950
Chapter 15 Multiple Integrals Clear[x, y, f] f[x_, y_]:= 1 / (x y) Integrate[f[x, Integrate[f[x, y], {x, 1, 3}, {y, 1, x}] To reverse the order of integration, integration, it is best to first plot the region over which the integration extends. This can be done with ImplicitPlot ImplicitPlot and all bounds involving involving both x and y can be plotted. A graphics graphics package must be loaded. loaded. Remember Remember to use the double equal sign for the equations of the bounding curves. Clear[x, y, f]
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67.
''
69.
''
1
1
0
x
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70.
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4
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x/2
0
ex dy dx
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1 .1494 ‚ 106 ¸ 1.
4
1
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0
x
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È1 erfia2b 2È 1 erfia4b‰
œ 4 4 e4 2
e
0
1
0
x
1 x
y
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È 1 x
3
#
y# dy dx ¸ 3.142
The following graphs was generated using
dx dy dy
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0
1
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71. Evaluate the integrals: 1
1
68.
Mathematica.
Section 15.2 Areas, Moments, and Centers of Mass 72. Evaluate the integrals: 3
'' 0
œ
9 x2
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x cosa y2 bdy dx œ
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9
y
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73. Evaluate the integrals: 2
'' 0
œ
4
2y
y3
67,520 693
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3
x
ax2 y xy2 bdx dy œ '0 'x /32 a x2 y xy2 bdy dx
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2
¸ 97.4315
74. Evaluate the integrals: 2
Mathematica.
0
exy dx dy œ
¸ 20.5648
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4 x
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0
Mathematica.
951
952
Chapter 15 Multiple Integrals
75. Evaluate the integrals: 2
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x
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1
1
2
0
4
1
1 ln
2
1
y
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76. Evaluate the integrals: 2
8
2
y3
1
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8
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2
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15.2 AREAS, MOMENTS, AND CENTERS OF MASS
' ' dy dx œ ' (2 x) dx œ ’2x x2 “ # œ 2, ! or ' ' dx dy œ ' (2 y) dy œ 2 2
1.
0
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2
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0
2
0
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2 y
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0
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2
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3.
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2.
Mathematica.
0
0
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0
4
0
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0
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2
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Mathematica.
Section 15.2 Areas, Moments, and Centers of Mass 2
4.
y y
'' 0
y
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' '
6.
''
0
e
1
œ
0
0
!
4 3
ln 2
dy dx œ
'
2 ln x
0
ln 2
ex dx œ cex d 0
e
dy dx œ
ln x
œ 21 œ1
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e 1
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1
7.
8 3
ex
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8.
y
1
2y
2
1
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1
1
1
6
9.
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3
10.
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216 9
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dx dy œ
' a3x x# b dx œ 32 x# 3" x$‘ !$ 3
dy dx œ 9
œ # 9 œ #
0
953
954
Chapter 15 Multiple Integrals
11.
' ' dy dx œ ' (cos x sin x) dx œ csin x cos xd
4
cos x
0
sin x 4
0
ŠÈ
œ
2
12.
' '
1
‹ (0 1) œ È 2 1 2
dx dy œ
y
' ay 2 y#b dy œ ’ y2
œ ˆ2 4 ‰ ˆ
1
" 2 3" #
8 3
1 x
' ' œ' 1
2
2x 0 1
0
0
x
0
1
9 " œ # #
1 x
0
x 2
0
x 2
“
0 1
’
“
x x 4
4
ˆ
2 0
"
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‰ (2 1) œ
3
#
x
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2
2
2
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y
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2
14.
#
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0
13.
È 2
2
#
4 0
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0
0
4
0
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x
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2
0
15. (a) average œ
23 x$Î# ‘ 04 œ ˆ8 83 ‰ 163
' '
" 1
0
sin (x y) dy dx œ
0
32
œ 3
" 1
'
0
"
c cos (x y)d 0 dx œ
1
'
[ cos (x 1) cos x] dx
0
œ 1" c sin (x 1) sin xd 0 œ 1" [( sin 2 1 sin 1) ( sin 1 sin 0)] œ 0
(b) average œ 2
œ
1
"
Š ‹
' ' 0
2 0
sin (x y) dy dx œ
sin ˆx 1# ‰ sin x‘ 0
2
œ
1
ˆ sin
2 1
'
31
#
0
c cos (x y) d 1! Î# dx
œ
‰ ˆ
2 1
1 sin 1 sin # sin 0
'
0
‰‘ œ
cos ˆ x 1# ‰ cos x ‘ dx 4 1
' ’ xy2 “ " dx œ ' x# dx œ 4" œ 0.25; ! average value over the quarter circle œ ˆ " ‰ ' ' xy dy dx œ 4 ' ’ xy2 “ dx " 2 ' œ ax x$b dx œ 2 ’ x2 x4 “ œ #" ¸ 0.159. The average value over the square is larger. 1
16. average value over the square œ
'' 0
1
0
1
xy dy dx œ 1
4
1
1
0
0
1 x
1 x
1
1
0
0
0
4
1
0
17. average height œ
0
1
2
'' 4 "
0
!
2
1
2
ax# y#b dy dx œ 4" '0 ’x#y y3 “ 0
# !
dx œ
' ˆ2x# 38 ‰ dx œ #" ’ x3 4 "
2
0
4x
3
“
# !
8
œ 3
Section 15.2 Areas, Moments, and Centers of Mass
ln 2
'
2 ln 2
"
2 ln 2
2 ln 2
"
dy dx œ
ln 2
ln 2
ln 2
2 ln 2
œ (ln"2) œ
2 ln 2
' ’ lnxy “ dx (ln 2) xy (ln 2) ' "x (ln 2 ln ln 2 ln ln 2) dx œ ˆ ln"2 ‰' dxx œ ˆ ln"# ‰ cln xd '
"
18. average œ
2 ln 2
ln 2
ln 2
ˆ ‰ (ln 2 ln ln 2 ln ln 2) œ 1
2 ln 2 ln 2
"
ln #
1
2 x
1
1
2 x
' ' 3 dy dx œ 3' a2 x# xb dx œ 7# ; M œ ' ' 5 3y dy dx œ 3# ' œ 3' a2x x$ x# b dx œ 4 ; M œ ' '
19. M œ
0
x
y
0
1
1
x
0
5
0
x
2 x
0
x
' dx œ 3# '
3x dy dx œ 3 1
cy#d x2 0
x
1 0 1
0
cxyd x2 x
dx
a 4 5x # x%b dx œ 195
38
Ê x œ 14 and y œ 35 3
'' œ$' '
20. M œ $ Iy
3
0
0
3
3
0
3
x
0
3
0
3
0
2
3
3
3
' 3 dx œ 9$ ; I œ $ ' ' y# dy dx œ $ ' ’ y3 “ dx œ 27$ ; R $ I x# dy dx œ $ ' cx# yd ! dx œ $ ' 3x# dx œ 27$ ; R œ É M œ È 3 dy dx œ $
œ
x
0
0
3
0
4 y
0
É
Ix M
œ
È 3;
y
y
0
2
2
4 y
2
' ' dx dy œ ' Š4 y y# ‹ dy œ 143 ; M œ ' ' x dx dy œ "# ' cx#d dy y 128 y dx dy œ ' Š4y y # y# ‹ dy œ 10 œ "# ' Š16 8y y # 4 ‹ dy œ 15 ; M œ ' ' 3
21. M œ
0
y 2
y
0
2
x
0
64 35
Ê xœ 3
22. M œ
0
2
'' 0
and y œ
0
4 y
0
2
y 2
0
5 7
3 x
'
dy dx œ
0
y 2
4 y
y 2
3 0
3
9
(3 x) dx œ
œ # ; My
3 x
' ' 0
'
x dy dx œ
0
3 0
cxyd 03
x
dx œ
'
3 0
a3x x# b dx œ 9#
Ê x œ 1 and y œ 1, by symmetry 1
''
23. M œ 2
0
1 x
' È 1x# dx œ 2 ˆ 4 ‰ œ # ; M 1
dy dx œ 2
0
1
0
' a1 x#b dx œ ’x x3 “ " œ 23 0
24. M œ
!
