solucion larzon cap13
Transcript of solucion larzon cap13
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C H A P T E R 1 3
Multiple Integration
Section 13.1 Iterated Integrals and Area in the Plane . . . . . . . . . . . . . 365
Section 13.2 Double Integrals and Volume . . . . . . . . . . . . . . . . . . . 369
Section 13.3 Change of Variables: Polar Coordinates . . . . . . . . . . . . . 375
Section 13.4 Center of Mass and Moments of Inertia . . . . . . . . . . . . . 379
Section 13.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Section 13.6 Triple Integrals and Applications . . . . . . . . . . . . . . . . . 388
Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates . . . . 393
Section 13.8 Change of Variables: Jacobians . . . . . . . . . . . . . . . . . . 397
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
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365
C H A P T E R 1 3
Multiple Integration
Section 13.1 Iterated Integrals and Area in the Plane
Solutions to Even-Numbered Exercises
2. x2
x
y
xdy 12
y2
x x2
x
1
2 x4
x
x2
x x
2x2 1 4.
cos y
0
ydx yxcos y
0
y cosy
6. x
x3
x2 3y2dy x2y y3x
x3 x2x x
3 x2x3 x33 x52 x32 x5 x9
8. 1y2
1y2x2 y 2dx 13x3 y2x
1y2
1y2 213 1 y232 y21 y212
21 y 2
31 2y 2
10.
cosx 13 cos3x cosy2
y
cosy 13 cos3y cosy
2
y
sin3x cosydx 2
y
1 cos2x sinx cosydx
12.
1
14x2 163 dx
4x3
3
16
3x
1
1
43 16
3 4
3
16
3 8
1
1
2
2
x2 y2dydx 1
1x2y y
3
3 2
2
dx 1
12x2 83 2x2
8
3dx
14.
4
4
64 x3x2dx 29 64 x3324
4
0 2
912832
2048
92
4
4x2
064 x
3
dydx 4
4y64 x3
x2
0dx
16.
2010y 143 y3 2y3dy 5y2
7y4
6
y4
2 2
0
20 56
3 8
140
3
20
2yy
10 2x2 2y2dxdy 2010x 2x
3
3 2y2x
2y
y
dy 2020y 163 y3 4y3 10y
2
3y3 2y3dy
18. 2
0
2yy2
3y26y
3ydxdy 2
0
3xy2yy2
3y26y
dy 32
0
8y2 4y3dy 383y3 y42
0
16
20. 2
0
2 cos
0
rdrd 2
0
r2
2 2 cos
0
d 2
0
2 cos2d 12 sin 22
0
2
22.
4
0
cos3 sin d cos4
4 4
0
1
41
24
1 316
4
0
cos
0
3r2 sin drd 4
0
r3sin cos
0
d
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24. 3
0
0
x2
1 y2dydx
3
0
x2 arctany
0
dx 3
0
x22dx
2x3
3 3
0
9
2
26. 0
0
xyex2y2dxdy
0
12yex2y2
0
dy 0
1
2yey
2
dy 14ey2
0
1
4
28.
A 3
1
3
1
dxdy 3
1
x3
1
dy 3
1
2 dy 2y3
1
4
2
2
31
3
1
x
y
A 3
1
3
1
dydx 3
1
y3
1
dx 3
1
2 dx 2x3
1
4
30.
3y12
0
1y y1
12 2
12
0
3 dy 1
12 1y 2 1dy
12
0
x5
2
dy 1
12x
11y2
2
dy
x2 5
3
3
5
4
2
41
1
1
x 1y =
y
A 12
0 5
2dxdy
1
1211y2
2dxdy
A 5
2
1x1
0
dydx 5
2
y1x1
0
dx 5
2
1
x 1dx 2x 1
5
2
2
32.
y 2 sin , dy 2 cos d, 4 y 2 2 cos
2 12 sin 22
0
42
0
cos2d 22
0
1 cos 2d
A 2
0
4y 2
0
dxdy 2
0
4 y2dy
x 2 sin , dx 2 cos d, 4 x2 2 cos
2 12 sin 22
0
22
0
1 cos 2d
42
0
cos2d
2
0
4 x2dx
x
2
1 2
1
y x= 4 2
y
A 2
04x2
0
dydx 34.
3
532 16
16
5
35y53 y2
4 8
0
8
0
y23 y2dy
A 8
0
y23
y2dxdy
16 2
532
16
5
x2 25x524
0
4
0
2x x32dx
40
y2x
x32dx
1 1
1
2
3
4
5
6
7
8
2 3 4 5 6 7 8
y
x
(4, 8)
y =x3/2
y = 2x
A 4
0
2x
x32dydx
366 Chapter 13 Multiple Integration
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36.
x
4
6
2 4 6 8
2
2
(3, 3)
(9, 1)
y x=
y =9x
y
9y 12y 21
0
9 lny 12y 23
1
9
21 ln 9
1
0
9 ydy 3
1
9y ydy
1
0
x9
y
dy 3
1
x9y
y
dy
A 1
0
9
y
dxdy 3
1
9y
y
dxdy
9
21 ln 9
12x23
0
9 lnx9
3
9
2 9ln 9 ln 3
3
0
yx
0
dx 9
3
y9x
0
dx 3
0
xdx 9
3
9
xdx
A 3
0
x
0
dydx 9
3
9x
0
dydx 38.
x
2
3
4
1 2 3 4
1
y x= 2
y x=
y
A 2
0
2x
x
dydx 2
0
2x xdx x2
2 2
0
2
1 4 3 2
y2
4 2
0
2y y2
4 4
2
2
0
y
2dy
4
2
2 y2dy
A 2
0
y
y2 dxdy
4
2
2
y2dxdy
40.
x
2
3
4
1 2 3 4
1
y x= 2
y
2
0 x2
0
fx,ydydx
4
0
2
y
fx,ydxdy, y x 2, 0 y 4 42.
1
1
1
2
3
4
1 2 3 4
y
x
4
0 4y
0
fx,ydxdy
2
0
4x
2
0
fx,ydydx, 0 y 4 x2, 0 x 2
44.
1 1 2
2
3
y
x
e2
0
2
1
fx,ydxdy e
e2
ln y
1
fx,ydxdy
2
1
ex
0
fx,ydydx, 0 y ex, 1 x 2 46.
2
x
2
1
23
4
4
y
1
0
arccosy
arccos y
fx,ydxdy
2
2
cosx
0
fx,ydydx, 0 y cosx,
2 x
2
Section 13.1 Iterated Integrals and Area in the Plane 367
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48.
x
2
3
4
1 2 3 4
1
y
2
1
4
2
dxdy 4
2
2
1
dydx 2 50.
1
1
1
1
y
x
2
2
4y2
4y2dxdy 4
2
2
4x2
4x2dydx
2
2
4 x2 4 x2dx 4
52.
2
0
6y
2y
dxdy 2
0
6 3ydy 6y 3y2
2 2
0
6y = x
2 y = 6 x
y
x
11
1
1
2
3
4
5
6
2 3 4 5 6
(4, 2)
4
0
x2
0
dydx 6
4
6x
0
dydx 4
0
x
2dx
6
4
6 xdx 4 2 6
54.
3
0
y2
0
dxdy 3
0
y2dy y3
3 3
0
9
1
2
1
2
3
4
5
3
4
5
1 2 3 4 5 6 7 8 9
y
x
y = x
9
0
3
xdydx
9
0
3 xdx 3x 23x329
0
27 18 9
56. 2
2
4y2
0
dxdy 4
0
4x
4x
dydx 32
3
x
2
1 2 3
1
1
2x y= 4 2
y
58. The first integral arises using vertical representative rectangles. The second integral arises using horizontal
representative rectangles.
1
4cos4
1
2sin4
1
4
4
0
y
y2x sinydxdy
4
0
12y siny 1
8y2 sinydy
1
4cos4
1
2sin4
1
4
x
1
1
2
2
3
4
(2, 4)
y
2
0
2x
x2
x sinydydx
2
0
x cos2x x cosx2dx
368 Chapter 13 Multiple Integration
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Section 13.2 Double Integrals and Volume
60.
