Calculo 2 De dos variables_9na Edición - Ron Larson

60.
SECCIÓN 14.6 Integrales triples y aplicaciones 1037
14.6 Triple Integrals and Applications 1037
14.6 Triple
14.6
Integrals
Triple Integrals
and Applications
and Applications 1037 1037
14.6 Triple Integrals and Applications 1037
z
60. I z
60. x 1 1
I
12 x 12m ma2
a 2 b 2
b 2 CAPSTONE
I y 1 12 mb2 c 2 z
12m a 2 1b 2
y mb2 c z
60.
a CAPSTONE
x 12m 1
2 2 c z
CAPSTONE
60. I c
a
68. Think About It Of the integrals (a)–(c), which one is
z ma2 3 2 1
I z 1 12 ma2 c 2
b
68. Para pensar De las integrales a c), ¿cuál es igual a
12 mb2 x 12m
c 2 a 2 b 2 c
2 b a
68. Think
CAPSTONE
1
About It Of the integrals (a)–(c), which one is
y 12 1 mb2 2 c
a
68. Think equal to fx, y, z dz dy dx? Explain.
2
1 0 1 Explicar.
12 ma2 c 2
3 2 About 1 It Of the integrals (a)–(c), which one is
I y 121
b
equal to f x, y, z dz dy dx? Explain.
z 12 m mb2 c 2
a2 2
c b
a
68. Think About 3 2 It1
Of the integrals (a)–(c), which one is
1
2
equal 1 0 to 1 fx, y, dz dy dx? Explain.
I 2
3 2
1
1
0 1
z 12 ma2 c 2
3 2 1
b
equal to fx, y, z dz dy dx? Explain.
2
1 0 1
b
3 2a)
1 fx, y, z dz dx dy
b a) fx, 1
3
0
2 1
y, z 1 dz dx dy
c
a b
1 0a)
3 2 1
1 f x, y, dz dx dy
c a
1 2 3
2 b
a) 1 0 1 fx, y, z dz dx dy
x 2 a
y
1
x 2 2
b) 2 3 fx, y, dx dy dz
c
b
1 0 1
c a y
1
0
2
1
3
x 2 2
b) fx, y, z dx dy dz
y
1 0 1
x 2c
2a
b) 1 2 3 f x, y, dx dy dz
y
21
03
11
x 2 2
b) fx, y, z dx dy dz
y
2 3c)
1 1 0 1 fx, y, z dy dx dz
c) fx, 0
2
1
3
y, z 1 dy dx dz
0 1c)
2 3 1
1 f x, y, dy dx dz
c) 0 1 1 fx, y, z dy dx dz
I x
1
I y
1
I z
1
Momentos de inerciaMEn los ejercicios 61 y 62, dar una integral
triple
Moments
que represente
of Inertia
el
In
momento
Exercises
de inercia
61 and
con
62,
respecto
set up a
al
triple
eje z Valor
Average
promedioMEn
Value In Exercises
los ejercicios
69–72,
69 a
find
72, hallar
the average
el valor
value
promedio
of
Moments
de
Moments integral
of Inertia
la región
that of
sólida
Inertia gives
In Exercises
Q
the
de
In moment
densidad
Exercises
61 and
of
.
inertia 61
62,
and
set
about
up
62,
a
the set
triple
zup -axis triple of the Average
Average
Value
the
de
function
In
la
Value
Exercises
función
over In
sobre
the Exercises
69–72,
given
el sólido
solid.
find
69–72,
dado.
The
the
find
El
average
average
valor
the value average
value
promedio
of
of
a value continuous
de una
of
integral that Moments
solid
gives of
region
the Inertia
Q
moment In
of density
of inertia Exercises
.
about 61 the and z-axis 62, set of the up a triple
integral that gives the moment of inertia about the -axis of the the function Average
61.
función
over Value
the given In
continua
f x,
f(x,
y,
solid. Exercises
z
y, z)
over
The
sobre
a
average 69–72, find
solid
una
region
value the
región
Q
of
sólida
is
a average continuous
function the function f x, y, zover over the a solid given region solid. The Q isaverage value of a contin-
value of
the function over the given solid. The average value
Q
of
es
continuous
function x, y, over solid region is
solid region integral Q of x, that density y, gives z: 1 . the ≤moment x ≤ 1, 1 of inertia ≤ y ≤ about 1, 0 ≤ the z ≤z-axis 1 x of the
solid region
of density .
solid 61. Qregion
x, Qy, of z : density 1 x.