125$ 6
5
; My œ $ 5
''
Mx œ $
0
a
25. M œ
0
0
y dy dx œ
x
x
dy dx œ
0
4a 31
Ê xœyœ
1a
4
' #
5
$
0
cy# d x6x
'
x
'' 0
5 0
cxyd x6x
a
x
' # $
dx œ
a
; My œ
5 0
y dy dx œ
0
0
'
5
0
'
1
1 y# d 0 c 0
x
dx
$ a 5x# x$ b dx œ 625 ; 1#
a35x # 12x$ x%b dx œ 6256 $
x dy dx œ
'
a
cxyd 0 a 0
x
5
Ê x œ # and y œ 5
' xÈ a # x# dx œ a3 a
dx œ
0
, by symmetry
sin x
0
dx œ $
x
' ' dy dx œ ' sin x dx œ 2; M œ ' ' œ "4 ' (1 cos 2x) dx œ 4 Ê x œ # and y œ 8
26. M œ
0
1 x
4
x dy dx œ $
x
''
Ê y œ 31 and x œ 0, by symmetry
6x x
6x x
a
''
''
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x
1
œ
1
1
0
x
0
1
0
1
sin x
y dy dx œ
0
" #
'
0
cy#d 0sin x dx œ "# '0
sin# x dx
1
0
2
27. Ix œ
4 x
' ' 2
y# dy dx œ
4 x
2
' ’ y3 “
4 x
dx œ
2
4 x
2 3
'
2 2
a4 x# b$Î# dx œ 41 ; Iy œ 41 , by symmetry;
Io œ I x Iy œ 81 2
sin x
28. Iy œ
' '
29. M œ
' '
0
0
œ
b Ä _
x# dy dx œ
ex
0
lim
x
dy dx œ
'
0 b
'
xex dx œ
0
b
'
2
asin# x 0b dx œ #" '
ex dx œ lim Ä _
lim
bÄ
_
'
0 b
2
(1 cos 2x) dx œ
ex dx œ 1
cxex ex d b0 œ 1
b
b
lim Ä _
lim Ä _
abeb
1
# 0
ex
' ' x dy dx œ ' e b œ 1; M œ ' ' y dy dx eb œ 1; My œ b
0
0
x
ex
0
0
xex dx
955
956
Chapter 15 Multiple Integrals
30. My
0
'
"
œ #
œ' '
Ix œ
32. M œ
y y y
2
y y
b
' xe
"
Ê x œ 1 and y œ 4
4
x 2
b
y y
2
É
œ
3 2
x
Ä _
1
2
‘
b 0
œ1
0
2
0
y
y y
2
'
e"
dx œ lim
0
0
3 2
'
2
"
dx œ
' ’ x2 xy“ dy œ ' Š y2 2y$ 2y# ‹ dy œ ’ 10y y# 2y3 “ # œ 158 ; ! x y y 64 y# (x y) dx dy œ ' ’ 2 xy$ “ dy œ ' Š 2 2y& 2y% ‹ dy œ 105 ;
y
Ix M
2x
b
(x y) dx dy œ
0
É
Rx œ
Ä _
' e
bÄ_
2
0
0
lim
x dy dx œ lim
0
''
x
e
0
''
31. M œ
" # b
e2x dx œ
8 7
2
0
y
É
œ2
2 7
12 4y 4y
3 2
'
5x dx dy œ 5
3 2
’ “ x 2
12 4y
5
dy œ
#
4y
a12 4y# 16y%b dy œ 23È 3
3 2
'
3 2
' ' (6x 3y 3) dy dx œ ' 6xy 3# y# 3y‘ dx œ ' a12 12x #b dx œ 8; M œ ' ' x(6x 3y 3) dy dx œ ' a12x 12x$ b dx œ 3; M œ ' ' y(6x 3y 3) dy dx 17 3 17 œ ' a14 6x 6x# 2x$ b dx œ # Ê x œ 8 and y œ 16 1
33. M œ
2 x
0
x
2 x
0
x
1
2y y
1
2 x
x
0
1
y
1
0
1
1
x
0
0
2 x
x
1
0
1
1
2y y
1
' ' (y 1) dx dy œ ' a2y 2y$b dy œ "# ; M œ ' ' y(y 1) dx dy œ ' a2y# 2y%b dy œ 154 ; 8 8 M œ' ' x(y 1) dx dy œ ' a2y# 2y% b dy œ 154 Ê x œ 15 and y œ 15 ; Ix œ ' ' y# (y 1) dx dy " œ 2 ' ay$ y& b dy œ 6
34. M œ
0
y
0
1
y
x
2y y
0
0
y
0
1
y
1
0
2y y
0
y
1
0
1
6
1
1
' ' (x y 1) dx dy œ ' (6y 24) dy œ 27; M M œ ' ' x(x y 1) dx dy œ ' (18y 90) dy œ 99 I y 11 œ 216 ' ˆ 3 6 ‰ dy œ 432; R œ É M œ 4
35. M œ
0
0
0
1
y
x
6
0
0
1
1
0
11 3
and y œ
14 27
0
1
; Iy
6
0
0
y
1
1
(y 1) dy dx œ
x
1
' Š 3x2
1
' ' 1
x
' Š x#
x
x# # 3
1
1
‹ dx œ
1
x(y 1) dy dx œ
2 x%
1
1
1
' y(6y 24) dy œ 14; 1) dx dy œ ' ' x# (x y
y(x y 1) dx dy œ
y
48 œ 35 ; My œ
œ
0
6
Ê x œ
0
0
' '
''
1
1
36. M œ
œ
‹ dx œ
16 35
x
' Š 3x# x#
; Ry œ
1
É
Iy M
œ
x$
32 15
1
; Mx œ
‹ dx œ 0
1
1
' Š 56 x3 x# ‹ dx 9 Ê x œ 0 and y œ 14 ; I œ ' ' x# (y 1) dy dx
' ' 1
x
y(y 1) dy dx œ
1
1
y
1
1
x
É
3 14
1
1
x
1
' ' (7y 1) dy dx œ ' Š 7x# x# ‹ dx œ 3115 ; M œ ' ' y(7y 1) dy dx œ ' Š 7x3 x2 ‹ dx œ 1315 ; 1) dy dx M œ ' ' x(7y 1) dy dx œ ' Š 7x# x$ ‹ dx œ 0 Ê x œ 0 and y œ 13 ; I œ ' ' x# (7y 31 I 7x 7 21 œ ' Š # x% ‹ dx œ 5 ; R œ É M œ É 31
37. M œ
1
0
1
1
y
1
x
x
0
1
0
1
y
1
1
1
x
0
y
y
1
' ' ˆ1 20x ‰ dy dx œ ' ˆ2 10x ‰ dx œ 60; M œ ' ' y ˆ1 20x ‰ dy dx œ ' ’ˆ1 #x0 ‰ Š y# ‹“ " dx œ 0; " x ‰ x x ˆ1 20 dy dx œ ' Š2x 10 œ' ' ‹ dx œ 20003 Ê x œ 1009 and y œ 0; I œ ' ' y# ˆ1 20x ‰ dy dx I 2 x œ 3 ' ˆ1 20 ‰ dx œ 20; R œ É M œ É 3" 20
38. M œ
1
0
20
1
20
My
1
1
0
1
0
20
x
20
1
0
20
1
20
x
0
20
0
0
1
x
x
0
1
1
Section 15.2 Areas, Moments, and Centers of Mass 1
''
39. M œ
y
0
y
1
0
y
9
1
y
0
'
Ix M
3
È6
1
É
Io M
œ
y
5
0
5
y
3
y
7 10
1
y(y 1) dx dy œ 2
' ay
$ # y
b dy œ 76 ;
0 1
; Ix œ
''
y
0
y# (y 1) dx dy œ
y
1
' a2y% 2y$b dy œ 103
Ê Ry œ
0
'
1 0
É
Iy M
a2y% 2y$b dy 3È 2 œ 10 ;
È 2
3
5
1
y
1
0
y
1
x
y
y
6 5
10,000ey
0
1
y
0
1
0
y
É
Io M
Ê Ro œ
œ
y
0
2
È 5
dy dx œ 10,000 a1 e# b
x
1
y
0
y
y
x
2
œ
y
Io œ Ix Iy œ
' '
0
"
x# (y 1) dx dy œ
a3x# 1b dx dy œ '0 a2y$ 2yb dy œ 3# ; Mx 1
41.