20
xey2y
0
dy 20
yey2
dy 12 ey2
2
0
1
2e4
1
2e0
1
2 1 1
e4 0.4908
20
2x
ey2
dydx 20
y0
ey2
dxdy
62.
40
yx sinxx
0
dx 40
x sinxdx sinx x cosx4
0
sin 4 4 cos 4 1.858
2
0 4
y 2x sinxdxdy
4
0 x
0x sinxdydx
64. 10
2yy
sin x ydxdy sin 2
2
sin 3
3 0.408 66. a
0
ax0
x2 y 2dydx a4
6
68. (a)
(b)
(c) Both orders of integration yield 1.11899.
2
0 2
4y 2
xy
x2 y 2 1dxdy
3
2 2
0
xy
x2 y 2 1dxdy
4
3 164y
0
xy
x2 y 2 1dxdy
y 4 x2
4x 16 4y
x
1
1
2
2
3
4
yy 4 x2x 4 y 2
70. 20
2x
16 x3 y3dydx 6.8520 72. 20
1sin 0
15rdrd452
32
135
8 30.7541
78. False, letfx,y x.
74. A region is vertically simple if it is bounded on the left and right by vertical lines, and bounded on the top and bottom by
functions ofx. A region is horizontally simple if it is bounded on the top and bottom by horizontal lines, and bounded on the
left and right by functions ofy.
76. The integrations might be easier. See Exercise 59-62.
For Exercises 2 and 4, and the midpoints of the squares are
12,1
2, 3
2,
1
2, 5
2,
1
2, 7
2,
1
2, 1
2,
3
2, 3
2,
3
2, 5
2,
3
2, 7
2,
3
2.
x
2
3
4
1 2 3 4
1
yxi y
i 1
2.
40
20
1
2x2ydydx 4
0
x2y2
4 2
0dx 4
0
x2dx x3
3 4
0
64
3 21.3
i1
fxi,yixiyi 1
16
9
16
25
16
49
16
3
16
27
16
75
16
147
16 21
fx,y 1
2x2y
Section 13.2 Double Integrals and Volume 369
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4.
40
ln 3
x 1dx ln 3 lnx 1
4
0 ln 3ln 5 1.768
40
20
1
x 1y 1dydx 4
0
1x 1 lny 12
0dx
8
i1
fxi,yixiyi 4
9
4
15
4
21
4
27
4
15
4
25
4
35
4
45
7936
4725 1.680
fx,y 1
x 1y 1
6. 20
20
fx,ydydx 4 2 8 6 20
8.
2
8
8x 1
2sin 2x
0
8
0
1 cos 2xdx
0
1
2
sin2x
2dx
2
2
31
3
1
x
y
0
20
sin2x cos2y
dy
dx
0
12 sin2xy 12 sin 2y
2
0dx
10.
1024
27
256
9
256
27
2y92
27
y6
1444
0
4
0 y
72
3 y
5
24dy
1
1
2
3
4
2 3 4
y
x
(2, 4)4
0
y12y
x2y2dxdy 40x
3y2
3 y
12ydy
12.
1
2e e1
ey 12 e2y11
0
10
e e2y1dy
x1 1
2
y x= + 1y x= + 1
y
10
0y1
exydxdy 10
1y0
exydxdy 10
exy0
y1dy 1
0
exy1y
0dy
370 Chapter 13 Multiple Integration
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14.
x
y
22
23
23
25
2
2
23
0
sinxdx
sinx cosy2
0dx
20
sinx sinydxdy
20
sinx sinydydx 16.
For the first integral, we obtain:
1
1
2
3
4
2 3 4
y
x
5 8 e4 e4 13.
e
4x
1
x
x2
2
4
0
40xey
4x
x0
dx 4x0
xe4 xdx
40
4x0
xeydydx 40
4y0
xeydxdy
18.
14 ln1 x24
0
1
4ln17
1
24
0
x
1 x2dx
1
24
0
y2
1 x2x
0 dx
x
2
3
4
1 2 3 4
1
y x=
y
2
0 4
y 2
y
1 x2dxdy 4
0 x
0
y
1 x2dydx
22.
x
2
3
4
1 2 3 4
1
y
40
8 dx 32
4
0 2
0 6
2ydydx
4
0 6y
y2
2
0dx 24.
1
1
2
2
y =x
y
x
2
0 x
0 4 dydx
2
0 4xdx
2x
2
2
0
8
20.
x4 4 x232 1
2 x4 x2 4 arcsinx
2 1
12x4 x232 6x4 x2 24 arctanx
22
2 4
22x24 x2 13 4 x232dx
22x2y 13y3
4x2
0dx
2
2
4
x2
0
x2 y 2dydx
x2 1 1
1
2
3
4
x y= 4 2x y= 4 2
y
20
4y2
4y 2x2 y 2dxdy
Section 13.2 Double Integrals and Volume 371
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26.
1
1
2
2
y = 2 x
y
x
1
6x 23
2
0
4
3
20
1
22 x2 dx
20
2x0
2 x ydydx 202y xy y
2
2 2x
0
dx 28.
x
2
1 2
1 y x=
y
4
2y2 y4
4 2
0
20
y0
4 y 2dxdy 20
4y y3dy
30. 0
2ex2dx 4ex2
0 4
0
0
exy2dydx 0
2exy2
0dx
32. 1
0 x
0
1 x2dydx 13
34.
13x35
0
125
3
50
xyx
0dx 5
0
x2dx
V 50
x0
xdydx
x2 5
3
3
5
4
2
41
1
y x=
y
36.
4r3
3
2r2x 13x3r
0
42r
0
r2 x2dx
4r0
yr2 x2 y 2 r2 x2 arcsin yr2 x2r2x2
0dx
x
r
r
y r x= 2 2
yV 8r
0r2x2
0
r2 x2 y 2dydx
38.
32 64
3
32
5
256
15
16x 8x3
3
x5
5 2
0
20
16 8x2 x4dx
20
4 x24 x2dx
1 2
1
2
3
4
3 4
y
x
y = 4 x2
V 2
0
4x2
0
4 x2dydx 40.
x2 2
0
20
2dx
20
arctany
0dx
x
2
1 2
1
yV 2
0
0
1
1 y 2dydx
372 Chapter 13 Multiple Integration
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42.
5
2
2 y5
0
50
2dy
x2 5
3
3
5
4
2
41
1
y
V 50
0
sin2xdxdy 44. V 90
9y0
9 ydxdy 81
2
46. V 160
4y0
ln1 x ydxdy 38.25
48.
cab2 ab
3 ab
2
ab
6 abc
6
cab2 1 x
a2
x2b
2a
x3b
3a2
ab
6 1 x
a3
a
0
ca0
b1 xa xb
a 1 x
a b2
2b 1 x
a2
dx
ca0
y xya y 2
2bb1xa
0dx
V
Rfx,ydA
a
0 b1xa
0 c1
x
a
y
bdydx
z c1 xa y
b
x
ya
a
a
R
z
x
a
y
b
z
c 1
50.
x2 5
6
3
10
8
4
41
2
y e= x
y
101
dy y10
1 9
101
xlnyln y
0dy
ln 10
0 10
ex
1lny
dydx 10
1 ln y
0
1lny
dxdy 52.
2cos 2 2 sin 2 1
2cosy y siny2
0
220
y cosydy(2, 2)
y = x212
1
1
2
2
y
x
20
y cosy2ydy
2
0 2
12x2y cosydydx
2
0 2y
0y cosydxdy
54. Average 1
84
0
20
xydydx 1
84
0
2xdx x2
8 4
0 2
56.
e 12
e2 2e 1
2 ex1 12e2x1
0 2e2 12e2 e
1
2
Average1
121
0
1x
exydydx 210
ex1 e2xdx
Section 13.2 Double Integrals and Volume 373
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58. The second is integrable. The first contains
which does not have an elementary antiderivation.
siny2dy 60. (a) The total snowfall in the countyR.(b) The average snowfall inR.
62. Average 1
15060
45
5040
192x 576y x2 5y 2 2xy 5000dxdy 13,246.67
64. for all and
P0 x 1, 1 y 2 10
21
1
4xydydx 1
0
3x
8dx
3
16.
fx,ydA 20
20
1
4xydydx 2
0
x
2dx 1
x,yfx,y 0
66. for all and
12 e2x ex11
0
1
2e2 e1
1
2
1
2e1 12 0.1998.