1, 1 y 1, 0 z 1 x 1uous 1 function f x, y, z over a solid region Q is
61. Q 1
fx, y, z dV
61.
x, y, z :
x,
1
62. x 2 y, x
y: 1,
z 2
1
1,
y 1, 0 z
1, 1 x
V V f x, y, z dV
Q x, y, z: x 2 y 2 ≤ 1, 0 ≤ z ≤ 4 x 2 y 2 fx, y, z dV
61.
Q
x 2 Q
y 2 x, y,
z 2 z : 1 x 1, 1 y 1, 0 z 1 x V 1 fx, y, dV
62. Q x, y, z : x 2 y 2 1, 0 z 4 x 2 y 2
Q
2 2 2
fx, y, z dV
Q
donde where
Ves is el the volumen de of la the región solid region sólida Q.
62. Q x, y, z : x 2 x 2 y
y 2 2 1,
z 2
V
0 z 4 x 2 y 2
Q
62.
kxx, 2 y, : 2 2 1, 2 2 where V is the Q volume of the solid region Q.
62.
En
kx
los 2 Q x, y, z : x 2 y 2 1, 0 z 4 x 2 y 2
where
is the volume of the solid region Q.
ejercicios 63 y 64, utilizando la descripción de región só-
where 69. f(x, fx, Vy, y, is z) zthe zvolume 2 z 2 4 sobre 4 over of the el the cubo solid cube en region in el the primer Q. first octante bounded acotado por by
kx
los planos coordenados, y los planos
lida, In Exercises kx 2
dar la integral 2
69. fx, y,
63 and para 64, using 1 y z 1.
a) la masa, the description b) el centro of the de masa solid region,
69.
z
fx, the
z
coordinate 2 y, 4 over planes and the planes x 1, y 1, and z 1
y c) el
2 the cube
over the
in the
cube
first
in
octant
the first
bounded
octant bounded
by
by
In Exercises
momento set up
63
the
and
de integral
64, using
inercia con for
the
(a)
description
respecto the mass, al eje (b)
of the
z. the
solid
center
region, the coordinate 69. fx, y, planes z zand the planes x 1, y 1, and z 1
In Exercises 63 and 64, using the description of the of solid mass, region, and 70. 70. f
the fx, x,
coordinate 2 4 over the cube in the first octant bounded by
y, z z xyz
planes
sobre over the and
el cubo cube the planes in en the el primer first octant 1,
octante bounded 1, and
acotado by por the
set up the In
(c)
integral Exercises
the moment
for 63 (a) and
of
the 64,
inertia
mass, using
about
(b) the the description
the
center
z-axis.
of mass, of the and solid region, 70. fx, y, z
the
los coordinate
xyz
coordinate
over
planos coordenados planes
the
planes
cube
and
in
and
the
the
the
y los planes
first
planes
octant
x
planos x x 4,
bounded
1, y
4, y y 4,
by
1,
and
the
and z 1
set up the integral for (a) the mass, (b) the center of mass, and 70. fx, y, xyz over the cube in the first octant bounded y z 4. by 4 the
(c) the moment 63. set El up sólido the of inertia integral acotado about for por (a) the z the z-axis.
4 mass, x 2 (b) y 2 the y zcenter 0 con of la mass, función and coordinate 70. fx,
63. de The densidad solid bounded by z 4 x 2 y 2 and z 0 with density 71. 71. f fx, planes y,
x, y, y, z
z z and xyz
x the over
y planes the
z
xcube 4, in ythe 4, first and octant z bounded 4 by the
(c) the moment of inertia about the -axis.
coordinate planes and the
sobre over planes the el tetraedro tetrahedron 4,
4,
en in and
el the primer first (c) the moment of inertia about the z-axis.
octante
63. The solid cuyos vértices son
y
64. El function
bounded by with vertices 0, 0, 2, 0, , 0, 2, 0 and 0, 2
sólido en el primer kz
and with density
71. fx, y, z
coordinate
x y
planes
z over
and
the
the
tetrahedron
planes x
in
4,
the
y
first
4,
octant
and z 4
63. The solid bounded
z 4
by 0, 0, 0, 2, 0, 0, 0, 2, 0 0, 0, 2
octante x 2 y 2
acotado 2 z por and
0
los planos with coordenados
function The y xsolid 2 yin 2 the kz
2 y 2 and z 0 with density
density
71. fx, y, over the tetrahedron in the first octant
function 63. The solid kz bounded by z 4 x 2
with vertices 0, 0, 0 , 2, 0, 0, 2, and 0, 0, 2
64.
