ œ � œ � � œ �' ' '
1 1
1 ) 1
1
1
Î Î
Î � Î
Î
Î
) ) ) ) )
" "
# # #
#a b � ‘sin 24 4) ) 1
27. M 2 r dr d (1 cos ) d ; M 2 r cos dr dœ œ � œ œ' ' ' ' '
0 0 0 0 0
1 cos 1 cos 
y
1 ) 1 1 )� �
 ) ) ) ) )
# #
#
31
 2 cos 2 sin cos d x and y 0, by symmetryœ � � � � œ Ê œ œ'
0
1ˆ ‰4 cos 15 cos 4 5 5
3 24 4 4 6
) ) 1
) ) ) )
#
28. I r dr d (1 cos ) do 0 0 0
2 1 cos 2
œ œ � œ' ' '
1 ) 1�
$ %"
) ) )4 16
351
29. average r a r dr d a dœ � œ œ4 4 2a
a 3 a 31 1# #
' ' '
0 0 0
2 a 21 1Î Î
 
È # # $) )
30. average r dr d a dœ œ œ4 4 2a
a 3 a 31 1# #
' ' '
0 0 0
2 a 21 1Î Î
# $
) )
31. average x y dy dx r dr d dœ � œ œ œ" "# # #
1 1 1a a 3 3
a 2a
# #
' ' ' ' '
� � �
�
a a x 0 0 0
a a x 2 a 2
È
È
# #
# # È 1 1) )
 Section 15.3 Double Integrals in Polar Form 961
32. average (1 x) y dy dx (1 r cos ) r sin r dr dœ � � œ � �" "# # # # #
1 1
' '
R
c d c d' '
0 0
2 11
) ) )
 r 2r cos r dr d dœ � � œ � œ � œ" " "$ # #1 1 1
) )' ' '
0 0 0
2 1 2 2
0
1 1
1a b ˆ ‰ � ‘) ) ) )3 2 cos 3 2 sin 34 3 4 3
33. r dr d 2 ln r dr d 2 r ln r r d 2 e 1 1 d 2 2 e' ' ' ' ' '
0 1 0 1 0 0
2 e 2 e 2 2
e
1
1 1 1 1
È ÈŠ ‹ c d È È� ‘ ˆ ‰ˆ ‰ln r
r
#
) ) ) ) 1œ œ � œ � � œ �
"Î# "
#
34. dr d dr d (ln r) d d 2' ' ' ' ' '
0 1 0 1 0 0
2 e 2 e 2 2
e
1
1 1 1 1Š ‹ ˆ ‰ c dln r 2 lnr
r r
#
) ) ) ) 1œ œ œ œ#
35. V 2 r cos dr d 3 cos 3 cos cos dœ œ � �' ' '
0 1 0
2 1 cos 21 ) 1Î � Î
# # $ %
) ) ) ) ) )
2
3 a b
 sin 2 3 sin sinœ � � � � œ �2 5 sin 4 4 53 8 32 3 8� ‘" $) ) 1) ) ) 1Î20
36. V 4 r 2 r dr d (2 2 cos 2 ) 2 dœ � œ � � �' ' '
0 0 0
4 2 cos 2 41 ) 1Î ÎÈ È � ‘# $Î# $Î#) ) )43
 1 cos sin d cos œ � � œ � � œ2 2 2 2 6 2 40 2 643 3 3 3 3 9
32 32 cos1 1 1)È È È È'
0
4 4
0
1
1
Î
Îa b ’ “# � �) ) ) )$
37. (a) I e dx dy e r dr d lim re dr d# � �œ œ œ' ' ' ' ' '
0 0 0 0 0 0
2 2 b_ _ Î _ Î
� �a bx y# #
1 1Š ‹ ” •r r# #) )bÄ_
 lim e 1 d d Iœ � � œ œ Ê œ" "# # #
�' '
0 0
2 21 1Î Î
bÄ_
Š ‹b 4# ) ) 1 1È
 (b) lim dt e dt 1, from part (a)
xÄ_
' '
0 0
x
2e 2 2t� # #tÈ È È È1 1 1 1œ œ œ
_
�
#Š ‹Š ‹
38. dx dy dr d lim dr lim ' ' ' ' '
0 0 0 0 0
2 b b
0
_ _ Î _
" "
� � � �# �a b a b a b1 x y 1 r 1 rr r 4 1 r# # # ## # # #œ œ œ �
1
)
1 1
b bÄ_ Ä_
� ‘
 lim 1œ � œ1 14 1 b 4bÄ_
ˆ ‰"
� #
39. Over the disk x y : dA dr d ln 1 r d# # #" "� � �� Ÿ œ œ � �
3 r
4 1 x y 1 r 2
' '
R
 
# # #
' ' '
0 0 0
2 3 2 2 3 2
0
1 1
È
È
Î
Î
) )� ‘a b
 ln d (ln 2) d ln 4œ � œ œ' '
0 0
2 21 1ˆ ‰" "
# 4 ) ) 1
 Over the disk x y 1: dA dr d lim dr d# # "� � � �� Ÿ œ œ' '
R a 1
1 x y 1 r 1 r
r r
# # # #
' ' ' '
0 0 0 0
2 1 2 a1 1
) )’ “
Ä
�
 lim ln 1 a d 2 lim ln 1 a 2 , so the integral does not exist overœ � � œ � � œ _'
0
21
a 1 a 1Ä Ä� �
� ‘ � ‘a b a b" "# ## #) 1 1† †
 x y 1# #� Ÿ
40. The area in polar coordinates is given by A r dr d d f ( ) d r d ,œ œ œ œ' ' ' ' '
! ! ! !
" ) " " "
)
0
f f
0
Ð Ñ
Ð Ñ
) ) ) ) )’ “r2# " "# ## #
 where r f( )œ )
41. average (r cos h) r sin r dr d r 2r h cos rh dr dœ � � œ � �" "# # # $ # #
1 1a a# #
' ' ' '
0 0 0 0
2 a 2 a1 1c d a b) ) ) ) )
 d dœ � � œ � � œ � �" " "# # #1 1 1
) ) ) ) )
a 4 3 4 3 4 3
a 2a h cos a h a 2ah cos h a 2ah sin h
#
% $ # # # # # #' '
0 0
2 2 2
0
1 1
1Š ‹ Š ‹ ’ “) )
 a 2hœ �"#
# #a b
962 Chapter 15 Multiple Integrals
42. A r dr d 4 sin csc dœ œ �' ' '
1 ) 1
1 ) 1
Î Î
Î Î
4 csc 4
3 4 2 sin 3 4
) ) ) )
"
#
# #a b
 2 sin 2 cot œ � � œ"# #c d) ) ) 3 4411 ÎÎ 1
 