P0 x 1,x y 1 10
1x
exydydx 10
exy1
x dx 1
0
e2x ex1dx
0
limb
exyb
0dx
0
exdx limb
exb
0 1
fx,ydA
0
0
exydydx
x,yfx,y 0
68. Sample Program for TI-82:
Program: DOUBLE
: Input A
: Input B
: Input M
: Input C
: Input D
: Input N
:
:
:
: For
: For
:
:
:
: End
: End
: Disp V
V sin x y G H VC 0.5H2J 1 y
A 0.5G2I 1 xJ, 1, N, 1
I, 1, M, 1
D CN H
B AM G
0 V
70.
(a) 129.2018
(b) 129.2756
m 10, n 2020
40
20ex38dydx
374 Chapter 13 Multiple Integration
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72.
(a) 13.956
(b) 13.9022
m 6, n 441
21
x3 y3dxdy 74.
Matches a.
x
y
3
4
3 3
z
V 50
76. True
78.
Thus,
21
1
ydy lny
2
1 ln 2.
21
exy
y
0dy
21
0
exydxdy
0e
x
e
2x
x dx
0 2
1exydxdy
21
exydy 1x exy2
1
ex e2x
x
Section 13.3 Change of Variables: Polar Coordinates
2. Polar coordinates 4. Rectangular coordinates
6. R r, : 0 r 4 sin , 0 8. R r, : 0 r rcos 3, 0
10.
02 3 41
2
16
3
643 sin2
2 4
0
40
40
r2 sin cos drd 40
r3
3sin cos
4
0d 12.
02 31
2
4 1 1
e9
12 e9 12
0
20
30
rer2
drd 20
12er23
0d
Section 13.3 Change of Variables: Polar Coordinates 375
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14.
16 1 cos 32
0
1
6
20
sin
21 cos 2d
01
( , ) = (0, 1)x y
2
20
1cos 0
sin rdrd 20
sin r2
2 1cos
0d
16. a0
a2x2
0
xdydx 20
a0
r2 cos drda3
32
0
cos d a3
3sin
2
0
a3
3
18.
40
22 3
3d 22
3
3
4
0
22 3
3
4
42
3
20
8y2
y
x2 y 2dxdy 40
22
0
r2drd
20.
6420
sin4 sin6d64
6sin5cos sin
3cos
4
3
8 sin cos
2
0 2
4
0 4yy2
0
x2dxdy 2
0 4 sin
0
r3 cos2drd 2
0
64 sin4cos2d
22.
625
16
625
8 sin2
4
0
40
625
4sin cos d
52 20
x0
xydydx 552 2
25x2
0
xydydx 40
50
r3 sin cos drd
24.
1 e2522
2 1 e252
22
1 e252d5 4 3 2
2
1
2
3
4
5
3
4
5
1 1 2 3 4
y
x
22
50
er22rdrd 2
2er22
5
0
d
26.
20
30
9r r3drd 20
92 r2 1
4r4
3
0d
81
42
0
d81
8
30
9x2
0
9 x2 y2dydx 20
30
9 r2rdrd
376 Chapter 13 Multiple Integration
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28.
74 7
4
440
7
4d
V 420
10
r2 3rdrd 420
r4
4
3r2
2 1
0
d 30.
4ln 4 34
220
ln 4 34d
2
2
0
r2
4
1 2 ln r
2
1
d
220
21
rln rdrd
R lnx2 y 2dA 2
0
21
ln r2rdrd
32. 20
515 d 1015V 2
0
41
16 r2rdrd 20
13 16 r23
4
1d
34.
(8 times the volume in the first octant)
820
a3
3d 8a
3
3
2
0
4a3
3
820
12 2
3a2 r232
a
0d
V 8
2
0 a
0
a2 r2rdrd
x2 y 2 z2 a2z a2 x2 y 2 a2 r2
36.
(a)
(b)
(c) V 220
121cos2
14
9
4r2 36rdrd 0.8000
Perimeter 2 0
14 1 cos22 cos2sin2d 5.21dr
d cos sin
xy
1
1
1
zr1
21 cos2
1
2
1
2cos2
Perimeter
r2 drd2
d.
9
4r2 36 z
9
4r2 36
0.7
1 1
0.7
1
4 r
1
21 cos2
9
4x2 y2 9 z
9
4x2 y2 9;
38. A 20
42
rdrd 20
6 d 12
40.
1
24 4 cos 1
2
1
4sin 2
2
0
1
28 4 4
9
2
20
2sin 0
rdrd1
22
0
2 sin 2
d1
22
0
4 4 sin sin2d1
22
04 4 sin 1 cos 22 d
Section 13.3 Change of Variables: Polar Coordinates 377
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42. 840
3 cos 20
rdrd 4 40
9 cos2 2d 1840
1 cos 4d 18 14 sin 44
0
9
2
44. See Theorem 13.3. 46. (a) Horizontal or polar representative elements
(b) Polar representative element
(c) Vertical or polar
48. (a) The volume of the subregion determined by the point is base height Adding up the 20
volumes, ending with you obtain
(b)
(c) 7.4824103.5 179,621 gallons
5624013.5 1,344,759 pounds
5
4150 555 1250 2135 2025
5
46115 24013.5 ft3
3512 15 18 16 459 10 14 12
V 10
857 9 9 5 158 10 11 8 2510 14 15 11
45 10 812, 5 10 87.5, 16, 7
50. 4
0 4
0
5er rdrd 87.130 52.
Answer (a)
94 3 21
x
y
4
4
2
2
4
6
zVolume base height
54. True
56. (a) Let then
(b) Let then
e4x2
dx
eu2
1
2du
1
2.u 2x,
ex2
dx
eu2
2 12
du 12 2 .u 2x,
58.
For to be a probability density function,
k4
.
k
4 1
fx,y
0
0
kex2y2dydx 2
0
0
ker2
rdrd 20
k2 er2
0d 2
0
k
2d
k
4
60. (a)
(b)
(c) 220
4 cos 0
f rdrd
420
4x22
0
fdydx x
2
1
2
1
3
1
( 2) + 2 = 4x y 2y
420
24y2
2
fdxdy
378 Chapter 13 Multiple Integration
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Section 13.4 Center of Mass and Moments of Inertia
2.
1
120 93
243
4
149 x23
3 3
0
3
0
x9 x22
2dx
m 30
9x2
0
xydydx 30xy
2
2 9x2
0
dx
4.
1
281
2
81
4 54 2978
1
229 x232 9x2
2
x4
4 3
0
12
3
06x9 x2 9x x3dx
30
x
23 9 x2 9dx
m 30
39x2
3
xydydx 30xy
2
2 39x2
3
dx
6. (a)
y Mxm
ka2b36
ka2b24
2
3b
x My
m
ka3b26
ka2b24
2
3a
My a0
b0
kx2ydydx ka3b2
6
Mx a0
b0
kxy2dydx ka2b3
6
m a0
b0
kxydydx ka2b2
4(b)
y Mxm
kab2122a2 3b2
kab3a2 b2
b
4 2a2 3b2
a2 b2
x My
m
ka2b123a2 2b2
kab3a2 b2
a
4 3a2 2b2
a2 b2
My a0
b0
kx3 xy2dydx ka2b
123a2 2b2
Mx a0
b0
kx2y y3dydx kab2
122a2 3b2
m a0
b0
kx2 y 2dydx kab
3a2 b2
8. (a)
CONTINUED
x y Mxm
ka36
ka22
a
3
My Mx by symmetry
Mx
a
0
ax
0
kydydx ka3
6
xa
ay a x=
ym a2k
2
Section 13.4 Center of Mass and Moments of Inertia 379
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10. Thex-coordinate changes by h units horizontally and kunits vertically. This is not true for variable densities.