y
''
1
0
Ê Rx
0
0 dy œ 0 Ê x œ 0 and y œ
0
y
''
1
œ 10 ; Iy œ
Ê R! œ
1
œ
' ' y a3x# 1b dx dy œ ' a2y% 2y#b dy œ 1615 ; $b dy œ 56 x a3x# 1b dx dy œ 0 Ê x œ 0 and y œ 32 ; I œ ' ' y# a 3x# 1b dx dy œ ' a 2y& 2y 45 È 5 I I 11 11 x# a3x# 1b dx dy œ 2 ' ˆ 35 y& 3" y$ ‰ dy œ 30 œ É M œ 3 ; I œ ' ' Ê R œ É M œ É 45 ; 1
My
É
6 5
Io œ Ix Iy œ
'' œ' '
a2y# 2yb dy œ 53 ; Mx 0
x(y 1) dx dy œ
œ 10 Ê Rx œ
40. M œ
1
y
''
My œ
'
(y 1) dx dy œ
'
5
œ 10,000 a1 e# b
dx
5
x
1
’' ‰‘
dx 1
5
x
5
'
dx 1 x
“
b 2 ln ˆ1 x# ‰‘ !& 10,000 a 1 e 2 b 2 ln ˆ1 x# !& 5 5 7 œ 10,000 a 1 e# b 2 ln ˆ1 # ‰‘ 10,000 a 1 e# b 2 ln ˆ1 # ‰‘ œ 40,000 a 1 e# b ln ˆ 2 ‰ ¸ 43,329 2
œ 10,000 a 1 e
1
42.
'' 0
2y y y
œ 200
’
y 2
y
4
1
' '
43. M œ
'
100(y 1) dx dy œ
1
"
“
0
c100(y 1)xd y2y
a 1 x
dy dx œ 2a
'
1
'
1 0
0
a1 x# b dx œ 2a ’x x3 “
1
œ #
dy œ
100(y 1) a2y 2y# b dy œ 200
'
1 0
ay y$b dy
ˆ ‰ œ 50
' a1 2x# x%b dx œ a# ’x 2x3
2a
y
" œ (200) 4
!
0
1
0
x 5
“
"
œ !
8a 15
"
1
4a
œ 3 ; Mx œ !
Ê y œ
Mx M
' ' 1
Š ‹ œ Š ‹ 8a 15 4a 3
a 1 x
y dy dx
0
2a
œ 5 . The angle ) between the
x-axis and the line segment from the fulcrum to the center of mass on the y-axis plus 45° must be no more than 90° if the center of mass is to lie on the left side of the line x œ 1 Ê ) 5
Thus, if 0 a Ÿ
#
4
44. f(a) œ Ia œ
'' 0
2 0
1
4
1 Ÿ # Ê tan "
ˆ ‰Ÿ 2a 5
1
4
5
Ê a Ÿ #.
, then the appliance will have to be tipped more than 45° to fall over.
(y a)# dy dx œ
4
' ’ (23a)
a
3
0
“ dx œ
4 3
c(2 a)$ a$d ; thus f w(a)
œ 0 Ê 4(2 a) # 4a #
w œ 0 Ê a# (2 a)# œ 0 Ê 4 4a œ 0 Ê a œ 1. Since f w (a) œ 8(2 a) 8a œ 16 0, a œ 1 gives a
minimum value of I a. 1
1
' ' 2x œ ' È 1x
45. M œ
0
1 x
1
1
0
1 x
L 2
L
(b)
47. (a)
0
" # œ
’
' È 12x
$ x#
$ x#
dx œ 1
Mœ
dx œ
'' 0
$L
12
$L
3
"
dx œ c2 sin" xd ! œ 2
0
dx œ 2 a1 x# b
L 2
' I œ '
46. (a) I œ
1
dy dx œ
"Î# "
“
!
É † Ê R œ É † Ê Rœ
2y y
y
œ2 Ê x œ
$L
"
12
$L
$L
"
3
$L
1
$
dx dy œ 2$
' 0
œ
œ
2 1
ˆ
1
‰
1
# 0 œ 1 ; My œ
' ' 0
1 1
1 x 1 x
x dy dx
and y œ 0 by symmetry
L
È 3
2
L
È 3
ay y #b dy œ 2$ ’ y2
y
3
“
" !