44-46. Example CAS commands:
 :Maple
 f := (x,y) -> y/(x^2+y^2);
 a,b := 0,1;
 f1 := x -> x;
 f2 := x -> 1;
 plot3d( f(x,y), y=f1(x)..f2(x), x=a..b, axes=boxed, style=patchnogrid, shading=zhue, orientation=[0,180], title="#43(a)
 (Section 15.3)" ); # (a)
 q1 := eval( x=a, [x=r*cos(theta),y=r*sin(theta)] ); # (b)
 q2 := eval( x=b, [x=r*cos(theta),y=r*sin(theta)] );
 q3 := eval( y=f1(x), [x=r*cos(theta),y=r*sin(theta)] );
 q4 := eval( y=f2(x), [x=r*cos(theta),y=r*sin(theta)] );
 theta1 := solve( q3, theta );
 theta2 := solve( q1, theta );
 r1 := 0;
 r2 := solve( q4, r );
 plot3d(0,r=r1..r2, theta=theta1..theta2, axes=boxed, style=patchnogrid, shading=zhue, orientation=[-90,0],
 title="#43(c) (Section 15.3)" );
 fP := simplify(eval( f(x,y), [x=r*cos(theta),y=r*sin(theta)] )); # (d)
 q5 := Int( Int( fP*r, r=r1..r2 ), theta=theta1..theta2 );
 value( q5 );
 : (functions and bounds will vary)Mathematica
 For 43 and 44, begin by drawing the region of integration with the command.FilledPlot
 Clear[x, y, r, t]
 <<Graphics`FilledPlot`
 FilledPlot[{x, 1}, {x, 0, 1}, AspectRatio 1, AxesLabel {x,y}];Ä Ä
 The picture demonstrates that r goes from 0 to the line y=1 or r = 1/ Sin[t], while t goes from /4 to /2.1 1
 f:= y / (x y )2 2�
 topolar={x r Cos[t], y r Sin[t]};Ä Ä
 fp= f/.topolar //Simplify
 Integrate[r fp, {t, /4, /2}, {r, 0, 1/Sin[t]}]1 1
 For 45 and 46, drawing the region of integration with the ImplicitPlot command.
 Clear[x, y]
 <<Graphics`ImplicitPlot`
 ImplicitPlot[{x==y, x==2 y, y==0, y==1}, {x, 0, 2.1}, {y, 0, 1.1}];�
 The picture shows that as t goes from 0 to /4, r goes from 0 to the line x=2 y. will find the bound for r.1 � Solve
 bdr=Solve[r Cos[t]==2 r Sin[t], r]//Simplify�
 f:=Sqrt[x y]�
 topolar={x r Cos[t], y r Sin[t]};Ä Ä
 fp= f/.topolar //Simplify
 Integrate[r fp, {t, 0, /4}, {r, 0, bdr[[1, 1, 2]]}]1
 Section 15.4 Triple Integrals in Rectangular Coordinates 963
15.4 TRIPLE INTEGRALS IN RECTANGULAR COORDINATES
 1. F x, y, z dy dz dx dy dz dx 1 x z dz dx' ' ' ' ' ' ' '
0 0 x z 0 0 x z 0 0
1 1 x 1 1 1 x 1 1 1 x� � �
� �
a b a bœ œ � �
 1 x x 1 x dx dxœ � � � � œ œ œ�' '0 0
1 1’ “ ’ “a b a b a b a b a b1 x 1 x2 2 61 x6 0 1� � �
"2 2 3
 2. dz dy dx 3 dy dx 6 dx 6, dz dx dy, dx dy dz, dx dz dy,' ' ' ' ' ' ' ' ' ' ' ' ' ' '
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 2 3 1 2 1 2 1 3 3 2 1 2 3 1
 œ œ œ
 dy dx dz, dy dz dx' ' ' ' ' '
0 0 0 0 0 0
3 1 2 1 3 2
 3. dz dy dx' ' '
0 0 0
1 2 2x 3 3x 3y 2� � � Î
 3 3x y dy dxœ � �' '
0 0
1 2 2x� ˆ ‰3
2
 3(1 x) 2(1 x) 4(1 x) dxœ � � � �'
0
1� ‘† †34 #
 3 (1 x) dx (1 x) 1,œ � œ � � œ'
0
1
# $ "
!c d
 dz dx dy, dy dz dx,' ' ' ' ' '
0 0 0 0 0 0
2 1 y 2 3 3x 3y 2 1 3 3x 2 2x 2z 3� Î � � Î � � � Î
 dy dx dz, dx dz dy,' ' ' ' ' '
0 0 0 0 0 0
3 1 z 3 2 2x 2z 3 2 3 3y 2 1 y 2 z 3� Î � � Î � Î � Î � Î
 dx dy dz' ' '
0 0 0
3 2 2z 3 1 y 2 z 3� Î � Î � Î
 
 4. dz dy dx 4 x dy dx 3 4 x dx x 4 x 4 sin 6 sin 1 3 ,' ' ' ' ' '
0 0 0 0 0 0
2 3 4 x 2 3 2 2
0
È
�
#
œ � œ � œ � � œ œÈ È È’ “# # ## #�" �"3 x 1
 dz dx dy, dy dz dx, dy dx dz, dx dy dz, dx dz dy' ' ' ' ' ' ' ' ' ' ' ' ' ' '
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 2 4 x 2 4 x 3 2 4 z 3 2 3 4 z 3 2 4 zÈ È È È È� � � � �# # # # #
 5. dz dy dx 4 dz dy dx' ' ' ' ' '
� � � � �
� � � � � �
2 4 x x y 0 0 x y
2 4 x 8 x y 2 4 x 8 x y
È
È È
# # # # #
# #
# # # #
œ
 4 8 2 x y dy dxœ � �' '
0 0
2 4 xÈ � #c da b# #
 8 4 x y dy dxœ � �' '
0 0
2 4 xÈ � #a b# #
 8 4 r r dr d 8 2r dœ � œ �' ' '
0 0 0
2 2 21 1Î Îa b ’ “# # #
!
) )
r
4
%
 32 d 32 16 ,œ œ œ'
0
21Î
) 1ˆ ‰1#