12. (a)
y 4a
3by symmetry
x My
m
ka33
ka24
4a
3
k3 a2 x232a
0
ka3
3
ka0
xa2 x2dx
My a
0
a2x2
0
kxdydx
m a
0
a2x2
0
kdydx ka2
4(b)
x y M
y
m ka
5
5 8
ka4 8a
5
My Mx by symmetry
20
a0
kr4 sin drdka5
5
Mx a0
a2x2
0
kx2 y 2ydydx
20
a0
kr3drdka4
8
m a
0a2x2
0
kx2 y 2dydx
8. CONTINUED
(b)
y 2a
5by symmetry
x My
m
a515
a46
2a
5
a0
ax3 x 4 13 a3x a2x2 ax3 1
3x 4dx 13
a
0
a3x 3a2x2 6ax3 4x4dx a5
15
My a
0 ax
0
x3 xy2dydx
a0
x2y y3
3 ax
0dx a
0
ax2 x3 13 a x3dx a4
6
m a0
ax0
x2 y 2dydx
14.
y Mxm
16k
32k5
5
2
x My
m
32k
3
5
32k
5
3
My 20
x3
0
kx2dydx 32k
3
Mx 2
0
x3
0
kxydydx 16k
m 20
x3
0
kxdydx 20
kx4dx 32k
516.
x
2
3
4
1 2 3 4
1
y =4x
y
y My
m
24k
30k
4
5
x
My
m
84k
30k
14
5
My 41
4x0
kx3dydx 84k
Mx 41
4x0
kx2ydydx 24k
m 41
4x0
kx2dydx 30k
380 Chapter 13 Multiple Integration
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18.
x6 3 3 6
6
3
12
y x= 9 2
y
y Mxm
139,968k
35
35
23,328k 6
Mx 3
3
9x2
0
ky3dydx 139,968k
35
m 33
9x2
0
ky2dydx 23,328k
35
x 0 by symmetry 20.
x
1
L2
y = cosxL
t
y Mxm
kL
8
kL
8
x My
m
L2 2k
22
kL
L 2
2
My L2
0
cos xL0
kxdydx L 2 2k
22
Mx L20
cos xL0
kydydx kL
8
m L20
cos xL0
kdydx kL
22.
y Mxm
ka42 2 8
12
ka3
32 2a2
x My
m
ka42
8
12
ka3
32a
2
My R
kx2 y 2dA 40
a0
kr3 cos dka42
8
Mx R
kx2 y 2ydA 40
a0
kr3 sin dka42 2
8
0a
r a=y x=
2m
Rkx2 y2dA 4
0
a0
kr2drdka3
12
24.
y Mxm
k
6
2
k
1
3
x M
ym k1 2k
2
My e
1
ln x0
k
xxdydx k
Mx e1
ln x0
k
xydydx
k
6
m e1
ln x0
k
xdydx
k
2
2
2
3e1
3
1
x
y x= ln
y
Section 13.4 Center of Mass and Moments of Inertia 381
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26.
x My
m
5k
4
2
3k
5
6
5k
4
k
32
0
cos 32 1 cos2 3 cos 1 sin2 1
41 cos 22d
k3
2
0
cos 1 3 cos 3 cos2 cos3d
My RkxdA 2
0
1cos 0
kr2 cos drd
m RkdA 2
0
1cos 0
krdrd3k
2
01
r= 1 + cos
2y 0 by symmetry
28.
y Ixm bh312
bh2
h
66
6h
x Iym b3h12
bh2
b
66
6b
Iy b0
hhxb0
x2dydx b3h
12
Ix b0
hhxb0
y2dydx bh3
12
m
b
0
hhxb
0
dydx bh
230.
x y Ixm a4
8
2
a2
a
2
I0 Ix Ix a4
8
a4
8
a4
4
Iy Rx2dA
0
a0
r3 cos2drda4
8
Ix Ry 2dA
0
a0
r3 sin2drda4
8
m a2
2
32.
y Ixm ab3
4
1
ab
b
2
x Iym a3b
4
1
ab
a
2
I0 Iy Ix a3b
4
ab3
4
ab
4a2 b2
Iy 4 b0
abb2y 2
0
x2dxdy a3b
4
4b3
3a3a
2
2 xa2 x2 a2 arcsinx
a 1
8x2x2 a2a2 x2 a4 arcsinxa
a
0
ab3
4
4a0
b3
3a3a2 x232dx
4b3
3a3a
0
a2a2 x2 x2a2 x2dx
Ix 4a
0
baa2x2
0
y 2dydx
m
ab
382 Chapter 13 Multiple Integration
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34.
y Ixm 4ka515
2ka332a
2
5
2a
10
x Iym 2ka515
2ka33a
2
5
a
5
I0 Ix Iy 2ka5
5
Iy ka
aa
2x2
0
x2ydydx 2ka5
15
Ix k
a
aa2x2
0y3dydx
4ka5
15
ka0
a2 x2dx 2ka3
3
m 2ka0
a2x2
0
ydydx
ky 36.
y Ixmk60
k2425
x Iym k48
k24
1
2
I0 Ix Iy 9k
240
3k
80
Iy k
1
0 x
x2x3ydydx
k
21
0x5 x7dx
k
48
Ix k1
0
xx2
xy3dydx k
41
0
x5 x9dx k
60
m k10
xx2
xydydx k
21
0
x3 x5dx k
24
kxy
38.
y Ixm x 395
891
x
Iy
m
158
2079
35
6
395
891
I0 Ix Iy 316
2079
Iy 1
0
xx2
x2 y 2x2dydx 158
2079
Ix 1
0
xx 2
x2 y 2y 2dydx 158
2079
m 10
xx2
x2 y 2dydx 6
35
x2 y 2 40.
y Ixm 32,768k
65
21
512k
81365
65
x
Iy
m
2048k
45
21
512k
28
15
2105
15
I0 Ix Iy 321,536k
585
Iy 22
0
4xx3
kx2ydydx 2048k
45
Ix 22
0
4xx3
ky3dydx 32,768k
65
m 220
4xx3
kydydx 512k
21
ky
42. I 40
20
kx 62dydx 4
0
2kx 62dx 2k3 x 6
34
0
416k
3
44.
2k7a5
15
a5
8 ka556 15
60 2k1
4a5 2
3a5
1
5a5 2a3
a4
4
a4
16 a2
2 a3 a3
3
1
8 x2x2 a2a2 x2 a4 arcsinx
a a2
2 a2x x3
3 a
a
k14a4x 2a2x3
3
x5
5 2a
3 a2
2 xa2 x2 a2 arcsinx
a
aa
k14 a4 2a2x2 x 4 2a
3a2a2 x2 x2a2 x2
a2
2a2 x2dx
a
a
ky4
4
2ay3
3
a2y 2
2
a2x20
dx
I aaa
2x2
0
kyy a2dydx
Section 13.4 Center of Mass and Moments of Inertia 383
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46.
2k
3 32 32
192
5
128
7
1408k
105
k
322
16 12x2 6x4 x6dx k316x 4x3 6
5x
5
1
7x7
2
2
I 22
4x2
0
ky 22dydx 22
k3 y 13
4x2
0dx 2
2
k
32 x2 8dx
48.
will be the same.x,y
x,y k2 x. 50. Both and will decreaseyxx,y k4 x4 y.
52. Moment of inertia aboutx-axis.
Moment of inertia abouty-axis.Iy Rx2x,ydA
Ix R
y2x,ydA
54. Orient thexy-coordinate system so thatL is along they-axis andR is in the first quadrant. Then the volume of the solid is
By our positioning, Therefore, V 2rA.x r.
2xA.
2RxdARdA RdA
2RxdA
x
L
R( , )x y
y
V R 2xdA
56.
ya a
2
a3b12
L a2ab
a3L 2a
32L a
Iy b0
a0
y a22
dydx a3b
12
y a
2,A ab, h L
a
258.
ya a44
La2
a2
4L
a4
4
2
0
a4
4
sin2d
20
a0
r3 sin2drd
Iy aaa
2x2
a2x2y 2dydx
y 0,A a2, h L
384 Chapter 13 Multiple Integration
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Section 13.5 Surface Area
2.
2
2
31
3
1
x
R
y
S 3
0 3
0
14 dydx 3
0
314 dx 914
1 fx 2 fy
214
fx 2,fy 3
fx,y 15 2x 3y 4.