œ 2$
ˆ ‰œ "
$
6
3
3
Ê $ œ #
957
958
Chapter 15 Multiple Integrals 1
2y y
' ' ' ' 0
(b) average value œ
(y1) dx dy
y
1
2y y
0
Š‹ Š‹
œ
dxdy
y
œ 3# œ $, so the values are the same
3
48. Let (xi ß yi ) be the location of the weather station in county i for i œ 1 ß á ß 254. The average temperature 254
! T(xißyi)
in Texas at time t! is approximately
?i A
, where T(xi ß y i) is the temperature at time t ! at the
i 1
A
weather station in county i, ?iA is the area of county i, and A is the area of Texas. My M
49. (a) x œ
R
œ ' ' (x h)
(b) IL
' ' x$ (xß y) dy dx œ 0
œ 0 Ê My œ #
$ (xß y)
dA œ
'' x
R
#
dA ' ' 2hx $ (xß y) dA ' ' h# $ (xß y) dA
$ (xß y)
R #
œ Iy 0 h
' ' $ (xß y) dA œ I
R
cm
mh
R
#
R
50. (a) Ic m œ I L mh# Ê Ixœ5/7 œ Iy mh# œ (b) Ix 51. Mxp
1
œ
p
œ Ix
''
5 7
thus c œ x i y j œ
m m
"
œ
9 5
; Iy
b a b a bd
R
m m
2 #
ˆ‰
5 #
14 7
c a
"
ca
23 35
œ
Mx Mx m m
bd
œ
11 14
m c m c m m
47 14
œ Ix mh # œ 12 14 14
ˆ‰
; likewise, y œ
ca
"
Mx Mx i My My j œ
m" x" i y" j m# x# i y# j
; Iy
œ Iyœ11Î14 mh # œ
2
y dA" ' ' y dA# œ Mx Mx Ê x œ
R
œ
ˆ‰
23
mh# œ 35 14 7
39 5
17 # 14
ˆ‰
11 # 14
œ
47 14
œ 24
My My m m
;
b a
bd
m" x" m#x # i m"y " m#y # j
m m
52. From Exercise 51 we have that Pappus's formula is true for n œ 2. Assume that Pappus's formula is true for k 1
! mi ci
n œ k 1, i.e., that c(k 1) œ
i 1 k 1
. The first moment about x of k nonoverlapping plates is
! mi
! ' '
i 1
k1
y dAi
iœ1
Ri
' ' y dAk œ Mx Rk
thus c(k) œ x i y j œ
œ
"
mi
!m k
i 1
i
ˆ
i 1
i 1
iœ1
(b) c œ (c) c œ (d) c œ
54. c œ
15
Ê x
ˆ
3 4
1
ˆ b—
i
mi
iœ1
Mxc
Mx
i 1
1
k
‰‘ •
œ k 1
! mi c(k1)mk ck i 1
k 1
! mi i 1
, and by mathematical induction the statement follows.
i 7j 48(12i j) 15(3i 28j) 48(48i 4j) œ 1548 4†63 261 17 œ #8 and y œ 7
œ
2349 i 612 j 4†63
yœ
œ
Myc k
43 16
261 i 68j 4 †7
1
My
k
k 1
! mi i 1
yc mk yk j
xc i yc j mk xk i yk j
; similarly, y œ
mk
Myk j
8(i 3j) 2(3i 3.5 j ) 14i 31j 31 œ Ê x œ 57 and y œ 10 82 10 8(i 3j) 6(5i 2 j) 38i 36j œ Ê x œ 19 and y œ 18 14 14 7 7 2(3i 3.5j) 6(5i 2 j ) 36i 19j 9 œ Ê x œ 2 and y œ 19 8 8 8 8(i 3j) 2(3i 3.5 j ) 6(5i 2j) 44i 43j 11 œ 16 Ê x œ 4 and 16
‰
k 1
k 1
! mi
Mxk i Myc k k1
m c m c á mk 1 ck 1 mk ck m m á mk 1 mk
53. (a) c œ
Mxk Ê x œ
‰ ŒŒ! b a
M xc k
xc mk xk
iœ1
! mi
k
! mi
œ
!m
k1
"
œ
”Œ Œ ! – a
" k
k1
i
ck 1
mk
;
Section 15.3 Double Integrals in Polar Form
ˆ ‰
ˆ ‰
55. Place the midpoint of the triangle's base at the origin and above the semicircle. Then the center of h 3
mass of the triangle is 0 ß
ˆ j‰Š ‹ ˆ j‰ Šah ‹ h 3
(ah)
Pappus's formula, c œ
a 2
4a 3
a
2a
3
È
È
from Exercise 25. From
Š ‹j Šah ‹ , so the centroid is on the boundary ah
œ
if ah# 2a$ œ 0 Ê h# œ 2a# Ê h œ a ah# 2a$ 0 or h a
4a 31
, and the center of mass of the disk is 0 ß a
2 . In order for the center of mass to be inside T we must have
2.
56. Place the midpoint of the triangle's base at the origin and above the square. From Pappus's formula,
Š ‹ Š j‹ s ˆ j‰ , so the centroid is on the boundary if Š s ‹ sh
cœ
h 3
s 2
sh 6
sh
s
È
œ 0 Ê h# 3s# œ 0 Ê h œ s
#
15.3 DOUBLE INTEGRALS IN POLAR FORM 1
1 x
1.
' ' ' '
1 x
2.
1
' '
1 y
3.
4.
' ' ' '
a
5.
2
''
4 y
6.
7.
''
8.
''
9.
' '
1
0
1
1
0
a
' '
x# y# dx dy œ
' '
2
x
a
2
0
b
2
6 csc
' '
y dy dx œ
' '
4
0
4
0
0
2 x y 1 x 1
È
1
2
2
0
1
r$ dr d) œ
0
1
r$ dr d) œ
0
2
' ' 0
2
0
3
2
' '
1
2r 1r
0
1
d) œ
0
1 #
d) œ
0
8
#
0
r# sin ) dr d) œ
dy dx œ
2
2
2
r# cos ) dr d) œ 72
2 sec
' 4
"
r$ dr d) œ 4
0
' 4
"
' d ) œ 1a
a 2
r dr d) œ
x# y# dx dy œ
x
0
0
0
1 #
' d) œ 1
a
' '
dy dx œ
x dx dy œ
0
2
x# y# dx dy œ
0
0
" #
r dr d) œ
a b a b
y 0
1 0
' d) œ
" #
r dr d) œ
0
0
x
0
6
0
2
1 y
a
0
' '
1 y
a
0
dy dx œ
0
1
0
' '
1 x
1
1
dy dx œ
8 3
'
'
' 4
d) œ 21
0
2
4 0
2
c d
cot ) csc# ) d) œ 36 cot# )
3
2
1
' '
dr d) œ 4
' '
0
ˆ
1
œ 36
4 3
tan ) sec# ) d) œ
dr d) œ 2
2 4
" 1r
‰
dr d) œ 2
'
3
2
(1 ln 2) d)
œ (1 ln 2)1 1
10.
' '
È
0
1
1
y
4 x y 1x y
3
dx dy œ
2
' ' 2
1
3
4r
0
1r
2
ln 2
2
2
1
0
ˆ
1
" 1 r
‰
dr d) œ 4
'
3
2 2
ˆ ‰ 1
œ 41 1# ln 2
11.
' '
12.
' '
13.
''
0
1
0
y
È x
e
0
1
x
e
0
2
0
(ln 2)
1
(x
1)
0
œ )
sin 2) 2
x
y
xy x y
y
dx dy œ
' ' 0
2
1
dy dx œ
' '
dy dx œ
' '
0
2
sin# )
‘
2 0
œ
0
12 #
œ
0
0
rer dr d) œ
2 cos 0 1 #
rer dr d) œ
1
r(cos ) sin ) ) r
2
'
" #
'
(2 ln 2 1) d) œ
0
2 0
ˆ ‰ ' a "
e
1 d) œ
2
r dr d) œ
0
1 #
(2 ln 2 1)
1(e 1)
4e
b
2 cos# ) 2 sin ) cos ) d)
1
4
d)
3.
959
960
Chapter 15 Multiple Integrals 2
0
14.
''
15.
16.
' '
17.