 dz dx dy,' ' '
� � � �
� � �
2 4 y x y
2 4 y 8 x y
È
È
# # #
#
# #
 dx dz dy dx dz dy,' ' ' ' ' '
� � � � � � �
� � � �
2 y z y 2 4 8 z y
2 4 z y 2 8 y 8 z y
 
# # #
# #
#
È È
È È
�
 
 dx dy dz dx dy dz, dy dz dx dy dz' ' ' ' ' ' ' ' ' ' ' '
0 z z y 4 8 z 8 z y 2 x z x 2 4 8 z x
4 z z y 8 8 z 8 z y 2 4 z x 2 8 x 8 z x
� � � � � � � � � � � � � � �
� � � � � � � �
È
È
È È
È È
È
È
È
È
È
È
# # # # #
# # # #
#
� � dx,
 dy dx dz dy dx dz' ' ' ' ' '
0 z z x 4 8 z 8 z x
4 z z x 8 8 z 8 z x
� � � � � � � �
� � � �
È
È
È
È
È
È
È
È
# #
# #
�
964 Chapter 15 Multiple Integrals
 6. The projection of D onto the xy-plane has the boundary
 x y 2y x (y 1) 1, which is a circle.# # # #� œ Ê � � œ
 Therefore the two integrals are:
 dz dx dy and dz dy dx' ' ' ' ' '
0 2y y x y 1 1 1 x x y
2 2y y 2y 1 1 1 x 2y
� � � � � � �
� � �
È
È
È
È
# # # # # #
# #
 
 7. x y z dz dy dx x y dy dx x dx 1' ' ' ' ' '
0 0 0 0 0 0
1 1 1 1 1 1a b ˆ ‰ ˆ ‰# # # # # #"� � œ � � œ � œ3 32
 8. dz dx dy 8 2x 4y dx dy 8x x 4xy dy' ' ' ' ' '
0 0 x 3y 0 0 0
2 3y 8 x y 2 3y 2 3y
0
È È È
# #
# #
�
� �
œ � � œ � �a b � ‘# # $ #23
 24y 18y 12y dy 12y y 24 30 6œ � � œ � œ � œ �'
0
2 2
0
È
Èa b � ‘$ $ # %"52
 9. dx dy dz dy dz dy dz dz dz 1' ' ' ' ' ' ' ' '
1 1 1 1 1 1 1 1 1
e e e e e e e e ee e
1 1
" " "
xyz yz yz z z
ln x ln yœ œ œ œ œ’ “ ’ “
10. dz dy dx (3 3x y) dy dx (3 3x) (3 3x) dx (1 x) dx' ' ' ' '' '
0 0 0 0 0 0 0
1 3 3x 3 3x y 1 3 3x 1 1� � � �
œ � � œ � � � œ �� ‘# # #"# #9
 (1 x)œ � � œ3 3# #$
"
!c d
11. y sin z dx dy dz y sin z dy dz sin z dz (1 cos 1)' ' ' ' ' '
0 0 0 0 0 0
1 1 11 1 1
œ œ œ � 1 1 1
$ $
# #
12. (x y z) dy dx dz xy y zy dx dz (2x 2z) dx dz x 2zx dz' ' ' ' ' ' ' '
� � � � � � � �1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
� � œ � � œ � œ �� ‘ c d" # #"
�"
"
�"2
 4z dz 0œ œ'
�1
1
13. dz dy dx 9 x dy dx 9 x dx 9x 18' ' ' ' ' '
0 0 0 0 0 0
3 9 x 9 x 3 9 x 3È È È� � �# # #
œ � œ � œ � œÈ a b ’ “# # $
!
x
3
$
14. dz dx dy (2x y) dx dy x xy dy 4 y (2y) dy' ' ' ' ' ' '
0 4 y 0 0 4 y 0 0
2 4 y 2x y 2 4 y 2 24 y
4 y� � � �
� � �
�
� �
È È
È È
È
È
# #
# #
#
#
œ � œ � œ �c d a b# # "Î#
 4 y (4)œ � � œ œ’ “a b2 2 163 3 3# $Î#$Î# #!
15. dz dy dx (2 x y) dy dx (2 x) (2 x) dx (2 x) dx' ' ' ' ' ' '
0 0 0 0 0 0 0
1 2 x 2 x y 1 2 x 1 1� � � �
œ � � œ � � � œ �� ‘# # #" "# #
 (2 x)œ � � œ � � œ� ‘" "$ "
!6 6 6 6
8 7
16. x dz dy dx x 1 x y dy dx x 1 x 1 x dx x 1 x dx' ' ' ' ' ' '
0 0 3 0 0 0 0
1 1 x 4 x y 1 1 x 1 1� � � �# # #
œ � � œ � � � œ �a b a b a b a b’ “# # # ## #" "# #
 1 xœ � � œ’ “a b" "# $ "
!12 12
17. cos (u v w) du dv dw [sin (w v ) sin (w v)] dv dw' ' ' ' '
0 0 0 0 0
1 1 1 1 1
� � œ � � � �1
 [( cos (w 2 ) cos (w )) (cos (w ) cos w)] dwœ � � � � � � �'
0
1
1 1 1
 Section 15.4 Triple Integrals in Rectangular Coordinates 965
 sin (w 2 ) sin (w ) sin w sin (w ) 0œ � � � � � � � œc d1 1 1 1!
18. ln r ln s ln t dt dr ds (ln r ln s) t ln t t dr ds (ln s) r ln r r ds s ln s s 1' ' ' ' ' '
1 1 1 1 1 1
e e e e e e
e e e
1 1 1œ � œ � œ � œc d c d c d
19. e dx dt dv lim e e dt dv e dt dv e dv' ' ' ' ' ' ' '
0 0 0 0 0 0 0
4 ln sec v 2t 4 ln sec v 4 ln sec v 4
x 2t b 2t 2 ln sec v
1 1 1 1Î Î Î Î
�_
œ � œ œ �
bÄ �_
a b ˆ ‰" "# #
 dvœ � œ � œ �'
0
41Î Š ‹ � ‘sec v tan v v2 2 8## # #" "Î%!1 1
20. dp dq dr dq dr 4 q dr dr' ' ' ' ' ' '
0 0 0 0 0 0 0
7 2 4 q 7 2 7 7È � # q
r 1 r 1 3(r 1) 3 r 1
q 4 q 8
� � � �
� " "# $Î#
#
!
œ œ � � œ
È # ’ “a b
 8 ln 2œ œ8 ln 83
21. (a) dy dz dx (b) dy dx dz (c) dx dy dz' ' ' ' ' ' ' ' '
� � � �
� � � � �
1 0 x 0 1 z x 0 0 y
1 1 x 1 z 1 1 z 1 z 1 1 z y#
# #
È
È
È
È
 