71. fx, y, z x y z over the tetrahedron in the first octant
z 2 first 25 octant con función bounded de by densidad the coordinate planes 72. 72. with
f(x, fx, y, y, vertices
z) z x 0, 0,
y
sobre over , 2,
el the 0, ,
sólido solid 0, 2,
acotado bounded and 0,
por la by 0, esfera the xsphere
2 y 2
64. The solid
function
and
in the
x 2 first
y 2 octant
kz
z 2 bounded
25 with
by
density
the coordinate
function
planes
72. fx, y, z
with
kxy
x 2 x
vertices
y 2 y over
0,
3. z 2 the
0, 0
3
solid
, 2, 0,
bounded
0 , 0, 2, 0
by
and
the
0,
sphere
0, 2
64. The solid in the first octant bounded by the coordinate planes
72. fx, y, over the solid bounded by the sphere
and x64. 2 The y 2 solid with density function
x 2 y 2 z 2 3
and
2 z 2 in 2 25the 2 first octant bounded by the kxycoordinate planes
72. fx,
with density function
2 y, 2 z 2 x y over the solid bounded by the sphere
25
kxy
and x 73. Hallar Find the la región solid region sólida Q
where donde the la integral triple integral triple
Desarrollo WRITING 2 y
ABOUT 2 z
de 2 25 with density function kxy CAS x 2 y 2 z 2 3
conceptos CONCEPTS
CAS 73. Find the solid region Q where the triple integral
WRITING ABOUT CONCEPTS
CAS 73. Find the solid region where the triple integral
WRITING 65. Define a ABOUT triple integral CONCEPTS and describe a method of evaluating CAS 73. Find the solid
65. Define 65. a Definir triple una integral triple y describir un método para evaluar
una integral triple.
2 y 2 3z 2 dV
11 2x 2x 2 region 2 y 2 Q where 2 3z 3z 2 the triple integral
WRITING 2 dV
a triple
integral
ABOUT
integral.
and describe
CONCEPTS
65. Define triple integral and describe
a method of
method
evaluating
of evaluating 1 2x
a triple 65. integral. Define a triple integral and describe a method of evaluating
Q
2x 2 2 3z 2 dV
66. Determine triple integral.
1 2x
whether the moment of inertia about the y-axis
Q es is un a maximum. máximo. 2 y
Utilizar 2 3z
Use a computer un sistema 2 dV
algebra algebraico system por to computadora approximate y
66. Determine 66. Determinar a triple si el momento de inercia con respecto al eje y del
of
whether
integral.
Q
66. Determine the cylinder whether
the in Exercise the moment
of inertia
59 will of
about
increase inertia
the
about
y-
or
axis
decrease the y-axis
for is a maximum. Q
aproximar the Use el a valor computer
value. máximo. What
algebra
is ¿Cuál the
system
exact es el to
maximum valor approximate máximo value? exacto?
of the 66. cylinder cilindro Determine del ejercicio 59 aumentará o disminuirá
the nonconstant Exercise whether
density
59 the will moment increase of
x, y, z
or inertia decrease
x 2 about
z 2 for con the la y- densidad
constante
y
74. 74. Hallar
axis
is maximum. Use computer algebra system to approximate
of the cylinder in Exercise 59 will increase or decrease and a for 4. the maximum is a maximum. value.
Find the la región
What Use
solid region sólida
is a the computer
Q
exact
where donde
maximum algebra system
the la integral
value? to approximate
the nonconstant of the cylinder density in x, Exercise y, z 59 xwill 2 increase z 2 and a 4.
the maximum value. What is the exact triple maximum integral triple value?
CAS
67. the Consider nonconstant two solids, density solid x, A y, and solid B,
2 or
of equal 2 decrease for CAS
and weight 4.
CASas74. Find the solid
the maximum
region Q
value.
where
What
the triple
is the
integral
exact maximum value?
67. Consider 67. Considerar the el sólido A y el sólido B de pesos iguales que se
shown
two
nonconstant
solids,
below.
solid
density
A and solid
x, y,
B,
z
of equal
x 2 weight
z 2 and
as
a CAS 74. Find the solid region where the triple integral
67. Consider two solids, solid and solid B, of equal weight as CAS 74. Find the
shown below. muestran en la figura.
11
solid
xx 22
region
yy 22
Q
zz 22
where
dV
the triple integral
67. Consider two solids, solid A and solid B, of equal weight as
dV
shown (a) Because below. the solids have the same weight, which has the 1 x 2 y 2 z 2 dV
Q
(a) Because a) shown Como the
below. los sólidos tienen el mismo peso, ¿cuál tiene la
greater
solids
density?
have the same weight, which has the
2 2 2 dV
(a) Because the solids have the same weight, which has the
1 x 2 y 2 z 2 dV
Q
greater (a) densidad density? Because mayor? the solids have the same weight, which has the es is Qun a maximum. máximo. Utilizar Use a computer un sistema algebraico system por to computadora approximate y
(b) greater Which density? solid has the greater moment of inertia? Explain. is a maximum. Q
aproximar the Use el a valor computer
value. máximo. What
algebra
is ¿Cuál the
system
exact es el to valor maximum
approximate máximo value? exacto?