' '
0
1
(y
' '
1
y
1
1
1
1
1
1
2
0
1)
a
a1 x
x
2
2
18. A œ 2
' '
19. A œ 2
' '
0
1
0
2
4
20. A œ
' '
21. A œ
' '
1
1
' '
1 cos
' '
a
a
' '
a
Io œ
a
x
a
2
6
3
26. Io œ
2
' ' 2
6 sin 3
1
1
ˆ
0
0
2
28. Io œ
' ' 0
" #
r dr d) œ cos
r dr d) œ
0
1 0
)
#
2
29. average œ
4 1a
' '
30. average œ
4 1a
' '
31. average œ
" 1a
' '
0
2
#
2
ˆ
cos 2 2 sin ) # )
3 #
0
a
2
dy dx œ k
' ' 0
a
a
x x
#
‘ " 1 r
"
!
d) œ 2
2
'
d) œ 1
0
‰
d) œ
1
31 2
4
' ' 0
a
0
r& dr d) œ
ka 6
'
ka 6
a
b
cos# ) 2 cos ) d) œ
(1 cos ))# d) œ
31 #
cos 4) 4
4 31a
' 0
2
'
2
0
a$ d) œ
x# y# dy dx œ
" 1a
‰
sin 2) 4
d) œ
51 4
0
ka 1 3
d) œ
0
1
1 cos 2) #
a$ d) œ
2 6
) #
œ6
2
0
a
2 sin )
cos
0
Ê xœ
r
#
5 6
‘
3
0
;
a 31
'
2 2
œ 2
1
4
cos ) dr d)
and y œ 0, by symmetry
2a 3
r# dr d) œ
ka 1 6
3 21
2a 3
' '
d) œ
d È
' '
; My œ 2
2
351 16
(1 cos ))% d) œ
4 31a
" #
'
0
2
c
2 2
0
31 8
d) œ
(6 sin ) 3) d) œ 6 2 cos ) )
6
2
81 4
r& sin# ) dr d) œ
2
'
È
a
0
2
dy dx œ k
r# dr d) œ
a
(ln 4 1) d) œ 1(ln 4 1)
0
0
a# r# dr d) œ
r
0
0
4
‰
cos 2)
2 cos )
È
a
a
"
œ
641 27
15
r$ dr d) œ
2
cos# 3) d) œ 121
24 cos 2) sin# ) cos )
cos
0
sin) )
' (1 cos ))$ sin ) d) œ 4
3
0
2
2
c d
6
3
0
'
'
ˆ
'
'
dr d) œ 2
1 c os
' ' œ 2 ' 4 cos 3
27. M œ 2
k x# y#
x
' '
25. M œ 2
0
y# k x# y#
x
b
d) œ 2
c a bd a b
x
a
#
3r# sin ) dr d) œ
0
'
dr d) œ 4
2 cos ) cos# ) d) œ
0
'
r dr d) œ 2
0
a
1 2
cos
0
24. Ix œ
)
a
'
0
r dr d) œ
0
a
2
'
8 9
sin
2
0
0
r dr d) œ 144
3
' '
2
'
12 cos 3 0
'
4 5
(2 sin 2)) d) œ 2(1 1)
0
cos
0
0
sin( ) cos ) d) œ
ln r# 1 r dr d) œ 2
1
0
2
2
'
r dr d) œ
0
0
23. Mx œ
b
r dr d) œ
2
22. A œ 4
y
1
6
0
'
32 5
a b
1
' ' a1 2rr b
dy dx œ 4
r dr d) œ 2
0
2
' '
2
2
sin 2
0
sin# ) cos ) r% dr d) œ
0
b
x
2
2
2 sin
ln x # y# 1 dx dy œ 4
y
1
' '
xy# dx dy œ
2
0
d) œ
2a 3
4 5
Section 15.3 Double Integrals in Polar Form
œ 2
2
' '
" 1
0
' '
34.
'
2 0
0
a
R
e
e
ln r r
1
2
œ
2 3
2
0
1 cos
' ' 0
È
21 2 3
0 4
'
32
0
' ' 0
" #
' ' a1 0
0
1
4 b
a
0
y
2
Ä _
2 0
t
ˆ
ˆ
" 1b
1
" #
ln
c
2 0
‘
2
‰
"
4
‰
œ
3 4
:
0
4
0
È
lim
aÄ1
x# y# Ÿ 1
d
d) œ 2
'
2
‘
2 sin ) 3
2
œ
0
3 #
È ˆ ‰ ‘
2
'
e 1
d) œ
3 ) 4
" #
e
0
ˆ È ‰
1 1 d) œ 21 2
d) œ 21
0
b
cos ) 3
1
d) œ 1
2
#
1
dr d) œ
2
'
0
Ê Iœ
4
Š È ‹Š È ‹ 0
cos )
r dr d) œ
2
'
" #
’
32
3
er
0
8
‘
(2 2 cos 2) )$Î# 2$Î# d)
Š ‹
0
e 1
" 1
d) œ
51
4 3
œ
0
21 2 3
dt œ
‰
r ln r r
(ln r)#
0
' ' a1 r r b
”
“
È2 40È 2 64
4
9
0
'
lim
b
0
bÄ_
•
rer dr d)
È 1 #
œ 1, from part (a)
b
1 #
61
œ
' a1 rr b
lim
bÄ
0
_
1
dr œ
4 b
‘ 1 " r
lim Ä _
' ' 1 R
2
" x y
dA œ
3 2
' ' 0
0
2
1
2
r
dr d) œ
1r
' 0
a b‘ 2" ln 1 r#
3 2 0
œ
" 1a
œ
" #
a
' #
2 0
2
" #
' '
Š
0
a 4 #
a 2h
b
d)
2
'
d) œ (ln 2)
0
' ' 1
ln 1 a#
d) œ 1 ln 4 "
' '
dA œ
x y
0
d ) œ 21 † lim a
Ä
1
r
0
1r
" #
ln 1 a#
a b‘
' '
f 0
'
dr d) œ
r dr d ) œ
'
2 0
’
lim
aÄ1
'
a 0
r 1 r
“
dr d)
œ 21 † _, so the integral does not exist over
’“ r 2
f
d) œ 0
" #
'
f #()) d) œ
'
where r œ f()) " 1a
0
4
40. The area in polar coordinates is given by A œ
41. average œ
b
1
a b‘
2
0
2 cos ) 3
c d
2
'
4 3
d) œ
0
R
'
0
t
1
dx dy œ
b
x y
Ä _
‹
'
d
(1 r cos ))# r# sin# ) r dr d)
2
"
lim
Over the disk x# y# Ÿ 1: œ
sin 4) 32
' '
dx dy œ
Š
3 4
c
3 cos# ) 3 cos$ ) cos% ) d)
2
b
39. Over the disk x# y# Ÿ
'
0
b
1
0
a
2
'
2 3
1 cos# ) sin ) d) œ
' lim e b 1 b ' 2Èe dt œ È2 ' e 0
0
dr d) œ
2 r# dr d) œ
r
e ax
x
(b) x lim Ä _
È
2 cos 2
3
œ
œ
2 ln r r
"5
37. (a) I# œ
œ
ˆ ‰
r# cos ) dr d) œ
1
4
38.