 (d) dx dz dy (e) dz dx dy' ' ' ' ' '
0 0 y 0 y 0
1 1 y y 1 y 1 y� �
� �
È È
È È
22. (a) dy dz dx (b) dy dx dz (c) dx dy dz' ' ' ' ' ' ' ' '
0 0 1 0 0 1 0 1 0
1 1 z 1 1 z 1 z 1
� � �
� � �
È È È
 
 (d) dx dz dy (e) dz dx dy' ' ' ' ' '
� �1 0 0 1 0 0
0 y 1 0 1 y# #
23. V dz dy dx y dy dx dxœ œ œ œ' ' ' ' ' '
0 1 0 0 1 0
1 1 y 1 1 1
� �
#
# 2 2
3 3
24. V dy dz dx (2 2z) dz dx 2z z dx 1 x dx xœ œ � œ � œ � œ � œ' ' ' ' ' ' '
0 0 0 0 0 0 0
1 1 x 2 2z 1 1 x 1 11 x
0
� � �
�c d a b ’ “# # "
!
x 2
3 3
$
25. V dz dy dx (2 y) dy dx 2 4 x dxœ œ � œ � �' ' ' ' ' '
0 0 0 0 0 0
4 4 x 2 y 4 4 x 4È È� � � ’ “È ˆ ‰4 x�#
 (4 x) (4 x) (4) (16) 4œ � � � � œ � œ � œ� ‘4 4 32 203 4 3 4 3 3$Î# # $Î#" "%!
26. V 2 dz dy dx 2 y dy dx 1 x dxœ œ � œ � œ' ' ' ' ' '
0 1 x 0 0 1 x 0
1 0 y 1 0 1
� � � �
�
È È# #
 a b# 23
27. V dz dy dx 3 3x y dy dx 6(1 x) 4(1 x) dxœ œ � � œ � � �' ' ' ' ' '
0 0 0 0 0 0
1 2 2x 3 3x 3y 2 1 2 2x 1� � � Î � ˆ ‰ � ‘3 3
4#
# #
†
 3(1 x) dx (1 x) 1œ � œ � � œ'
0
1
# $ "
!c d
28. V dz dy dx cos dy dx cos (1 x) dxœ œ œ �' ' ' ' ' '
0 0 0 0 0 0
1 1 x cos x 2 1 1 x 1� Ð Î Ñ �1 ˆ ‰ ˆ ‰1 1x x
# #
 cos dx x cos dx sin u cos u du cos u u sin uœ � œ � œ � �' ' '
0 0 0
1 1 2
2
0
ˆ ‰ ˆ ‰ � ‘ c d1 1 1
1 1 1 1
x x 2 x 4 2 4
# # #
"
! # #
1
1
Î
Î
 1œ � � œ2 4 4
1 1 1
1
# #
ˆ ‰
#
29. V 8 dz dy dx 8 1 x dy dx 8 1 x dxœ œ � œ � œ' ' ' ' ' '
0 0 0 0 0 0
1 1 x 1 x 1 1 x 1È È È� � �# # #
 
È a b# # 163
30. V dz dy dx 4 x y dy dx 4 x 4 x dxœ œ � � œ � � �' ' ' ' ' '
0 0 0 0 0 0
2 4 x 4 x y 2 4 x 2� � � �# # #a b a b a b’ “# # ## #"#
 4 x dx 8 4x dxœ � œ � � œ"# #
# ##' '
0 0
2 2a b Š ‹x 12815%
966 Chapter 15 Multiple Integrals
31. V dx dz dy (4 y) dz dy (4 y) dyœ œ � œ �' ' ' ' ' '
0 0 0 0 0 0
4 16 y 2 4 y 4 16 y 2 4ˆ ‰ ˆ ‰È È� Î � � Î# # È16 y�
#
#
 2 16 y dy y 16 y dy y 16 y 16 sin 16 yœ � � � œ � � � �' '
0 0
4 4È È È� ‘ ’ “a b# # #" "# �" #%! $Î# %!y4 6
 16 (16) 8œ � œ �ˆ ‰1# " $Î#6 3321
32. V dz dy dx (3 x) dy dx 2 (3 x) 4 x dxœ œ � œ � �' ' ' ' ' '
� � � � � � �
� � �
2 4 x 0 2 4 x 2
2 4 x 3 x 2 4 x 2
È È
È È
# #
# # È #
 3 2 4 x dx 2 x 4 x dx 3 x 4 x 4 sin 4 xœ � � � œ � � � �' '
� �2 2
2 2È È È’ “ ’ “a b# # # �" ## #
�# �#
$Î#x 2
2 3
 12 sin 1 12 sin ( 1) 12 12 12œ � � œ � � œ�" �" # #ˆ ‰ ˆ ‰1 1 1
33. dz dy dx 3 dy dx' ' ' ' '
0 0 2 x y 2 0 0
2 2 x 4 2x 2y 2 2 x� � � �
Ð � � ÑÎ
œ � �ˆ ‰3x 3y# #
 3 1 (2 x) (2 x) dxœ � � � �'
0
2� ‘ˆ ‰x 3
4#
#
 6 6x dxœ � � �'
0
2’ “3x 3(2 x)4# ## �
 6x 3x (12 12 4 0) 2œ � � � œ � � � � œ’ “# � #
!
x 2
2 4 4
(2 x)$ $$
 
34. V dx dy dz (8 2z) dy dz (8 2z)( z) dz 64 24z 2z dzœ œ � œ � ) � œ � �' ' ' ' ' ' '
0 z z 0 z 0 0
4 8 8 z 4 8 4 4� a b#
 64z 12z zœ � � œ� ‘# $ %
!
2 320
3 3
35. V 2 dz dy dx 2 (x 2) dy dx (x 2) 4 x dxœ œ � œ � �' ' ' ' ' '
� � �
� Î � � Î
2 0 0 2 0 2
2 4 x 2 x 2 2 4 x 2 2È È# # È #
 2 4 x dx x 4 x dx x 4 x 4 sin 4 xœ � � � œ � � � � �' '
� �2 2
2 2È È È’ “ ’ “a b# # # �" ## # #�# �#" $Î#x 3
 4 4 4œ � � œˆ ‰ ˆ ‰1 1# # 1
36. V 2 dz dx dy 2 x y dx dy 2 xy dyœ œ � œ �' ' ' ' ' '
0 0 0 0 0 0
1 1 y x y 1 1 y 1 1 y
0
� � �
�
# # # #
#a b ’ “# # #x3$
 2 1 y 1 y y dy 2 1 y y y dy 1 y dyœ � � � œ � � � œ �' ' '
0 0 0
1 1 1a b a b a b a b’ “ ˆ ‰# # # # # % '" " " "#3 3 3 3 32
 yœ � œ œ2 2 6 43 7 3 7 7
y’ “ ˆ ‰ ˆ ‰( "
!
37. average x 9 dz dy dx 2x 18 dy dx 4x 36 dxœ � œ � œ � œ" " "# # #8 8 8 3
31' ' ' ' ' '
0 0 0 0 0 0
2 2 2 2 2 2a b a b a b
38. average (x y z) dz dy dx (2x 2y 2) dy dx (2x 1) dx 0œ � � œ � � œ � œ" " "2 2 2' ' ' ' ' '0 0 0 0 0 0
1 1 2 1 1 1
39. average x y z dz dy dx x y dy dx x dx 1œ � � œ � � œ � œ' ' ' ' ' '
0 0 0 0 0 0
1 1 1 1 1 1a b ˆ ‰ ˆ ‰# # # # # #"3 32
40. average xyz dz dy dx xy dy dx x dx 1œ œ œ œ" " "8 4 2' ' ' ' ' '0 0 0 0 0 0
2 2 2 2 2 2
41. dx dy dz dy dx dz dx dz z dz' ' ' ' ' ' ' ' '
0 0 2y 0 0 0 0 0 0
4 1 2 4 2 x 2 4 2 44 cos x 4 cos x x cos x
2 z 2 z z
sin 4a b a b a bÈ È È
# # #
œ œ œ
Î ˆ ‰
#
�"Î#
 (sin 4)z 2 sin 4œ œ� ‘"Î# %
!
 Section 15.4 Triple Integrals in Rectangular Coordinates 967
42. 12xz e dy dx dz 12xz e dx dy dz 6yz e dy dz 3e dz' ' ' ' ' ' ' ' '
0 0 x 0 0 0 0 0 0
1 1 1 1 1 y 1 1 1
#
zy zy zy zy# # # #œ œ œ
È ’ “ "
!
 3 e z dz 3 e 1 3e 6œ � œ � œ �'
0
1
z za b c d "!
43. dx dy dz dy dz dz dy' ' ' ' ' ' '
0 z 0 0 z 0 0
1 1 ln 3 1 1 1 y
$ $
$
È È
1 1 1 1 1 1e sin y 4 sin y 4 sin y
y y y
2x a b a b a b# # #
# # #
œ œ 
 4 y sin y dy 2 cos y 2( 1) 2(1) 4œ œ � œ � � � œ'
0
1
1 1 1a b c da b# # "!
44. dy dz dx dz dx x dx dz (4 z) dz' ' ' ' ' ' ' '
0 0 0 0 0 0 0 0
2 4 x x 2 4 x 4 4 z 4� � �# #
 