1
2
1
2
2 1 1
x
y x= 9 2
y x= 9 2
R
y
20
30
14 rdrd 914
S 339x29x2
14 dydx
1 fx 2 fy 2 14
fx 2, fy 3
R x,y:x2 y 2 9
fx,y 10 2x 3y
6.
square with vertices
34 2y1 4y2 ln2y 1 4y23
0
3
4637 ln6 37
S 30
30
1 4y 2dxdy 30
31 4y 2dy
1 fx 2 fy
21 4y2
fx 0, fy 2y
0, 0, 3, 0, 0, 3, 3, 3R
2
2
31
3
1
x
R
yfx,y y 2
8.
x
2
1 2
1 R
y = 2 x
y
12
53
8
5
2
332
2
5
352
2
2
5
21 y32 25 1 y522
0
S 202y
0
1 ydxdy 20
1 y 2 ydy
1 fx2fy
21 y
fx 0, fy y12
fx,y 2 2
3y32 10.
1
1
1 1
x
R
y
2
0
1
12
1732 1d
6
1717 1
20
112 1 4r2 322
0d
S 20
20
1 4r2rdrd
1 fx 2 fy
21 4x2 4y 2
fx 2x, fy 2y
fx,y 9 x2 y 2
Section 13.5 Surface Area 385
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12.
204
0
1 r2rdrd2
31717 1
S
4
4
16x2
16x2
1 y2 x2dydx
1 fx 2 fy
21 y 2 x2
fx y, fy x
R x,y:x2 y2 162
2
2 2
x
x y2 2+ = 16
yfx,y xy
14. See Exercise 13.
20
a0
a
a2 r2rdrd 2a2S a
aa2x2a2x2
a
a2 x2 y 2dydx
16.
42
4
2
6
6
x
y x= 16 2
y
20
40
1 4r2rdrd
246565 1
S 4
0 16x
2
0
1 4x2 y 2dydx
1 fy2 fy
21 4x2 4y 2
z 16 x2 y 2 18.
1
1
1 1
x
y
x2+y2= 4
S 202
0
5rdrd 45
1 fx2fy
2
1
4x2
x2
y2
4y2
x2
y25
z 2x2 y2
20.
triangle with vertices
2
2
31
3
1
x
R
y x=
y
S 20
x0
5 4y2dydx 1
122121 55
1 fx 2 fy
25 4y2
0, 0, 2, 0, 2, 2R
fx,y 2x y2 22.
2
2
2 2
x
x y22+ = 16
y
2
0
4
0 1
4r2
drd
6565 16
S 4416x216x2
1 4x2 4y 2dydx
1 fx 2 fy
21 4x2 4y 2
fx 2x, fy 2y
0 x2 y 2 16
R x,y: 0 fx,y 16
fx,y x2 y 2
386 Chapter 13 Multiple Integration
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24.
S
1
0
1
0
1 x sinx2dydx 1.02185
1 fx 2 fy
21 x sinx2
fx x12 sinx, fy 0
R x,y: 0 x 1, 0 y 1
fx,y 23 x
32 cosx 26. Surface
Matches (c)
xy33
3
2
z
area 9
28.
1
0
1 y3dy 1.1114
S 10
10
1 y3dxdy
1 fx 2 fy
21 y3
fx 0, fy y32
R x,y: 0 x 1, 0 y 1
fx,y 25y
52 30.
S
4
0
x
0
1 13x2 y 2dydx
1 13x2 y 2
1 fx fy 21 2x 3y2 3x 2y2
fx 2x 3y, fy 3x 2y 3x 2y
R x,y: 0 x 4, 0 y x
fx,y x2 3xy y 2
32.
S 22
(2)x2(2)x2
1 4x2 y 2 sin2x2 y 2dydx
1 fx 2 fy
21 4x2 sin2x2 y 2 4y 2sin2x2 y 2 1 4sin2x2 y 2x2 y 2
fx 2x sinx2y 2, fy 2y sinx
2y 2
R x,y: x2 y 2 2fx,y cosx2 y 2
34.
S 40
x0
1 e2xdydx
1 e2x
1 fx 2 fy
21 e2x sin2y e2x cos2y
fx ex siny, fy e
x cosy
R x,y: 0 x 4, 0 y x
fx,y ex siny 36. (a) Yes. For example, letR be the square given by
and S the square parallel toR given by
(b) Yes. LetR be the region in part (a) and S the surface
given by
(c) No.
fx,y xy.
0 x 1, 0 y 1,z 1.
0 x 1, 0 y 1,
38.
S R1 fx 2 fy 2dA
Rk2 1 dA k2 1
RdA Ak2 1 r2k2 1
1 fx 2 fy
21 k2x2x2 y 2 k
2y 2
x2 y 2k2 1
fx,y kx2 y 2
Section 13.5 Surface Area 387
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42. False. The surface area will remain the same for any vertical translation.
40. (a)
(c)
S 2500
150
1 fy2fx
2dydx 3087.58 sq ft
fx 0,fy 1
25
y2 8
25
y 16
15
fx,y 1
75y3
4
25y2
16
15y 25
z 1
75y3
4
25y2
16
15y 25 (b)
cubic feet
(d) Arc length
Surface area of roof 25030.8758 3087.58 sq ft
30.8758
100266.25 26,625
V 250150 175y3
4
25y2
16
15y 25dy
Section 13.6 Triple Integrals and Applications
2.
2
3
1
1
1
1
y 2z2dydz 2
9
1
1
y3z21
1dz
4
9
1
1
z2dz 427z31
1
8
27
11
11
11
x2y 2z2dxdydz 1
311
11
x3y 2z21
1dydz
4.
1
29
0
xy 2 3x3y3
0dy
2
189
0
y3dy 136y 49
0
729
4
90
y30
y29x2
0
zdzdxdy 1
29
0
y30
y 2 9x2dxdy
6.
41
1xlnz2
2 e2
1dx 4
1
2
xdx 2 ln x
4
1 2 ln 4
41
e2
1
1xz0
lnzdydzdx 41
e2
1
lnzy1xz
0dzdx 4
1
e2
1
lnz
xzdzdx
8. 20
y20
1y0
sinydzdxdy 20
y20
siny
ydxdy
1
22
0
sinydy 12 cosy2
0
1
2
10. 20
2x2
0
4y2
2x2y2ydzdydx 2
0
2x2
0
4y 2x2y 2y3dydx 162
15
12.
60
3(x2)0
1
2 6 x 2y
3 2
ex2y 2dydx 2.118
3
0 2(2y3)
0 62y3z
0
zex2y 2dxdzdy
6
0 (6x)2
0 (6x2y)3
0
zex2y 2dzdydx
14. 30
2x0
9x2
0
dzdydx
388 Chapter 13 Multiple Integration
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16.
44
16x2
16x280x
2y2
12x2y2dzdydx
z 8x2 y2 2z 16x2 y2 z2 2z z2 80z2 2z 80 0 z 8z 10 0
z 1
2x2 y2 2z x2 y2
18. 1
0 1
0 xy
0
dzdydx 1
0 1
0
xydydx 1
0
x
2dx x
2
4 1
0
1
4
20.
49x36 x2 324 arcsinx6 1
6x36 x232
6
0
4162 648
4603636 x2 x236 x2 1336 x232dx
460
36x2
0
36x2y
2
0
dzdydx 460
36x2
0
36 x2 y2dydx 46036y x2y y
3
3 36x2
0
dx
22.
20
18 9x 2x2 x3dx 18x 92x2 2
3x3
1
4x 4
2
0
50
3
2
0 2x
0 9x2
0dzdydx
2
0 2x
09 x2dydx
2
09 x22 xdx
24. Top plane:
Side cylinder:
30
9y2
0
6xy0
dzdxdy
x2 y2 9
x
y
6
3
6
63
zx y z 6 26. Elliptic cone:
40
4zy
2z22
0
dxdydz
x
y
5
5
4
3
3
2
2
1
1
z
4x2 z2 y 2
28.
40
2y0
y0
xyzdxdzdy 40
22y
2z0
dxdzdy 10421
20
(2z)2
0
y0
xyzdxdydz 20
4(2z)22z
0
xyzdxdydz
20
2z0
4x2
xyzdydxdz
2
0
2x
0
4
x2xyzdydzdx
40
y0
2x0
xyzdzdxdy
Q
xyzdV 20
4x22x
0
xyzdzdydx
x
y44
2
4
2
(2, 4)
zQ x,y,z: 0 x 2,x2 y 4, 0 z 2 x
Section 13.6 Triple Integrals and Applications 389
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30.