1
ˆ
2
'
) 8 sin 2) 3 sin ) sin$ )
36. V œ 4 œ
0
e
0
2 ln r dr d) œ 2
1
' '
dr d) œ
' '
35. V œ 2
0
0
'
" 1
1
e
' '
r dr d) œ
2
' '
" 1
y# dy dx œ
2
ln r r
1
d b
#
r$ 2r# cos ) r dr d) œ
Š ‹ 'Š ‹
33.
0
1
c
' ' (1 x)
" 1
32. average œ
a 0
c
d ' Š
(r cos ) h)# r# sin# ) r dr d) œ
2a h cos ) 3
a h #
‹
d) œ
" 1
2
0
a 4
" 1a
2ah cos ) 3
2
' ' 0
h #
‹
a 0
a
b
r$ 2r#h cos ) rh# dr d)
d) œ
" 1
’
a ) 4
2ah sin ) 3
h) #
“
2 0
" #
r# d),
e
961
962
Chapter 15 Multiple Integrals 3
42. (a) A œ œ
" #
4
' '
c
4
2 sin csc
44-46.
3
'
d ˆ ‰ˆ ‰
2) sin 2) cot )
(b) V œ 21yA œ 21
" #
r dr d) œ
3
31 4 31
4
4
1 #
œ
4 4 1 #
a
b
4 sin# ) csc# ) d)
œ 1#
41 3
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)" ); q1 := eval( x=a, [x=r*cos(theta),y=r*sin(theta)] );
# (a) # (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 ); Mathematica : (functions and bounds will vary) For 43 and 44, begin by drawing the region of integration with the FilledPlot command. Clear[x, y, r, t]
<
<
f:=Sqrt[x y] topolar={x Ä r Cos[t], y Ä r Sin[t]}; fp= f/.topolar //Simplify Integrate[r fp, {t, 0, 1/4}, {r, 0, bdr[[1, 1, 2]]}]
Section 15.4 Triple Integrals in Rectangular Coordinates 15.4 TRIPLE INTEGRALS IN RECTANGULAR COORDINATES 1
1.
a b ’a b a b
1 x
0
0
0
1
2.
1
x z
1
1
1 x
1
1
2
3
''' ''' 0 3
0 1
0 2
0
0
0
1
1
0
“
2
0
1
2
1
0
1
0
2 2x
0
0
3
2
1
'''
6 dx œ 6,
0
0
0
'''
0 1 0
0
ˆ
‰
#
1
0
3 3x 3y 2
0
3
0
2
3
0
4 x
0
dz dxdy,
4 x
' ' 2
'
4 x 2
'' 0
3 0
2
0
2
8 x x
'' 0
È
4 x 0
y
0
0
y
0
2
0
0
2
0
2
4 y
2
4 y
4
' '' y
z
' ' ' 0
4
z
z
'' ' 0
0
0
2
dx dy dz,
3
1
' ' ' dx dz dy, 0
0
0
‘
z
#
#
2
'
3 0
'' 0
0
2
''
dy dzdx, 4 x 0
0
8 x
' x
È
' 3 4x
4 x# dy dx œ
c a bd a b a b ’ ˆ‰
4 x
2
4
0
1
0
2
dz dy dx œ 4
4 x
0
2
0
2
1 y 2 z 3
' ' 8 2 x y dy dx œ 8 ' ' 4 x y dy dx œ 8' ' 4 r r dr d) œ 8 ' 2r œ 32 ' d) œ 32 œ 161 , ' ' ' dzdxdy, 2
2
0
0
2
3
'''
dz dx dy,
0
3 3y 2
0
dz dy dx œ
' ''
œ4
0
1 y 2 z 3
4 x
2
0
3
"
0
0
3
3
2 2x 2z 3
0
2
0
2 2z 3
0
dz dx
"
3 3x
0
2 2x 2z 3
0
3
d
1
0
1 z 3
0
c
#
0 1 y 2
2
b
3 3x 3y 2
0 2 2x
1
5.
a “
0
4.
0
0
2
0
0
’
2
0
' ' 3 dy dx œ ' dydx dz, ' ' ' dydz dx dz dy dx œ
x z
' ' ' dzdy dx œ ' ' 3 3x 32 y dy dx œ ' 3(1 x) † 2(1 x) 34 † 4(1 x) dx œ 3 ' (1 x) dx œ (1 x)$ ! œ 1, '' ' ' dydz dx, dzdx dy, ' ' '' ' ' dxdz dy, dydx dz, ' ' ' ' ' dxdydz
3.
1 x
' ' ' F x, y, z dy dz dx œ ' ' ' dy dz dx œ ' ' 1 x z a1 xb dx œ ' a1 xb dx œ a1 xb œ 1 œ' 1x x 1x 2 2 6 6
4 z
0
'
#
dx œ
3 0
’ È
3 #
2
dydx dz,
4 x# 4 sin"
x
3
''' 0
0
0
4 z
x #
“
3
dxdy dz,
2
œ 6 sin" 1 œ 31,
0 2
4 z
''' 0
0
0
dx dzdy
y
dz dy dx
y
#
#
2
#
#
0
r
4
“
#
!
d)
1 #
8 x
x
y
y
z y z y
2
dx dz dy
z y z y
8 z y
2
4
8
8 z y
''
8 z
dx dy dz
8
''
8 z
dy dx dz
z x z x
8 y
' ' ' 4
4
8 z
8 z
'
dx dz dy,
8 z y 8 z y
2
dx dy dz,
8 z x
'
8 z x
dy dx dz
4
' '' 2
x
z x z x
2
dy dz dx
8 x
8 z x
' ' ' 2
4
8 z x
dy dz dx,
963
964
Chapter 15 Multiple Integrals
6. The projection of D onto the xy-plane has the boundary x# y# œ 2y Ê x# (y 1)# œ 1, which is a circle. Therefore the two integrals are: 2
2y
'' 0
y
2y
1
1
'
y
a
1
2y x
1
y
1
b
'''
8.
' ' ' dz dx dy œ ' ' œ ' 24y 18y$ 12y$ dy œ
0
0
2
3y
0
0
2
0
e
9.
e
1
1
1
10.
'' 0
x
a
e
'''
c
0
e
0
#
1
1
''
dz dy dx œ "
!
œ
0
0
x# y#
"5
2
y%
‘
e
dy dz œ
1
2 0
'
'' 1
e 1
0
2
'
1 0
ˆ ‰
2 3
8x
0
2 3
x#
dx œ 1
x$ 4xy#
‘
3y
dy
0
œ 24 30 œ 6
"
yz
dy dz œ
'
(3 3x y) dy dx œ
' ' ' (x y z) dy dx dz œ ' ' œ ' 4z dz œ 0
'' 0
1y
0
1
1 1
dy dx œ
b
1
1
3
1
0
'
e
1
’ “ ln y z
e
e
dz œ 1
(3 3x)#
' z dz œ1 "
1
‘
" #
(3 3x)# dx œ
9 #
'
1 0
(1 x) # dx
#
y sin z dx dy dz œ
0
‰
"
3 3x
12.
1
ˆ
1
y
3
1
1
x
e
''' 0
2y
' dzdy dx
8 2x# 4y# dx dy œ
11.
0
x
12y#
ln x yz
1
d
0
0
e
3 3x y
(1 x)$
3
œ
'
1
3y
x
1
''
a b ''’ “ 2
dx dy dz œ
xyz
3 3x
y
3y
"
1
0
x# y# z# dz dy dx œ
8 x
1
1
7.