sin 2z x sin 2z sin 2z sin 2z
4 z 4 z 4 z 4 z� � � � #
"œ œ œ �
È ˆ ‰ ˆ ‰
 cos 2z sin zœ � œ � � œ� ‘ � ‘" " "% %
! # #
#
!4 4
sin 4#
45. dz dy dx 4 x y a dy dx' ' ' ' '
0 0 a 0 0
1 4 a x 4 x y 1 4 a x� � � � � �# # #
œ Ê � � � œ4 415 15a b#
 4 a x 4 a x dx 4 a x dx (4 a) 2x (4 a) x dxÊ � � � � � œ Ê � � œ Ê � � � �' ' '
0 0 0
1 1 1’ “a b a b a b c d# # # # # %# # #" "# #4 415 15
 (4 a) x x (4 a) (4 a) (4 a) 15(4 a) 10(4 a) 5 0œ Ê � � � � œ Ê � � � � œ Ê � � � � œ8 2 x 8 2 815 3 5 15 3 5 15’ “# $ # #"! "&
 3(4 a) 2(4 a)1 0 [3(4 a) 1][(4 a) 1] 0 4 a or 4 a 1 a or a 3Ê � � � � œ Ê � � � � œ Ê � œ � � œ Ê œ œ# "3 313
46. The volume of the ellipsoid 1 is so that 8 c 3.x z 4abc
a b c 3 3
y 4(1)(2)(c)# #
# # #
#
� � œ œ Ê œ1
1
1
47. To minimize the integral, we want the domain to include all points where the integrand is negative and to
 exclude all points where it is positive. These criteria are met by the points (x y z) such thatß ß
 4x 4y z 4 0 or 4x 4y z 4, which is a solid ellipsoid centered at the origin.# # # # # #� � � Ÿ � � Ÿ
48. To maximize the integral, we want the domain to include all points where the integrand is positive and to
 exclude all points where it is negative. These criteria are met by the points (x y z) such thatß ß
 1 x y z 0 or x y z 1, which is a solid sphere of radius 1 centered at the origin.� � �   � � Ÿ# # # # # #
49-52. Example CAS commands:
 :Maple
 F := (x,y,z) -> x^2*y^2*z;
 q1 := Int( Int( Int( F(x,y,z), y=-sqrt(1-x^2)..sqrt(1-x^2) ), x=-1..1 ), z=0..1 );
 value( q1 );
 : (functions and bounds will vary)Mathematica
 Due to the nature of the bounds, cylindrical coordinates are appropriate, although Mathematica can do it as is also.
 Clear[f, x, y, z];
 f:= x y z2 2
 Integrate[f, {x, 1,1}, {y, Sqrt[1 x ], Sqrt[1 x ]}, {z, 0, 1}]� � � �2 2
 N[%]
 topolar={x r Cos[t], y r Sin[t]};Ä Ä
 fp= f/.topolar //Simplify
 Integrate[r fp, {t, 0, 2 }, {r, 0, 1},{z, 0, 1}]1
 N[%]
968 Chapter 15 Multiple Integrals
15.5 MASSES AND MOMENTS IN THREE DIMENSIONS
 1. I y z dx dy dz a y z dy dz a yz dzx
c 2 b 2 a 2 c 2 b 2 c 2
c 2 b 2 a 2 c 2 b 2 c 2 b 2
b 2
œ � œ � œ �' ' ' ' ' '
� Î � Î � Î � Î � Î � Î
Î Î Î Î Î Î
Î
� Î
a b a b ’ “# # # # #y3$
 a bz dz ab z ab b c b c ;œ � œ � œ � œ � œ �'
� Î
Î
Î
� Î
c 2
c 2 c 2
c 2
Š ‹ ’ “ Š ‹ a b a bb b z b c c abc M12 12 3 12 12 1 12$ # $ # $# # # # ##
 R ; likewise R and R , by symmetryx y zœ œ œÉ É Éb c a c a b12 12 12# # # # # #� � �
 2. The plane z is the top of the wedge I y z dz dy dxœ Ê œ �4 2y3
� # #
x 3 2 4 3
3 4 4 2y 3' ' '
� � � Î
Ð � ÑÎ a b
 dy dx dx 208; I x z dz dy dxœ � � � œ œ œ �' ' ' ' ' '
� � � � � � Î
Ð � ÑÎ
3 2 3 3 2 4 3
3 4 3 3 4 4 2y 3
y’ “ a b8y 2y 8(2 y)3 3 81 81 364 104# $ $� # #
 dy dx 12x dx 280;œ � � � œ � œ' ' '
� � �3 2 3
3 4 3’ “ ˆ ‰(4 2y) x (4 2y)81 3 3 81 34x 64 32� � #$ # #
 I x y dz dy dx x y dy dx 12 x 2 dx 360z 3 2 4 3 3 2 3
3 4 4 2y 3 3 4 3
œ � œ � � œ � œ' ' ' ' ' '
� � � Î � � �
Ð � ÑÎ a b a b a bˆ ‰# # # # #83 32y
 3. I y z dz dy dx cy dy dx dxx 0 0 0 0 0 0
a b c a b a
œ � œ � œ � œ' ' ' ' ' 'a b Š ‹ Š ‹# # # �c cb c b3 3 3 3abc b c$ $ $ # #a b
 b c where M abc; I a c and I a b , by symmetryœ � œ œ � œ �M M M3 3 3a b a b a b# # # # # #y z
 4. (a) M dz dy dx (1 x y) dy dx x dx ;œ œ � � œ � � œ' ' ' ' ' '
0 0 0 0 0 0
1 1 x 1 x y 1 1 x 1� � � � Š ‹x 6## #" "
 M x dz dy dx x(1 x y) dy dx x 2x x dxyz 0 0 0 0 0 0
1 1 x 1 x y 1 1 x 1
œ œ � � œ � � œ' ' ' ' ' '
� � � �
" "
#
$ #a b 24
 x y z , by symmetry; I y z dz dy dxÊ œ œ œ œ �" # #4 x 0 0 0
1 1 x 1 x y' ' '� � � a b
 y xy y dy dx (1 x) dx I I , by symmetryœ � � � œ � œ Ê œ œ' ' '
0 0 0
1 1 x 1
y x
� ’ “# # $ %� � " " "(1 x y)3 6 30 30$
 (b) R 0.4472; the distance from the centroid to the x-axis is 0x œ œ œ ¸ � � œ œÉ É É ÉIM 5 5 16 16 8 45 2x " " " "#È È
 0.3536¸
 5. M 4 dz dy dx 4 4 4y dy dx 16 dx ; M 4 z dz dy dxœ œ � œ œ œ' ' ' ' ' ' ' ' '
0 0 4y 0 0 0 0 0 4y
1 1 4 1 1 1 1 1 4
xy
# #
a b# 2 323 3
 2 16 16y dy dx dx z , and x y 0, by symmetry;œ � œ œ Ê œ œ œ' ' '
0 0 0
1 1 1a b% 128 128 125 5 5
 I 4 y z dz dy dx 4 4y 4y dy dx 4 dx ;x 0 0 4y 0 0 0
1 1 4 1 1 1
œ � œ � � � œ œ' ' ' ' ' '
#
a b ’ “ˆ ‰ Š ‹# # # %64 1976 79043 3 105 10564y'
 I 4 x z dz dy dx 4 4x 4x y dy dx 4 x dxy 0 0 4y 0 0 0
1 1 4 1 1 1
œ � œ � � � œ �' ' ' ' ' '
#
a b ’ “ˆ ‰ ˆ ‰Š ‹# # # # # #64 8 1283 3 3 764y'
 ; I 4 x y dz dy dx 16 x x y y y dy dxœ œ � œ � � �483263 z 0 0 4y 0 0
1 1 4 1 1' ' ' ' '
#
a b a b# # # # # # %
 16 dxœ � œ'
0
1Š ‹2x 2 2563 15 45#
 6. (a) M dz dy dx (2 x) dy dx (2 x) 4 x dx 4 ;œ œ � œ � � œ' ' ' ' ' '
� � � Î � � � Î �
� Î � � Î
2 4 x 2 0 2 4 x 2 2
2 4 x 2 2 x 2 4 x 2 2
Š ‹ Š ‹
È È
Š ‹ Š ‹
È È
# #
# # Š ‹È # 1
 M x dz dy dx x(2 x) dy dx x(2 x) 4 x dx 2 ;yz 2 4 x 2 0 2 4 x 2 2
2 4 x 2 2 x 2 4 x 2 2
œ œ � œ � � œ �' ' ' ' ' '
� � � Î � � � Î �
� Î � � Î
Š ‹ Š ‹
È È
Š ‹ Š ‹
È È
# #
# # Š ‹È # 1
 M y dz dy dx y(2 x) dy dxxz 2 4 x 2 0 2 4 x 2
2 4 x 2 2 x 2 4 x 2
œ œ �' ' ' ' '
� � � Î � � � Î
� Î � � Î
Š ‹ Š ‹
È È
Š ‹ Š ‹
È È
# #
# #
 (2 x) dx 0 x and y 0œ � � œ Ê œ � œ" � � "# #'
�2
2 ’ “4 x 4 x4 4# #
 Section 15.5 Masses and Moments in Three Dimensions 969
 (b) M z dz dy dx (2 x) dy dx (2 x) 4 x dxxy 2 4 x 2 0 2 4 x 2 2
2 4 x 2 2 x 2 4 x 2 2
œ œ � œ � �' ' ' ' ' '
� � � Î � � � Î �
� Î � � Î
Š ‹ Š ‹
È È
Š ‹ Š ‹
È È
# #
# #
" "
# #
# # #Š ‹È
 5 zœ Ê œ1 54
 7. (a) M 4 dz dy dx 4 r dz dr d 4 4r r dr d 4 4 d 8 ;œ œ œ � œ œ' ' ' ' ' ' ' ' '
0 0 x y 0 0 r 0 0 0
2 4 x 4 2 2 4 2 2 2È � Î Î Î
�
#
# # #
1 1 1
) ) ) 1a b$
 M zr dz dr d 16 r dr d d z , and x y 0,xy 0 0 r 0 0 0
2 2 4 2 2 2
œ œ � œ œ Ê œ œ œ' ' ' ' ' '
1 1 1
#
) ) )r 32 64 83 3 3#
%a b 1
 by symmetry
 (b) M 8 4 r dz dr d cr r dr d d c 8 c 2 2,œ Ê œ œ � œ œ Ê œ Ê œ1 1 ) ) )' ' ' ' ' '
0 0 r 0 0 0
2 c c 2 c 21 1 1È È
 