(b) Which b) ¿Cuál solid
greater
has sólido the
density?
greater tiene moment el momento of inertia? de Explain. inercia mayor?
is maximum. Use computer algebra system to approximate
(b) the maximum value. What is the exact maximum value?
(c) Which
Explicar. The solids solid are has rolled the greater down moment an inclined of inertia? plane. They Explain.
is a maximum. Use a computer algebra system to approximate
are 75. 75. Encontrar the Solve maximum for a en in value. the la integral triple What integral. triple. is the exact maximum value?
(c) The solids
(b) Which
(c) The started
are
solids
rolled
solid
at the are
down
has the
same rolled
an
greater
time down
inclined
moment
and an at
plane.
of
inclined the same
They
inertia?
plane. height.
are
Explain.
They Which are75. Solve for
the
a in
maximum
the triple
value.
integral.
What is the exact maximum value?
11
3ay 3 a y 2 4 x y 2
will reach the bottom first? Explain.
14
dz dx dy
0
2 4xy
started c) (c) Los at The the sólidos same solids se time are hacen rolled and rodar at down the hacia same an inclined abajo height. en Which plane. un plano They inclinado.
Empiezan al mismo tiempo y a la misma altura.
2 4
are
Solve for in the triple 2 integral.
started at the same time and at the same height. Which
dz dx dy 14
1 375. a
Solve
y
for
x a in the triple integral.
1 3 a y 2
will reach started the bottom at the first? same Explain. time and at the same height. Which
2 4 x y 14
0 00
dz a
15
a dx dy
2
will reach the bottom first? Explain.
1 3 a y 2 4 x y 2 14 15
¿Cuál will llegará reach the abajo bottom primero? first? Explicar. Explain.
dz dx
0 0 a
15 dy 14
0 0 a dz dx dy 15
76. 76. Determinar Determine 0 0
el the avalor value de of b de b such manera that 15que the el volumen of del the elipsoide ellipsoid
76. Determine
76. Determine x 2 the value
y 2 bthe 2 of b
value
such
z 2 9of that such 1
the
es 16
volume
that . the
of
volume
the ellipsoid
of the ellipsoid
x 2 76. y 2 Determine b 2 z 2 9 1 es 16 .
Axis of
2 2 2 the value 2 of b such that the volume of the ellipsoid
es 16 .
x es 16 .
Axis of
Axis revolution of
PUTNAM 2 y 2 b
EXAM 2 z 2 9 1
CHALLENGE
revolution PUTNAM EXAM CHALLENGE
Axis of
revolution
Axis Eje de of
Preparación PUTNAM 77. EvaluateEXAM del CHALLENGE examen Putnam
Axis of
revolution
revolution
revolución 77. Evaluate
PUTNAM EXAM CHALLENGE
Axis of
77. Evaluate 1 1 1
revolution
Eje revolution
Axis
de
of
1
77. 1límEvaluar
Evaluate 1 . . . cos 2
n→ 0 0 0 2n x 1 x . . .
2 x n dx 1 dx . . . 2 dx n .
lím . . . 1 1
cos
revolución
2 1
revolution
n→ 0 0 0 2n x 1 x . . .
lím . . . cos 2 x n dx 1 dx . . . 1 1 1 2 2 dx n .
Solid A
Solid B
n→ 0 0 0 2n 1 . . .
2 n dx 1 dx . . . 2 dx n .
lím . . . cos 2
Solid A
Solid B
n→ This problem 0 0 was composed 0 2n x 1 x . . .
2 x n dx 1 dx . . . 2 dx n .
by the Committee on the Putnam Prize Competition.
Solid Solid This problem © was The composed Mathematical by the Association Committee of on America. the Putnam All rights Prize reserved. Competition.
Sólido Solid A A
Sólido Solid BB
This problem was composed by the Committee on the Putnam Prize Competition.
x 2 y 2 z 2
kx 2
kz
Para discusión
0 1 1
x, y, z x 2 z 2
kxy
© The Mathematical Este Association of America. All rights reserved.
This problema
The Mathematical was fue preparado composed por
Association by the el on the Prize of Committee America. All on rights the Putnam reserved. Prize Competition.
© The The Mathematical Association of of America. Todos All rights los derechos reserved. reservados.