0
1
' '
dzdx dy and
1
1
1
sin z dy dz œ 1 1
xy
"
2
1 #
'
y# zy
1 0
‘
1 #
sin z dz œ
(1 cos 1) 1
" "
dx dz œ
' ' 1
1 1
(2x 2z) dx dz œ
'
1 1
c
x# 2zx
d
" "
dz
1
3
13.
''
14.
''
0
9
0
4 y
2 3
3
dz dy dx œ
''
dz dx dy œ
''
0
2x y
0
0
$Î#
6
1 x
'' ' 0
12
2 3
œ
4 x 3
‘
"
!
œ
0
"
6
1
x dz dy dx œ $
"
!
œ
3
0
2
4 y
(2x y) dx dy œ
0
“ 'a b
9 x# dx œ 9x
4 y
x# xy
4 y
2
dy œ
(2 x y) dy dx œ
'
1
0
(2 x)#
" #
0
0
0
0
$ !
œ 18
"Î#
4 y#
‘
(2 x)# dx œ
'' 0
1 x 0
a
b
x 1 x# y dy dx œ
'
1 0
x
" #
'
1 0
’a b a b“ 1 x#
#
" #
"
12
' ' ' cos (u v w) du dv dw œ ' ' [sin (w v 1) sin (w v)] dv dw œ ' [( cos (w 21) cos (w 1)) (cos (w 1) cos w)] dw 0
x 3
(2y) dy
16 3
2 x 0
a b ’ ' c d '
9 x# dy dx œ
(2 x)# dx
8 7 6 œ 6
y
1 x#
È
4 y
(4)$Î# œ
''
x
0
0
1
’ a b“ "
!
dz dy dx œ
(2 x)$
"
#
2 x y
0
0
9
2
4 y#
2 x
œ
17.
x
’ a b“
œ
0
9
'
'' ' 1
16.
4 y
œ
1
15.
'
0
2
0
x
0
0
1 x#
dx œ
'
1 0
" #
a b
x 1 x#
#
dx
Section 15.4 Triple Integrals in Rectangular Coordinates
c
d d
1
œ sin (w 21) sin (w 1) sin w sin (w 1) ! œ 0 e
e
18.
'''
19.
' ' œ '
1
1
e
1
4
0
ln sec v 0
Š
4
0
7
20.
e
4 q
œ
8 ln 8 3
1
1 x
1
0
1
1
0
y
0
1
'' 0
4 x 0
(e)
''
(b)
''' ' ''
1
''
2 2x
0
'
1
2
1 x
"
1 #
4
1
1
29. V œ 8
'' 0
2
30. V œ œ
" #
cos
0
4 q#
ln sec v
e 1
ds œ s ln s s
'
e2t dt dv œ
0
#
$Î#
!
dr œ
8 3
'
7 0
4 0
"
r 1
ˆ " #
d
e 1
œ1
" #
e2 ln sec v
‰
dv
dr
'' ' ' 4x 0
0
2
0
% !
'
4 x 0
a b
# #
0
1
c
d
2 2x
''
dz dy dx œ
0
0
0
z 1
'
1
"
4
(16) œ
'
c d 'a b ’ “ ' ’ È ˆ ‰“ 1
0
1
1 x
2z z#
32 3
3 #
2
0
dx œ
0
0
1 x# dx œ x
4x
4 x
x 3
dx
#
20 4 œ 3
a b ‰ '
y dy dx œ
3 3x
1 0
x cos
1x #
0
dx œ
2 1
y
2
dz dy dx œ 2
' 0
Š
'' 0
4 x
'
1
0
1 x# dx œ 1
y dy dx œ
1x #
dy dx œ
‘
"
!
1 x
0
4 1
' '
1
0
0
0
x
#
dx œ
ˆ ‰ cos
1x #
2 3
6(1 x)#
128 15
(1 x) dx
2
u cos u du œ
3 4
‘
† 4(1 x)# dx
2 1
4 1
c
d
cos u u sin u
a b ' ’a b a b“
1 x# dy dx œ 8
4 x# y dy dx œ
0
8 4x#
1x #
È a b ‹
'' 0
ˆ‰
sin
1
dz dy dx œ 8
dx œ
cos
ˆ‰ 0
1 x 0
1 x
1 0
dx dy dz
"
''
dx dy dz
dzdx dy
(1 x)$ ! œ 1
dz dy dx œ
y
0
1 x
ˆ
' '
0
1
dy dx dz
0
0
' '
(c)
0
y
1
(4) $Î#
4 3
œ
'' '
4
0
1 z
(c)
dzdx dy
0
y
(2 y) dy dx œ
1
4 1 œ 1
0
4 x
‘
0
x
1
1
0
1
dy dx dz
z
0
1
1 z
1 y
1
4 x
''
x 2
dx
1 x
0
dz dy dx œ 2
0
0
1x #
cos
(4 x)#
4
ˆ‰ ' ˆ ‰
1
0
'
y
(2 2z) dz dx œ
y
#
'' '
y
1 x
4
0
0
''
dz dy dx œ
0
0
1
3 3x 3y 2
0
1
0
2 y
1 x
1
œ
4
' '
d c
' dx œ 23
2 3
y# dy dx œ
1
0
'
1
''
dy dz dx œ
'' ' œ ' 3(1 x) dx œ 0
0
'
1 z
0
(e)
2 2z
0
0
1
0
1
(4 x)$Î#
4 3
1 z
dx dz dy
1
0
0
4
1
''
dz dy dx œ
1 x
0
0
c
(ln s) r ln r r
1
’a b“
"
0 3(r 1)
(b)
y
25. V œ
1
1
'
0
7
'
dq dr œ
0
0
'' '
œ
0
dx dzdy
24. V œ
28. V œ
2
1
1
0
27. V œ
' ' qÈ r4 1 q
e
18
" #
œ
1
1
'' '
26. V œ 2
1Î%
dy dz dx
23. V œ
‘
2 !
_
z
0
1
œ
v
bÄ
'
e2t eb dt dv œ
dy dz dx y
1
0
tan v 2
0
dr ds œ
a b
lim
y
0
''' ' ' '
(d)
0
dp dq dr œ
x
1 y
0
22. (a)
ln sec v
1 z
'' '
(d)
4
e 1
œ 8 ln 2
' ' '
21. (a)
dv œ
(ln r ln s) t ln t t
1
7
q r1
0
‹
" #
c
e
1
' '
ex dx dt dv œ
#
2
0
2t
'
sec v
''' 0
''
ln r ln s ln t dt dr ds œ
2
0
'
1
0
1 x# dx œ
4 x#
#
" #
16 3
4 x#
#
dx
2 0
"
!
œ
2 3
965
966
Chapter 15 Multiple Integrals
31. V œ
''
4
œ
'
16 y
È ˆ‰
16 y# dy 6
2
4 x
È
2
2
2
"
œ 12 sin 2 x
'
4 x
0
È
4
' y 0
œ 81
1 12 sin
"
È
2
'
x
2
4 x
2
(3 x) dy dx œ 2
’ È ˆ‰ ˆ ‰ ˆ ‰ ‘ 2
4 x
1 #
1 #
12
2 x y 2
’
4
8
0
z
#
z
z$
2
'
œ
'
œ4
4 x
2
2
0
4
1
0
0
1
0
37. average œ
"
38. average œ
"
40. average œ 1
''' 0
% !