#
a b È$ ##c c4# #1
 since c 0�
 8. M 8; M z dz dy dx dy dx 0; M x dz dy dxœ œ œ œ œxy yz1 3 1 1 3 1 3 1
1 5 1 1 5 1 5 1' ' ' ' ' ' ' '
� � � � �
’ “z2# "�"
 2 x dy dx 4 x dx 0; M y dz dy dx 2 y dy dx 16 dx 32œ œ œ œ œ œ œ' ' ' ' ' ' ' ' '
� � � � � �1 3 1 1 3 1 1 3 1
1 5 1 1 5 1 1 5 1
xz
 x 0, y 4, z 0; I y z dz dy dx 2y dy dx 100 dx ;Ê œ œ œ œ � œ � œ œx 1 3 1 1 3 1
1 5 1 1 5 1' ' ' ' ' '
� � � �
a b ˆ ‰# # # 2 2 4003 3 3
 I x z dz dy dx 2x dy dx 3x 1 dx ;y 1 3 1 1 3 1
1 5 1 1 5 1
œ � œ � œ � œ' ' ' ' ' '
� � � �
a b a bˆ ‰# # # #2 4 163 3 3
 I x y dz dy dx 2 x y dy dx 2 2x dx R Rz x z1 3 1 1 3 1
1 5 1 1 5 1
œ � œ � œ � œ Ê œ œ' ' ' ' ' '
� � � �
a b a b ˆ ‰ É# # # # # 98 400 503 3 3
 and Ry σ 23
 9. The plane y 2z 2 is the top of the wedge I (y 6) z dz dy dx� œ Ê œ � �L 2 2 1
2 4 2 y 2' ' '
� � �
Ð � ÑÎ c d# #
 dy dx; let t 2 y I 4 5t 16t dt 1386;œ � � œ � Ê œ � � � œ' ' '
� � �2 2 2
2 4 4
L’ “ Š ‹(y 6) (4 y) (2 y)24 3 24 313t 49� � �# " ## $ $
 M (3)(6)(4) 36 Rœ œ Ê œ œ"# #L É ÉIM 77L
10. The plane y 2z 2 is the top of the wedge I (x 4) y dz dy dx� œ Ê œ � �L 2 2 1
2 4 2 y 2' ' '
� � �
Ð � ÑÎ c d# #
 x 8x 16 y (4 y) dy dx 9x 72x 162 dx 696; M (3)(6)(4) 36œ � � � � œ � � œ œ œ" "# ## # #' ' '
� � �2 2 2
2 4 2a b a b
 RÊ œ œL É ÉIM 358L
11. M 8; I z (y 2) dz dy dx y 4y dy dx dxœ œ � � œ � � œ œL 0 0 0 0 0 0
4 2 1 4 2 4' ' ' ' ' 'c d ˆ ‰# # # 13 10 403 3 3
 RÊ œ œL É ÉIM 35L
12. M 8; I (x 4) y dz dy dx (x 4) y dy dx 2(x 4) dxœ œ � � œ � � œ � � œL 0 0 0 0 0 0
4 2 1 4 2 4' ' ' ' ' 'c d c d � ‘# # # # # 8 1603 3
 RÊ œ œL É ÉIM 320L
13. (a) M 2x dz dy dx 4x 2x 2xy dy dx x 4x 4x dxœ œ � � œ � � œ' ' ' ' ' '
0 0 0 0 0 0
2 2 x 2 x y 2 2 x 2� � � � a b a b# $ # 43
 (b) M 2xz dz dy dx x(2 x y) dy dx dx ; M byxy xz0 0 0 0 0 0
2 2 x 2 x y 2 2 x 2
œ œ � � œ œ œ' ' ' ' ' '
� � � �
# �x(2 x)
3 15 15
8 8$
 symmetry; M 2x dz dy dx 2x (2 x y) dy dx 2x x dxyz 0 0 0 0 0 0
2 2 x 2 x y 2 2 x 2
œ œ � � œ � œ' ' ' ' ' '
� � � �
# # # #a b 1615
 x , and y zÊ œ œ œ4 25 5
970 Chapter 15 Multiple Integrals
14. (a) M kxy dz dy dx k xy 4 x dy dx 4x x dxœ œ � œ � œ' ' ' ' ' '
0 0 0 0 0 0
2 x 4 x 2 x 2È È� # a b a b# # %#k 32k15
 (b) M kx y dz dy dx k x y 4x dy dx 4x x dxyz 0 0 0 0 0 0
2 x 4 x 2 x 2
œ œ � œ � œ' ' ' ' ' '
È È
�
#
# # # $ &
#a b a bk 8k3
 x ; M kxy dz dy dx k xy 4 x dy dx 4x x dxÊ œ œ œ � œ �5 k4 3xz 0 0 0 0 0 0
2 x 4 x 2 x 2' ' ' ' ' 'È È�
#
# # # &Î# *Î#a b ˆ ‰
 y ; M kxyz dz dy dx xy 4 x dy dxœ Ê œ œ œ �256 2k 40 2231 77
È È
xy 0 0 0 0 0
2 x 4 x 2 x' ' ' ' 'È È�
# a b# #
 16x 8x x dx zœ � � œ Ê œk 256k 84 105 7'0
2a b# % '
15. (a) M (x y z 1) dz dy dx x y dy dx (x 2) dxœ � � � œ � � œ � œ' ' ' ' ' '
0 0 0 0 0 0
1 1 1 1 1 1ˆ ‰3 5
# #
 (b) M z(x y z 1) dz dy dx x y dy dx x dxxy 0 0 0 0 0 0
1 1 1 1 1 1
œ � � � œ � � œ � œ' ' ' ' ' '" "# #ˆ ‰ ˆ ‰5 13 43 6 3
 M M M , by symmetry x y zÊ œ œ œ Ê œ œ œxy yz xz 4 83 15
 (c) I x y (x y z 1) dz dy dx x y x y dy dxz 0 0 0 0 0
1 1 1 1 1
œ � � � � œ � � �' ' ' ' 'a b a b ˆ ‰# # # # #3
 x 2x x dx I I I , by symmetryœ � � � œ Ê œ œ œ'
0
1
x y zˆ ‰$ # "3 4 6 63 11 11
 (d) R R Rx y z IM 1511œ œ œ œÉ Éz
16. The plane y 2z 2 is the top of the wedge.� œ
 (a) M (x 1) dz dy dx (x 1) 2 dy dx 18œ � œ � � œ' ' ' ' '
� � � � �
Ð � ÑÎ
1 2 1 1 2
1 4 2 y 2 1 4 ˆ ‰y
#
 (b) M x(x 1) dz dy dx x(x 1) 2 dy dx 6;yz 1 2 1 1 2
1 4 2 y 2 1 4
œ � œ � � œ' ' ' ' '
� � � � �
Ð � ÑÎ ˆ ‰y
#
 M y(x 1) dz dy dx y(x 1) 2 dy dx 0;xz 1 2 1 1 2
1 4 2 y 2 1 4
œ � œ � � œ' ' ' ' '
� � � � �
Ð � ÑÎ ˆ ‰y
#
 M z(x 1) dz dy dx (x 1) y dy dx 0 x , and y z 0xy 1 2 1 1 2
1 4 2 y 2 1 4
œ � œ � � œ Ê œ œ œ' ' ' ' '
� � � � �
Ð � ÑÎ
"
# Š ‹y4 31#
 (c) I (x 1) y z dz dy dx (x 1) 2y dy dx 45;x 1 2 1 1 2
1 4 2 y 2 1 4
œ � � œ � � � � " � œ' ' ' ' '
� � � � �
Ð � ÑÎ a b ’ “ˆ ‰# # # " "# $3 3 2y y$
 I (x 1) x z dz dy dx (x 1) 2x dy dx 15;y 1 2 1 1 2
1 4 2 y 2 1 4
œ � � œ � � � � " � œ' ' ' ' '
� � � � �
Ð � ÑÎ a b ’ “ˆ ‰# # # " "# $3 3 2x y y#
 I (x 1) x y dz dy dx (x 1) 2 x y dy dx 42z 1 2 1 1 2
1 4 2 y 2 1 4
œ � � œ � � � œ' ' ' ' '
� � � � �
Ð � ÑÎ a b a bˆ ‰# # # ##y
 (d) R R and Rx y zI IM M 6 M 35 5 7
I
œ œ ß œ œ ß œ œÉ ÉÉ É ÉÉx zy#
17. M (2y 5) dy dx dz z 5 z dx dz 2 z 5 z (1 z) dzœ � œ � œ � �' ' ' ' ' '
0 z 1 0 0 z 1 0
1 1 z z 1 1 z 1
� �
� �
È ˆ ‰ ˆ ‰È È
 2 5z z 5z z dz 2 z z 2z z 2 3œ � � � œ � � � œ � œ'
0
1ˆ ‰ � ‘ ˆ ‰"Î# $Î# # $Î# # &Î# $" "
# #
"
!
10 9 3
3 3 3
18. M x y dz dy dx x y 16 4 x y dy dxœ � œ � � �' ' ' ' '
� � � � � � �
� � � �
2 4 x 2 x y 2 4 x
2 4 x 16 2 x y 2 4 x
È È
È È
a b
ˆ ‰
# # # #
# #
# # È È c da b# # # # # #
 4 r 4 r r dr d 4 d 4 dœ � œ � œ œ' ' ' '
0 0 0 0
2 2 2 21 1 1a b ’ “# #
!
) ) )4r r 64 5123 5 15 15
$ & 1
19. (a) Let V be the volume of the ith piece, and let (x y z ) be a point in the ith piece. Then the work done? i i i iß ß
 by gravity in moving the ith piece to the xy-plane is approximately W m gz (x y z 1)g V zi i i i i i i iœ œ � � � ?
 the total work done is the triple integral W (x y z 1)gz dz dy dxÊ œ � � �' ' '
0 0 0
1 1 1
 g xz yz z z dy dx g x y dy dx g xy y y dxœ � � � œ � � œ � �' ' ' ' '
0 0 0 0 0
1 1 1 1 1� ‘ ˆ ‰ � ‘" " " " " " " "
# # # # #
# # $ # #" "
! !3 6 2 4 6
5 5
 g x dx g x g gœ � œ � œ œ'
0
1ˆ ‰ ˆ ‰’ “"# "!13 x 13 16 412 4 12 12 3#
 Section 15.5 Masses and Moments in Three Dimensions 971
 (b) From Exercise 15 the center of mass is and the mass of the liquid is the work done byˆ ‰8 8 8 515 15 15ß ß Ê#