60
10
1x2
0
xyzdydxdz
10
60
1x2
0
xyzdydzdx
60
10
1y2
0
xyzdxdydz
1
0 6
0 1y
2
0
xyzdxdzdy
10
1y2
0
60
xyzdzdxdy
Q
xyzdV 10
1x2
0
60
xyzdzdydx
x
y2
1
6
21
zQ x,y,z: 0 x 1,y 1 x2, 0 z 6
32.
y Mxz
m 2
Mxz k5
0
5x0
15153x3y0
y2dzdydx 125
4k
m k
5
0
5x
0
15153x3y
0
ydzdydz 125
8
k 34.
y Mxz
m
kab2c24
kabc6
b
4
Mxz kb
0
a[1(yb)]0
c[1(yb)(xa)]0
ydzdxdy kab2c
24
m k
b
0
a1(yb)
0
c1(yb)(xa)
0
dzdxdy kabc
6
36.
z Mxy
m
kabc33
kabc22
2c
3
y Mxz
m
kab2c24
kabc22
b
2
x Myz
m
ka2bc24
kabc22
a
2
Mxz ka0
b0
c0
yzdzdydx kab2c2
4
Myz ka
0 b
0 c
0
xzdzdydx ka2
bc2
4
Mxy ka0
b0
c0
z2dzdydx kabc3
3
m ka0
b0
c0
zdzdydx kabc2
2
38. will be greater than whereas and will be unchanged.yx85,z
40. and will all be greater than their original values.zx,y
390 Chapter 13 Multiple Integration
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42.
z Mxy
m
k
16k3
3
16
y Mxz
m
2k
16k3
3
8
x Myz
m
0
16k3 0
Mxy 2k2
0
4x2
0
y0
zdzdydx k
Mxz 2k2
0
4x2
0
y0
ydzdydx 2k
Myz k2
2
4x2
0
y0
xdzdydx 0
k20
4 x2dx 16k
3
m 2k20
4x2
0
y0
dzdydx
44.
z Mxy
m k12
4k2
4
y Mxzm kln 4
k ln 4
k20
10
1
y 2 12dydx k2
0
y2y 2 1 1
2arctany
1
0dx k14
82
0
dx k12
4
Mxy 2k2
0
10
1(y21)
0
zdzdydx
Mxz 2k2
0
10
1(y21)
0
ydzdydx 2k20
10
y
y 2 1dydx k2
0
ln 2dx kln 4
m 2k20
10
1(y21)
0
dzdydx 2k20
10
1
y 2 1dydx 2k4
2
0
dx k
x 0
46.
z Mxy
m
10k
10k 1
y Mxz
m
15k2
10k
3
4
x Myz
m
25k2
10k
5
4
Mxy k5
0
(35)x30
(115)(6012x20y)0
zdzdydx 10k
Mxz k5
0 (35)x3
0 (115)(6012x20y)
0
ydzdydx 15k2
Myz k50
(35)x30
(115)(6012x20y)0
xdzdydx 25k
2
m k50
(35)x30
(115)(6012x20y)0
dzdydx 10k
x2 5
3
3
5
4
2
41
1
y x= (5 - )35
yfx,y 1
1560 12x 20y
Section 13.6 Triple Integrals and Applications 391
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48. (a)
(b)
Iz Iyz Ixz 7ka7
180
Iy Ixy Iyz a7k
30
Ix Ixy Ixz a7k
30
Iyz Ixz by symmetry
Ixz ka2
a2a2a2a2a2
y 2x2 y 2dzdydx kaa2a2a2a2
x2y 2 y 4dydx 7ka7
360
Ixy ka2
a2a2a2a2a2
z2x2 y 2dzdydx a3k
12a2a2a2a2
x2 y 2dydx a7k
72
Ix Iy Iz ka5
12
ka5
12
ka5
6
Ixz Iyz ka5
12by symmetry
Ixy ka2a2a2a2a2a2
z2dzdydx ka5
12
50. (a)
CONTINUED
Ix Ixz Ixy 2048k
9,Iy Iyz Ixy
8192k
21,Iz Iyz Ixz
63,488k
315
k40
20
1
2x216 8y 2 y4dydx
k
24
0
x216y 8y3
3
y5
5 2
0dx
k
24
0
256
15x2dx
8192k
45
Iyz k40
20
4y2
0
x2zdzdydx k40
20
1
2x24 y 22dydx
k40
20
1
216y 2 8y4 y6dydx
k
24
016y
3
3
8y5
5
y7
7 2
0dx
k
24
0
1024
105dx
2048k
105
Ixz k4
0 2
0 4y2
0
y 2zdzdydx k4
0 2
0
12y 24 y 22dydx
k
44
0
256y 256y3
3
96y5
5
16y7
7
y9
9 2
0dx k4
0
16,384
945dx
65,536k
315
k
44
0
20
256 256y 2 96y4 16y6 y8dydx
Ixy k40
20
4y2
0
z3dzdydx k40
20
1
44 y 24dydx
392 Chapter 13 Multiple Integration
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Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates
50. CONTINUED
(b)
Ix Ixz Ixy 48,128k
315, Iy Iyz Ixy
118,784k
315, Iz Ixz Iyz
11,264k
35
k40
20
4y2
0
4x2dzdydx k40
20
4y2
0
x2zdzdydx 4096k
9
8192k
45
4096k
15
Iyz k4
0
20
4y2
0
x24 zdzdydx
k40
20
4y2
0
4y 2dzdydx k40
20
4y2
0
y 2zdzdydx 1024k
15
2048k
105
1024k
21
Ixz 4
0 2
0 4y2
0
y 24 zdzdydx
k40
20
4y2
0
4z2dzdydx k40
20
4y2
0
z3dzdydx 32,768k
105
65,536k
315
32,768k
315
Ixy 40
20
4y2
0
z24 zdzdydx
52.
Iz Ixz Iyz 1
12ma2 c2
Iy Ixy Iyz 1
12mb2 c2
Ix Ixy Ixz 1
12ma2 b2
Iyz c2
c2a2a2b2b2
x2dzdydx abc2c2
x2dx abc3
12
1
12c2abc
1
12mc2
Ixz c2
c2a2a2b2b2
y 2dzdydx bc2c2a2a2
y 2dydx ba3
12c2c2
dx ba3c
12
1
12a2abc
1
12ma2
Ixy c2
c2a2a2b2b2
z2dzdydx b3
12c2c2a2a2
dydx 1
12b2abc
1
12mb2
54. 11
1x2
1x24x
2y2
0
kx2x2 y 2dzdydx 56. 6
58. Because the density increases as you move away from the axis of symmetry, the moment of intertia will increase.
2.
1
24
0
20
4r 4r2 r3drd1
24
0
2r2 4r3
3
r4
4 2
0
d2
34
0
d
6
4
0
2
0
2r
0
rzdzdrd 4
0
2
0rz
2
2 2r
0
drd
4. 20
0
20
e3
2ddd 20
0
13 e3
2
0
dd 20
0
1
31 e8
dd
2
61 e8
Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates 393
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6.
52
36
4
0
sin cos d 52
36sin2
2 4
0
52
144
1
34
0
sin cos sin sin3
3 4
0
d
1
34
0
40
sin cos cos 1 sin2dd
40
40
cos 0
2 sin cos ddd1
34
0
40
cos3sin cos dd
8. 20
0
sin 0
2 cos 2ddd8
9
10.
2
0
94
d 92
20
3r2
2
r4
4 3
0
d
y
x
2233
4
z20
30
3r2
0
rdzdrd 20
30
r3 r2drd
12.
468
3
117
32
0cos
0
d
yx 7
7
7 r= 5
r= 2
z20
0
52
2 sin ddd117
32
0
0
sin dd
14. (a)
(b) 20
60
40
3 sin2ddd 20
262 csc
4
3 sin2 ddd82
3 23
20
20
16r2
0
r2dzdrd82
3 23
16. (a) 20
10
1r2
0
rr2 z2dzdrd
8(b) 2
0
20
10
3 sin ddd
8
18.