0
œ
z
2
2
È
2
'
x
2
0
2
2 3
8
2
0
1
0
1
''' 0
0
1
0
2
1 0
2
0
0
a b È
4 cos x 2 z 2y
%
#
''
œ
y#
$Î# % !
2
(3 x) x 2
È “ ’a b “ 4 x# dx
#
2 3
#
4 x#
$Î#
# #
2 4
œ2
4
(8 2z)() z) dz œ
0
4
'
0
a
b
64 24z 2z# dz
4 x 2 0
È “ ’ a b“
(x 2) dy dx œ
'
4 x# 4 sin"
x
’ È
#
2
2
4 x# dx
(x 2)
#
$Î#
"
3 4 x#
#
# #
0
a
1 y 0
dy œ 2
'
1 0
2
0
(x y z) dz dy dx œ
"
2
b
2
"
xyz dz dy dx œ 4
dx dy dz œ
œ (sin 4)z"Î# ! œ 2 sin 4
1 y#
'' 8
'
"
3
1 0
’ “ ‰ x 3
xy#
3 y# 3 y% "
2
'' ' 0
0
4
x 2
0
2
'' 0
a
2
"
1
''
1 y
dy
'
a b
b
0
dy œ
0
1
1
0
ˆ
x# y#
a b È
4 cos x 2 z
xy dy dx œ
"
2
2 3
1
0
1 y' dy
3
‰
dy dx œ
a
4x# 36 dx œ
'
"
2
'
1
0
1 0
31 3
(2x 1) dx œ 0
ˆ ‰ 2 3
x#
dx œ 1
2
' x dx œ 1 0
4
dy dx dz œ
"
2
0
(2x 2y 2) dy dx œ
2 0
' 8 "
1
0
0
b
2x# 18 dy dx œ
0
''
x# y# z# dz dy dx œ
0
a bˆ
"
x# 9 dz dy dx œ
2 0
b
x# y# dx dy œ 2
4 7
a
'''
‘
6 7
2
0
2
1
1 y#
3
''' "
b“
œ 41
0
2
' '
4 x# dx œ x
dz dx dy œ 2
œ
0
6
16 y#
320 3
dz dy dx œ 2
0
'
(8 2z) dy dz œ
y
"
8
"
dy dx
#
8
x 2
"
1
39. average œ
4
x
1 y# !
% !
œ 121
œ (12 12 4 0)
!
a b’ a b “ ’ “ ˆ ‰ˆ ‰ ''' a b
'
y 7
2 3
1 #
1 y
'' '
y
œ
‘
4 x# dx
2
1 #
36. V œ 2 œ2
2
È ˆ‰ ˆ ‰ 2
#
“
4
2 3
' '
#
0
''
dx dy dz œ
œ 64z 12z#
35. V œ 2
“
(2x) 4
8 z
'''
34. V œ
‘ ’a
#
x 2
œ 6x 3x#
y 4
(4 y) dy
2 x
0
#
0
2
'
4 x# 4 sin"
4 x# dx œ 3 x
(1) œ 12
4 2x 2y
È
#
0
16 y# dy œ y 16 y# 16 sin"
2
4 x# dx 2
4
(4 y) dz dy œ
0
' '
dz dy dx œ
0
' È 16 y
2
32 3
3 x
ˆ ‰ ’
0
0 2
41.
" #
0
16 y
'' ' dz dy dx œ ' ' 3 3x 3y œ ' 3 1 x (2 x) 34 (2 x) dx 3(2 x) œ ' 6 6x 3x dx 4 2
0
(16)$Î#
"
' ' œ3' 2 2
4
''
dx dz dy œ
2
32. V œ
33.
4 y
0
1 #
œ 16
'
0
4 0
2
'' 0
2 0
a b È
x cos x z
dx dz œ
'
4
0
ˆ ‰ sin 4 #
z"Î# dz
Section 15.4 Triple Integrals in Rectangular Coordinates 1
42.
1
1
0
0
x
1
0
a b
1
0
ln 3
z
z
1 2x
0
0
y
0
0
2
0
4 x
0
œ 1
'' 0
Ê œ
"
4
0
8 15
0
1 0
’a
Ê
'
4 x a
''
dy dz dx œ
‘ % !
œ
"
4
0
" #
y
dz dy dx œ
4 a x#
’
b a #
" #
41 sin 1 y y z
#
2
sin 2z 0 4z
4 x
0
‘
sin# z
4 15
Ê
"
!
1
0
0
6yz ezy dy dz œ
'
% !
4
sin 4
1
4 a x
b“ “
x
0
y
0
3ezy
"
!
dz
a b dz dy
41 sin 1 y y
'' 0
4 z 0
ˆ ‰ sin 2z 4z
'
x dx dz œ
4 0
ˆ ‰ sin 2z 4z
0
#
œ
b
4 x# y a dy dx œ
dx œ
"
!
a
8 15
4 15
Ê
" #
'
1 0
a
4 a x#
Ê (4 a) #
2 3
b
4 15
#
dx œ
(4 a)
"
5
4 15
œ
1
Ê
8 15
'
" #
(4 z) dz
0
c
d
(4 a)# 2x# (4 a) x % dx
Ê 15(4 a) # 10(4 a) 5 œ 0
Ê 3(4 a)# 2(4 a) 1 œ 0 Ê [3(4 a) 1][(4 a) 1] œ 0 Ê 4 a œ x a
0
’ “
#
'' 0
1
''
dy dz œ
dz dx œ
œ
(4 a)# x 3 x$ (4 a) 5
46. The volume of the ellipsoid
1
œ 2(1) 2(1) œ 4
x sin 2z 4z
4 a x#
2
a b
1
a bd
x
cos 2z
4 a x
'
1
1
#
'' '
1
''
dx dy dz œ
"
a b c
0
45.
c d
z
1
44.
1
' ' ' e siny a y b dx dy dz œ ' ' œ ' 41 y sin 1y dy œ 2 cos 1y 1
43.
1
' ' ' 12xz ezy dy dx dz œ ' ' ' 12xz ezy œ 3' e z dz œ 3 e 1 ! œ 3e 6
"
3
or 4 a œ 1 Ê a œ
13 3
or a œ 3
y 4(1)(2)(c)1 z 4abc œ 81 Ê c œ 3. b c œ 1 is 3 1 so that 3
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 ß y ß z) such that 4x# 4y# z# 4 Ÿ 0 or 4x# 4y# z# Ÿ 4, which is a solid ellipsoid centered at the origin. 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 ß y ß z) such that 1 x# y# z# 0 or x# y# z# Ÿ 1, which is a solid sphere of radius 1 centered at the origin. 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 ); Mathematica : (functions and bounds will vary) 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:= x2 y2 z Integrate[f, {x,1,1}, {y,Sqrt[1 x2 ], Sqrt[1 x2 ]}, {z, 0, 1}]
N[%] topolar={x Ä r Cos[t], y Ä r Sin[t]}; fp= f/.topolar //Simplify Integrate[r fp, {t, 0, 21}, {r, 0, 1},{z, 0, 1}]
N[%]
967