 gravity in moving the center of mass to the xy-plane is W mgd (g) g, which is the same asœ œ œˆ ‰ ˆ ‰5 8 42 15 3
 the work done in part (a).
20. (a) From Exercise 19(a) we see that the work done is W g kxyz dz dy dxœ ' ' '
0 0 0
2 x 4 xÈ � #
 kg xy 4 x dy dx x 4 x dx 16x 8x x dxœ � œ � œ � �' ' ' '
0 0 0 0
2 x 2 2È
"
#
# # # # % '# #a b a b a bkg kg4 4
 x x xœ � � œ
kg 256k g
4 3 5 7 105
16 8� ‘$ & (" #
!
†
 (b) From Exercise 14 the center of mass is and the mass of the liquid is the work done byŠ ‹5 8 32k4 77 7 1540 2ß ß ÊÈ
 gravity in moving the center of mass to the xy-plane is W mgd (g)œ œ œˆ ‰ ˆ ‰32k 815 7 105256k g†
21. (a) x 0 x (x y z) dx dy dz 0 M 0œ œ Ê ß ß œ Ê œMMyz ' ' '
R
$ yz
 (b) I h dm (x h) y dm x 2xh h y dmL œ � œ � � œ � � �' ' ' ' ' ' ' ' '
D D D
k k k k a bv i i j# # # # #
 x y dm 2h x dm h dm I 0 h m I h mœ � � � œ � � œ �' ' ' ' ' ' ' ' '
D D D
a b# # # # #x c mÞ Þ
22. I I mh ma ma maL c mœ � œ � œÞ Þ # # # #2 75 5
23. (a) (x y z) I I abc I Iß ß œ ß ß Ê œ � � Ê œ �ˆ ‰ Š ‹Éa b c a b4 4 4c m z abc a b# # # # Þ Þ �z c mÞ Þ # # # #a b
 ; Rœ � œ œ œabc a b abc a b abc a b3 4 1 M 12c m
I a ba b a b a b# # # # # #
Þ Þ
# #� � �
# Þ Þ
�É Éc m
 (b) I I abc 2bL c m a b4 12 4 1abc a b abc a 9b abc 4a 28bœ � � � œ � œÞ Þ # #
#
#
� � �Œ �É ˆ ‰# # # # # # #a b a b a b
 ; Rœ œ œabc a 7b3 M 3L
I a 7ba b# # # #� �É ÉL
24. M dz dy dx (4 y) dy dx 4y dx 12 dx 72;œ œ � œ � œ œ' ' ' ' ' ' '
� � � Î � � � �
Ð � ÑÎ
3 2 4 3 3 2 3 3
3 4 4 2y 3 3 4 3 3
2 2
3 3 2
y’ “# %
�#
 x y z 0 from Exercise 2 I I 72 0 0 I I I 72 16œ œ œ Ê œ � � œ Ê œ � �x c m L 169c m c mÞ Þ Þ ÞŠ ‹ Š ‹È É# # # Þ Þ #
 208 72 1488œ � œˆ ‰1609
25. M x dV x dV M M x M M ; similarly,yz yz yz yz yz m mB B
" #
�
" # " # " #
œ � œ � Ê œ �' ' ' ' ' '
B B
" #
" # Ð Ñ Ð Ñ Ð Ñ Ð Ñ �
 y M M and z M M x y zœ � œ � Ê œ � �Ð Ñ Ð Ñ � Ð Ñ Ð Ñ �xz xz m m xy xy m m
" # " # " # " #
c i j k
 M M M M M Mœ � � � � �"� Ð Ñ Ð Ñ Ð Ñ Ð Ñ Ð Ñ Ð Ñm m yz yz xz xz xy xy
" #
" # " # " #
� ‘ˆ ‰ ˆ ‰ ˆ ‰i j k
 m x m x m y m y m z m zœ � � � � �"� " " # # " # " " # #" #m m
" #
c da b a b a bi j k
 m x y z m x y zœ � � � � � œ"� �" " " # # #" #
�
m m m m
m m
" # " #
" " # #c da b a bi j k i j k c c
26. (a) 12 2 4 12 2 x , y , zc i j k i j kœ � � � � � � œ Ê œ œ œˆ ‰ ˆ ‰3 13 13 132 7 14 7 1413" "# # � �13 132# i j k
 (b) 12 12 12 12 x 1, y , zc i j k i j kœ � � � � � � œ Ê œ œ œˆ ‰ ˆ ‰3 11 7 12 2 2 42 7# #" � �i j k12
 (c) 2 4 12 2 12 x , y , zc i j k i j kœ � � � � � � œ Ê œ œ œ �ˆ ‰ ˆ ‰" " "# # # # � �11 13 37 513 74 514 14 7 14i j k
 (d) 12 2 4 12 12 2 12 x , y , zc i j k i j k i j kœ � � � � � � � � � � œ Ê œ œ œˆ ‰ ˆ ‰ ˆ ‰3 11 25 46 72 26 26 13 2625 92 7" " "# # # # � �i j k
972 Chapter 15 Multiple Integrals
27. (a) , where m and m ; ifc œ œ œ œŠ ‹
1 1
1
a h h 2 a 3a
3 4 3 8
a h 3a
3 4
# $
# # #
�Š ‹ Š ‹Š ‹ Š ‹Š ‹k k�
� � " #
�
m m m m 3 3
a h 2 a
" # " #
# $
k
1 1
 0, or h a 3, then the centroid is on the common baseh 3a4
# #� œ œ È
 (b) See the solution to Exercise 55, Section 15.2, to see that h a 2.œ È
28. , where m and m s ; if h 6s 0,c œ œ œ œ � œ
Š ‹ ˆ ‰ a b
s h h s
3 4
s
12
#
$
#
#
# #Š ‹ Š ‹ � ‘k k�
� � " #
$ # #
s h 6s� �
m m m m 3
s h
" # " #
#
k
 or h 6s, then the centroid is in the base of the pyramid. The corresponding result in 15.2, Exercise 56, is h 3s.œ œÈ È
15.6 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
 1. dz r dr d r 2 r r dr d 2 r d' ' ' ' ' '
0 0 r 0 0 0
2 1 2 r 2 1 21 1 1È � #
) ) )œ � � œ � � �’ “ ’ “a b a b# # #"Î# $Î#" "
!3 3
r$
 dœ � œ'
0
21Š ‹2 23 3 34 2$Î# ) 1 Š ‹È �"
 2. dz r dr d r 18 r dr d 18 r d' ' ' ' ' '
0 0 r 3 0 0 0
2 3 18 r 2 3 21 1 1
#
#
Î
�
È
) ) )œ � � œ � � �’ “ ’ “a b a b# #"Î# $Î#" $
!
r r
3 3 12
$ %
 œ
9 8 2 7
2
1 Š ‹È �
 3. dz r dr d 3r 24r dr d r 6r d d' ' ' ' ' ' '
0 0 0 0 0 0 0
2 2 3 24r 2 2 2 221 ) 1 1 ) 1 1 1) 1Î � Î Î
!
#
) ) ) )œ � œ � œ �a b � ‘ Š ‹$ # % #3 3 42 4 16) )1 1# %# %
 œ � œ3 1712 520# ’ “) ) 11 1$ &# % 21
!
 4. z dz r dr d 9 4 r 4 r r dr d 4 4r r dr d' ' ' ' ' ' '
0 0 4 r 0 0 0 0
3 4 r1 ) 1 1 ) 1 1 ) 1Î � Î Î
� �
È
È
#
#
) ) )œ � � � œ �"#
# # $c d a ba b a b
 4 2r 4 dœ � œ � œ' '
0 0
1 1’ “ Š ‹# Î!
r 2 37
4 154
% # %
# %
) 1
) ) 1
1 1
)
 5. 3 dz r dr d 3 r 2 r r dr d 3 2 r d' ' ' ' ' '
0 0 r 0 0 0
2 1 2 r 2 1 21 1 1ˆ ‰� # �"Î#
) ) )œ � � œ � � �’ “ ’ “a b a b# # #�"Î# "Î# "
!
r
3
$
 3 2 d 6 2 8œ � œ �'
0
21Š ‹ Š ‹È È43 ) 1
 6. r sin z dz r dr d r sin dr d d' ' ' ' ' '
0 0 1 2 0 0 0
2 1 1 2 2 1 21 1 1
� Î
Î a b ˆ ‰ Š ‹# # # $ # ") ) ) ) )� œ � œ � œr sin12 4 24 3# ) 1
 7. r dr dz d dz d d' ' ' ' ' '
0 0 0 0 0 0
2 3 z 3 2 3 21 1 1Î
$ ) ) )œ œ œz 3 3324 20 10
% 1
 8. 4r dr d dz 2(1 cos ) d dz 6 d 12' ' ' ' ' '
� � �
�
1 0 0 1 0 1
1 2 1 cos 1 2 11 ) 1
) ) ) 1 ) 1œ � œ œ#
 9. r cos z r d dr dz z r dr dz r 2 rz dr dz' ' ' ' ' ' '
0 0 0 0 0 0 0
1 z 2 1 z 1 zÈ È È1 a b a b’ “# # # # $ ##
!
) ) ) 1 1� œ � � œ �r r sin 22 4
# #) )
1
 r z dz z dzœ � œ � œ � œ' '
0 0
1 1z’ “ Š ‹ ’ “1 1 1 1 1r z z z4 4 12 4 3% # $ %1 1# # $ "!
È
!
10. (r sin 1) r d dz dr 2 r dz dr 2 r 4 r r 2r dr' ' ' ' ' '
0 r 2 0 0 r 2 0
2 4 r 2 2 4 r 2
� �
� �
È È# #
1
) ) 1 1� œ œ � � �’ “a b# #"Î#
 2 4 r r 2 4 (4) 8œ � � � � œ � � � œ1 1 1’ “a b � ‘" "# # $Î#$Î# #
!3 3 3 3
r 8$
 Section 15.6 Triple Integrals in Cylindrical and Spherical Coordinates 973
11. (a) dz r dr d' ' '
0 0 0
2 1 4 r1 È � #
)
 (b) r dr dz d r dr dz d' ' ' ' ' '
0 0 0 0 3 0
2 3 1 2 2 4 z1 1È
È
È
) )�
�
#
 