(Volume of lower hemisphere) (Volume in the first octant)
128
3
642
3
64
32 2
128
3 4823
82
3
128
3 4823
2
013 16 r232
4
22d
V128
3 4
2
0
220
r2drd 20
422
r16 r2drd
4
yx7
7
7
zV2
343 4
2
0
220
r0
rdzdrd 20
42216r
2
0
rdzdrd
394 Chapter 13 Multiple Integration
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20.
8
32 2
2
0
1
3
4 r232 r3
3
2
0
d
20
20
r4 r2 r2drd
V 20
20
4r2
r
rdzdrd 22.
3k1 e4
20
6ke4 6kd
20
6ker2
2
0
20
20
12er
2
0
krdzdrd 20
20
12ker
2
rdrd
24.
z Mxy
m
kr02h2
12 3
r02hk
h
4
2kh2
r02
r04
12
2 kr0
2h2
12
2kh2
r02
20
r00
r02r 2r0r
2 r3drd
Mxy 4k2
0
r00
hr0rr00
zrdzdrd
m 1
3r0
2hk from Exercise 23
x y 0 by symmetry 26.
z Mxy
m
kr02h330
kr02h212
2h
5
1
30kr0
2h3
Mxy 4k2
0 r0
0 h(r
0r)r
0
0
z2rdzdrd
1
12kr0
2h2
m 4k20
r00
h(r0r)r00
zrdzdrd
x y 0 by symmetry
kz
28.
1
15r0
5kh
4kh1
30r0
5
2
4kh20
1
30r0
5d
4kh20
r05
5
r05
6 d
4kh20
r5
5
r6
6r0r0
0
d
4kh20
r00
r0 r
r0r4drd
4k
2
0
r0
0
hr0rr
0
0
r4dzdrd
Iz Q
x2 y2x,y,zdV 30.
3
2ma2
3
2ka4h
Iz 2k2
0
2a sin 0
h0
r3dzdrd
m ka2h
Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates 395
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34.
ka4
4
1
4k2a4
ka4121
4sin 2
2
0
ka4
2
0 sin2
d
2ka420
20
sin2dd
m 8k20
20
a0
3 sin2ddd 36.
z Mxy
m
kR4 r44
2kR3 r33
3R4 r4
8R3 r3
1
4kR4 r4 18 kR4 r4 cos 2
2
0
1
4kR4 r42
0
sin 2d
1
2kR4 r42
02
0
sin 2dd
Mxy 4k2
0
20
Rr
3 cos sin ddd
m k23R3 2
3r3 23 kR3 r3
x y 0 by symmetry
38.
4k
15R5 r5
2k5 R5 r5cos cos3
3 2
0
2k
5R5 r52
0
sin 1 cos2d
4k
5R5 r52
0
20
sin3dd
Iz 4k
2
0
2
0
R
r
4 sin3ddd 40.
z cos cos z
x2 y2 z2
y sin sin tan y
x
x sin cos 2 x2 y2 z2
42.
2
1
2
1
2
1
fsin cos , sin sin , cos 2
sin ddd
44. (a) You are integrating over a cylindrical wedge.
46. The volume of this spherical block can be determined as follows. One side is length
Another side is Finally, the third side is given by the length of an arc of angle in a
circle of radius Thus:
2 sin
V sin
sin .
.
x
y
i i isin
i i i
z.
32. (includes upper and lower cones)
1 22 43 b3 a3 23 2 2 b3 a3 43 b3 a3cos
4
0
4
3b3 a34
0
sin dy
x
aa
b
b
b
z
8
3b3 a34
0
20
sin dd
V 840
20
ba
2 sin ddd
(b) You are integrating over a spherical block.
396 Chapter 13 Multiple Integration
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Section 13.8 Change of Variables: Jacobians
2.
x
uy
v
y
ux
v
ad cb
y cu dv
x au bv 4.
x
uy
vy
ux
v v 2u vu 2u
y uv
x uv 2u
6.
x
uy
vy
ux
v 11 00 1
y v a
x u a 8.
x
uy
vy
ux
v 1v1 1
u
v2 1
v
u
v2
u v
v2
y u v
x u
v
3,66, 3
0,62, 2
3, 04, 1
0, 00, 0
u, vx,y10.
v x 4y
u x y
y
1
3 u v
x 1
34u v
(0, 0) (3, 0)
(3, 6)(0, 6)
11
1
2
3
4
5
6
1 2 4 5 6
v
u
12.
1523 26
3 120
152 23u3 26
3u
1
1
11
15
2 2u2 26
3 du
11152
v3
3 u2v
3
1
du
1
1
3
1
15
2v2 u2dvdu
1
1
3
1
6012 u v1
2u v12dvdu
R60xydA
x
uy
vy
ux
v
1
2 1
2 1
21
2 1
2
y 1
2u v, v x y
x 1
2u v, u x y
(1, 3)(1, 3)
(1, 1) (1, 1)
2 1
1
2
1 2
v
u
1, 11, 0
1, 31, 2
1, 32, 1
1, 10, 1
u, vx,y
Section 13.8 Change of Variables: Jacobians 397
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18.
0
12 u3
3 1 cos 2v
2 2
dv 73
12 v 1
2sin 2v
0
74
12
R
x y2 sin2x ydA
0
2
u2 sin2v12dudv
x,y
u, v
1
2
x 1
2u v,
u x y 2,
2
x
x y = 0
x y =x y+ =
x y+ = 2
2
2
3
2
3
yu x y ,
y 1
2u v
v x y
v x y 0
20.
8
0
16v 32dv 2516v528
0
4096
52
R3x 2y2y x32dA
8
0 16
0
uv3218dudv
x
uy
vy
ux
v
1
43
8 1
81
4 1
8
x 1
4u v,
u 3x 2y 16,
x2
3
3
5
2
41
1
12
2 = 0y x
2 = 8y x
3 + 2 = 16x y
3 + 2 = 0x y
y
(0, 0)
(2, 3)
(2, 5)
(4, 2)
u 3x 2y 0,
y 1
8u 3v
v 2y x 8
v 2y x 0
16.
Ry sinxydA 4
1 4
1
vsin u1vdvdu
4
1
3 sin udu 3 cos u4
1 3cos 1 cos 4 3.5818
x
uy
vy
ux
v
1
v
y v
x u
v
14.
2
0
2u1 eu2du 2u2
2 ueu2 eu2
2
0 21 e2
R 4x yexydA
2
0
0
u2
4uev12dvdu
x
uy
vy
ux
v
1
2
y 1
2u v
x 1
2u v
u1 2
1
2
v u= 2
v
398 Chapter 13 Multiple Integration
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22.
4
112 ln1 v2
4
11
udu 12 ln 17 ln 2 ln u
4
1
1
2 ln17
2 ln 4
R xy
1 x2y2dA
4
1
4
1
v
1 v2 1
udvdu
x
uy
vy
ux
v
1
u
x u,
u x 4,
u x 1,
y v
u
v xy 4
v xy 1
x
2
3
4
1 2 3 4
1
x = 4
x = 1
xy = 4
xy = 1
y
24. (a)
Let and
Let
(b)
Let and
Let
2Aab2 0 0 4
2 4 2Aab
Aab2
0
1
0
cos2 rrdrdAab2r
sinr2
4
2cosr2
1
02
u rcos ,v rsin .
Rfx,ydA
1
1
1u2
1u2A cos2u2 v2abdvdu
y bv.x au
R:x2
a2
y2
b2 1
fx,y A cos 2x2
a2
y 2
b2
12398 7
16sin 2
2
0 12394 117
122
0
8 41 cos 22 9
4 1 cos 2
2 d 122
0
398 7
8cos 2d
122
0
8r2 4r4 cos2 94 r4 sin21
0d 12
2
0
8 4 cos2 94 sin2d
2
0
1
0
16 16r2 cos2 9r2 sin2 12rdrd
u rcos , v rsin .R16 x2 y 2dA
1
11u2
1u216 16u2 9v2 12dvdu
y 3v.x 4u
V
R
fx,ydA
R:x2
16
y 2
9 1
fx,y 16 x2 y 2
Section 13.8 Change of Variables: Jacobians 399
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Review Exercises for Chapter 13
2. 2yy
x 2 y 2dx x3
3xy 2
2y
y
10y3
3
4.
2
0
4x 2 2x3 2x4dx
4
3
x3 1
2
x4 2
5
x5
2
0
88
15
2
0
2x
x2
x 2 2ydydx
2
0
x 2y y 2
2x
x2
dx
6. y4 y 2 4 arcsiny23
03
4
33
024y
2
24y2dxdy 23
0
4 y 2dy
8.
1
2
2
0
6 3ydy 126y 3
2y2
2
0 3
A 20
6y2y
dxdy
20
x0
dydx 32
62x0
dydx 20
6y2y
dxdy
10.
A 40
6xx2
x22x
dydx 40
8x 2x2dx 4x2 23x34
0
64
3
40
6xx2
x22x
dydx 01
11y11y
dydx 80
11y39y
dxdy 98
39y39y
dxdy
12. A 20
y21
0
dxdy 10
20
dydx 51
2x1
dydx 14
3
14. A
3
0
2yy2
y
dxdy
0
3
11x
x
dydx
1
0
11x
11x
dydx 9
2
16. Both integrations are over the common regionR shown in the figure. Analytically,
30
2x30
exydydx 53
5x0
exydydx 35 e5 e3 2
5 e5 e3 8
5e5
2
5
20
5y3y2
exydxdy 2
5
8
5e5
x
1
1
2
2
3
3
4
4
5
5
(3, 2)
y
26. See Theorem 13.5. 28.
x,y,z
u, v, w
4
0
1
1
4
0
0
1
1 17x 4u v, y 4v w, z u w
30.
1rcos2 rsin2 rx,y,z
r, ,z
cos
sin
0
rsin
rcos
0
0
0
1
x rcos , y rsin , z z
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22.
Since we have
P 0.50
0.250
8xydydx 0.03125
k 8.k8 1,
kx4
8 1
0
k
8
10
kx3
2dx
10
x0
kxydydx 10kxy
2
2 x
0dx 24. False, 1
0
10
xdydx 21
21
xdydx
26. True, 10
10
1
1 x2 y2dxdy < 1
0
10
1
1 x2dxdy
4
28. 20
r4
4 4
0d 2
0
64 d 3240
16y20
x2 y 2dxdy 20
40
r3drd
30.
4
3R2 b232
8
3R2 b2322
0
d
8
32
0R2 r232
R
bd
V 820
Rb
R2 r2rdrd 32.
The polar region is given by and
Hence,
x
1
1
2
3
4
4, )( 12/ 138/ 13
8/ 13
x y2 2+ =16
23
xy =
y
arctan320
40
rcos rsin rdrd288
13
0 0.9828.
0 r 4
tan 1213
813
3
2 0.9828
18.
12x33
0
27
2
3
2
3
0
x2dx
30xy 12y2
x
0dx
V 30
x0
x ydydx 20. Matches (c)
x
y
2
2
1
2
3
z
Rev iew Exercises for Cha pte r 13 401
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36.
y Ixm 16,384k315128k15 12821
x Iym 512k105128k15 47
m Rx,ydA 2
0
4x2
0
kydydx 128
15k
I0 Ix Iy 16,384k
315
512k
105
17,920
315k
512
9k
Iy Rx2x,ydA 2
0
4x2
0
kx2ydydx 512
105k
Ix Ry2x,ydA 2
0
4x2
0
ky3dydx 16,384
315k
38.
62 ln4 32 92
2 ln2
2
6
52
3 ln22 3
12418 2 ln4 18 1
121818 ln2 2212
122y2 4y2 2 ln2y 2 4y2 112 2 4y2322
0
S
2
0 2
y 2 4y2dxdy
2
0 22 4y2 y2 4y2dy
1 fx2 fy
2 2 4y2
fx 1,fy 2y
R x,y: 0 x 2, 0 y x
fx,y 16 x y2
34.
y Mxm 17kh
2
L80
127khL
51h140
x My
m
5khL2
24
12
7khL
5L
14
kh
2L
02x x
2
L
x3
L2dx kh
2 x2 x3
3L
x4
4L2L
0
kh
2
5L2
12
5khL2
24
My kL0
h22xLx2L2
0
xdydx
kh2
8
17L
10
17kh2L
80
kh2
8 4x 2x2
L
x3
L2
x 4
2L3
x5
5L4L
0
kh2
8
L
04 4xL
3x2
L2
2x3
L3
x4
L4dx
kh2
8L
02 xL
x2
L22
dx
Mx kL
0
h22xLx2L2
0
ydydx
x
h
L
y = 2 h2
xL
x2
L2( (
ym kL0
h22xLx2L2
0
dydx kh
2L
02 xL
x2
L2dx 7khL
12
402 Chapter 13 Multiple Integration
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44.
20
5 arctan 5cos 2
0d
25 arctan 5
20
20
arctan 5
0sin dd
50
25x2
0
25x2y2
0
1
1 x2 y2 z2dzdydx 2
0
20
50
2
1 2sin ddd
46. 20
4x2
0
4x2y2
0
xyzdzdydx 4
3
48.
84 2 sin 2 14sin3cos 3
41
2
1
4sin 2
2
0
29
2
220
32 sin2 4 sin4d 820
8 sin2 sin4d
220
2 sin 0
r16 r2drdV 220
2 sin 0
16r2
0
rdzdrd
50.
z Mxy
m
kc2a416
2kca33
3ca
32
y Mxz
m
kca48
2kca33
3a
16
x 0
Mxy 2k2
0
a0
crsin 0
rzdzdrd kc220
a0
r3 sin2drd1
4kc2a42
0
sin2d1
16kc2a4
Mxz 2k2
0
a
0
crsin
0
r2 sin dzdrd 2kc2
0
a
0
r3 sin2drd1
2kca4
2
0
sin2d1
8kca4
m 2k20
a0
crsin 0
rdzdrd 2kc20
a0
r2 sin drd2
3kca32
0
sin d2
3kca3
40. (a) Graph of
over regionR
x
y50
50
50
R
z
25 1 ex2y21000 cos2x2 y 2
1000
fx,y z
(b) Surface area
Using a symbolic computer program, you obtain surface
area sq. ft. 4,540
R
1 fxx,y2 fyx,y2dA
42. 1
22
0
20
r5drd16
32
0
d32
322
4x24x2
(x2y2)2
0
x2 y 2dzdydx 20
20
r22
0
r3dzdrd
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52.
z Mxy
m
81
4
1
162
1
8
20
30
12 r3 9
2rdrd
2
0
18 r4 9
4r2
3
0
d 818 2
0
81
4
Mxy 2
0 3
0 4
25r2zrdzdrd
2
0 5
3 25r2
25r2zrdzdrd
2
0 3
0 8 12 25 r2rdrd 0
x y 0 by symmetry
500
3 213 25 r232 2r2
3
0
500
3 2643 18
125
3 500
3
14
3 162
m 500
3 3
0
20
25r24
rdzddr500
3 3
0
20
r25 r2 4rddr
54.
4ka6
9
Iz k
0
20
a0
2 sin22 sin ddd
56.
8
15a
aa
1z2a2
1z2a21y
2z2a2
1y2z2a2
x2 y2dxdydz
Iz Qx2 y2dV
x2 y2 z2
a2 1 58.
Since represents a paraboloid with vertex
this integral represents the volume of the solid
below the paraboloid and above the semi-circle
in thexy-plane.y 4 x2
0, 0, 1,z 1 r2
0
20
1r2
0
rdzdrd
60.
2u2v 2u2v 8uv
x,yu, v
xu
yv yu
xv
62.
Boundary inxy-plane Boundary in uv-plane
4 arctan 5 4 arctan v5
51
4
1 v2dv 5
1
51
1
1 v2dudv
R x
1 x2y2dA 5
1
51
u
1 u2vu21ududv
v 5xy 5
v 1xy 1
u 5x 5
u 1x 1
x u,y v
u u x, v xy
y = 1x
y = 1x
x = 5
x = 1
1
1
2
3
4
5
4 5
y
x
x,y
u, vx
uy
vx
vy
u 11u 0
1
u
404 Chapter 13 Multiple Integration