 (c) r d dz dr' ' '
0 0 0
1 4 r 2È � # 1
)
 
12. (a) dz r dr d' ' '
0 0 r
2 1 2 r1 � #
)
 (b) r dr dz d r dr dz d' ' ' ' ' '
0 0 0 0 1 0
2 1 z 2 2 2 z1 1
) )�
È
�
 (c) r d dz dr' ' '
0 r 0
1 2 r 2� # 1
)
 
13. f(r z) dz r dr d' ' '
� Î
Î
1
1 )
2 0 0
2 cos 3r#
ß ß) )
14. r dz dr d r cos dr d cos d' ' ' ' ' '
� Î � Î � Î
Î Î Î
1 1 1
1 ) 1 1
2 0 0 2 0 2
2 1 r cos 2 1 2
$ % ") ) ) ) )œ œ œ5 5
2
15. f(r z) dz r dr d 16. f(r z) dz r dr d' ' ' ' ' '
0 0 0 2 0 0
2 sin 4 r sin 2 3 cos 5 r cos 1 ) ) 1 ) )
1
� Î �
� Î
ß ß ß ß) ) ) )
17. f(r z) dz r dr d 18. f(r z) dz r dr d' ' ' ' ' '
� Î � Î
Î � Î �
1 1 )
1 ) 1 ) )
2 1 0 2 cos 0
2 1 cos 4 2 2 cos 3 r sin 
 ß ß ß ß) ) ) )
19. f(r z) dz r dr d 20. f(r z) dz r dr d' ' ' ' ' '
0 0 0 4 0 0
4 sec 2 r sin 2 csc 2 r sin 1 ) ) 1 ) )
1
Î � Î �
Î
ß ß ß ß) ) ) )
21. sin d d d sin d d sin d d' ' ' ' ' ' '
0 0 0 0 0 0 0
2 sin 1 1 9 1 1 1 1
3 9 3 9 ) 9 9 ) 9 9 )# % #
!
œ œ � �8 8 33 3 4 4
sin cos Š ‹’ “$ 9 9 1
 2 sin d d d dœ œ � œ œ' ' ' '
0 0 0 0
1 1 1 1
# #
# !
9 9 ) ) ) 1 ) 1� ‘sin 2) 1
22. ( cos ) sin d d d 4 cos sin d d 2 sin d d 2' ' ' ' ' ' '
0 0 0 0 0 0 0
2 4 2 2 4 2 21 1 1 1 1 1Î Î
 3 9 3 9 3 9 ) 9 9 9 ) 9 ) ) 1# #
Î%
!œ œ œ œc d 1
23. sin d d d (1 cos ) sin d d (1 cos ) d' ' ' ' ' '
0 0 0 0 0 0
2 1 cos 2 2 21 1 9 1 1 1Ð � ÑÎ
3 9 3 9 ) 9 9 9 ) 9 )# $ %" " !œ � œ �24 96 c d 1
 2 0 d d (2 )œ � œ œ œ" "%96 96 6 316' '0 0
2 21 1a b ) ) 1 1
24. 5 sin d d d sin d d sin d d' ' ' ' ' ' '
0 0 0 0 0 0 0
3 2 1 3 2 3 21 1 1 1 1 1Î Î Î
3 9 3 9 ) 9 9 ) 9 9 )$ $ $
!
œ œ � �5 5 24 4 3 3
sin cos Š ‹’ “# 9 9 1
 cos d dœ � œ œ5 5 56 3' '0 0
3 2 3 21 1Î Îc d9 ) )1 1! #
974 Chapter 15 Multiple Integrals
25. 3 sin d d d 8 sec sin d d 8 cos sec d' ' ' ' ' '
0 0 sec 0 0 0
2 3 2 2 3 21 1 1 1 1
9
Î Î
3 9 3 9 ) 9 9 9 ) 9 9 )# $ #"
Î$
!
œ � œ � �a b � ‘2 1
 ( 4 2) 8 d d 5œ � � � � � œ œ' '
0 0
2 21 1� ‘ˆ ‰"
# #) ) 1
5
26. sin cos d d d tan sec d d tan d' ' ' ' ' '
0 0 0 0 0 0
2 4 sec 2 4 21 1 9 1 1 1Î Î
3 9 9 3 9 ) 9 9 9 ) 9 )$ # #" " "
Î%
!
œ œ4 4 2
� ‘1
 dœ œ"8 4'0
21
) 1
27. sin 2 d d d d d d d d' ' ' ' ' ' ' '
0 4 0 0 0
2 0 2 2 0 2 0 2
� Î � �
Î
1 1 1 1
1
3 9 9 ) 3 3 ) 3 ) 3 3$ $
Î#
Î% # #
œ � œ œ� ‘cos 22 9 11 3 3 1$ $
 2œ œ’ “13%8 #! 1
28. sin d d d 2 sin d d sin d' ' ' ' ' '
1 9 1 9 1
1 9 1 1 9 1
9
9
Î Î Î
Î Î Î
6 csc 0 6 csc 6
3 2 csc 2 3 2 csc 3 2 csc 
csc 
3 9 ) 3 9 1 3 9 3 9 3 9 9# # $œ œ 23
1 c d
 csc dœ œ14 283 3 3
1 1'
1
1
Î
Î
6
3
# 9 9 È
29. 12 sin d d d 12 8 sin d d d' ' ' ' ' '
0 0 0 0 0 0
1 4 1 41 1 1 1Î Î
3 9 9 ) 3 3 3 9 9 ) 3$ �
Î%
!
œ �Œ �’ “sin cos 3# 9 9 1
 8 cos d d 8 d d 8 d 4œ � � œ � œ � œ �' ' ' ' '
0 0 0 0 0
1 1 11 1Š ‹ Š ‹ Š ‹ ’ “c d2 10 10 5
2 2 2 2
3 3 3 31È È È È3 9 ) 3 3 ) 3 1 3 3 1 3Î%! #
"
!
#
 œ
Š ‹È
È
4 2 5 
2
� 1
30. 5 sin d d d 32 csc sin d d 32 sin csc d d' ' ' ' ' ' '
1 1 9 1 1 1 1
1 1 1 1 1 1
Î � Î Î � Î Î � Î
Î Î Î Î Î Î
6 2 csc 6 2 6 2
2 2 2 2 2 2 2
3 9 3 ) 9 9 9 ) 9 9 9 ) 9% $ & $ $ #œ � œ �a b a b
 32 sin csc d sin d cot œ � œ � � �1 9 9 9 1 9 9 1 9' '
1 1
1 1
Î Î
Î Î
6 6
2 2a b c d’ “$ # Î#
Î'
Î#
Î'
32 sin cos 
3 3
64# 9 9
1
1
1 1
1
 cos 3 11 3œ � � œ � œ œ1 9 1 1 1Š ‹ Š ‹ Š ‹c d È Èˆ ‰32 3 3 3 33 324 3 3 3 364 64È È È È1 111 1Î#Î' #
31. (a) x y 1 sin 1, and sin 1 csc ; thus# # # #� œ Ê œ œ Ê œ3 9 3 9 3 9
 sin d d d sin d d d' ' ' ' ' '
0 0 0 0 6 0
2 6 2 2 2 csc 1 1 1 1 9
1
Î Î
Î
3 9 3 9 ) 3 9 3 9 )# #�
 (b) sin d d d sin d d d' ' ' ' ' '
0 1 6 0 0 0
2 2 sin 1 2 2 61 3 1 1
1Î
Ð Î Ñ Î
�"
3 9 9 3 ) 3 9 9 3 )# #�
32. (a) sin d d d' ' '
0 0 0
2 4 sec 1 1 9Î
3 9 3 9 )#
 (b) sin d d d' ' '
0 0 0
2 1 41 1Î
3 9 9 3 )#
 sin d d d� ' ' '
0 1 cos
2 2 41 1
3
È
�"
Ð"Î Ñ
Î
3 9 9 3 )#
 
33. V sin d d d 8 cos sin d dœ œ �' ' ' ' '
0 0 cos 0 0
2 2 2 2 21 1 1 1
9
Î Î
3 9 3 9 ) 9 9 9 )# $"3 a b
 8 cos d 8 d (2 )œ � � œ � œ œ" " "
Î#
!3 4 3 4 12 6
cos 31 31' '
0 0
2 21 1’ “ ˆ ‰ ˆ ‰9 ) ) 1% 9 1 1
34. V sin d d d 3 cos 3 cos cos sin d dœ œ � �' ' ' ' '
0 0 1 0 0
2 2 1 cos 2 21 1 9 1 1Î � Î
3 9 3 9 ) 9 9 9 9 9 )# # $"3 a b
 cos cos cos d 1 d d (2 )œ � � � œ � � œ œ œ" " "# # $ %
Î#
!3 4 3 2 4 12 12 6
3 1 3 11 11 11' ' '
0 0 0
2 2 21 1 1� ‘ ˆ ‰ ˆ ‰9 9 9 ) ) ) 11 1
 Section 15.6 Triple Integrals in Cylindrical and Spherical Coordinates 975
35. V sin d d d (1 cos ) sin d d dœ œ � œ' ' ' ' ' '
0 0 0 0 0 0
2 1 cos 2 21 1 9 1 1 1�
3 9 3 9 ) 9 9 9 ) )# $" " "�
!3 3 4
( cos )’ “9 1%
 (2) d (2 )œ œ œ" %12 3 34 8'0
21
) 1 1
36. V sin d d d (1 cos ) sin d d dœ œ � œ' ' ' ' ' '
0 0 0 0 0 0
2 2 1 cos 2 2 21 1 9 1 1 1Î � Î
3 9 3 9 ) 9 9 9 ) )# $" " "�
Î#
!3 3 4
( cos )’ “9 1%
 d (2 )œ œ œ" "12 12 6'0
21
) 1 1
37. V sin d d d cos sin d d dœ œ œ �' ' ' ' ' '
0 4 0 0 4 0
2 2 2 cos 2 2 21 1 9 1 1 1
1 1Î Î
Î Î
3 9 3 9 ) 9 9 9 ) )# $
Î#
Î%
8 8
3 3 4
cos’ “% 9 1
1
 d (2 )œ œ œˆ ‰ ˆ ‰83 16 6 3" "'021 ) 1 1
38. V sin d d d sin d d cos d dœ œ œ � œ œ' ' ' ' ' ' '
0 3 0 0 3 0 0
2 2 2 2 2 2 21 1 1 1 1 1
1 1Î Î
Î Î
3 9 3 9 ) 9 9 ) 9 ) )#
Î#
Î$
8 8 4 8
3 3 3 3c d 11 1
39. (a) 8 sin d d d (b) 8 dz r dr d' ' ' ' ' '
0 0 0 0 0 0
2 2 2 2 2 4 r1 1 1Î Î Î �
3 9 3 9 ) )#
È #
 (c) 8 dz dy dx' ' '
0 0 0
2 4 x 4 x yÈ È� � �# # #
40. (a) dz r dr d (b) sin d d d' ' ' ' ' '
0 0 r 0 0 0
2 3 2 9 r 2 4 31 1 1Î Î � Î ÎÈ È #
) 3 9 3 9 )#
 (c) sin d d d 9 sin d d 9 1 d' ' ' ' ' '
0 0 0 0 0 0
2 4 3 2 4 21 1 1 1 1Î Î Î Î Î
3 9 3 9 ) 9 9 ) )# "
�
œ œ � � œŠ ‹È Š ‹È2 9 2 241
41. (a) V sin d d d (b) V dz r dr dœ œ' ' ' ' ' '
0 0 sec 0 0 1
2 3 2 2 3 4 r1 1 1
9
Î �
3 9 3 9 ) )#
È
È #
 
 (c) V dz dy dxœ ' ' '
� � �
� � �
È
È
È
È
È
3 3 x 1
3 3 x 4 x y
#
# # #
 (d) V r 4 r r dr d d dœ � � œ � � œ � � �' ' ' '
0 0 0 0
2 3 2 21 1 1È ’ “ Š ‹a b ” •# "Î# � # #
$
!
") ) )
a b È4 r
3 3 3
r 3 4#
$Î#