Calculo 2 De dos variables_9na Edición - Ron Larson

34 PM Page 1044
1044 Chapter CAPÍTULO 14 14Multiple Integración Integration múltiple
1044 Chapter 14 Multiple Integration
1044 Chapter 14 Multiple Integration
Cilindro recto: I z 3 3
2 ma2 3
CAS
1044 32. Right circular Chapter cylinder: 14 Multiple I z 2ma Integration
2
CAPSTONE Para discusión
ultiple CAS Integration 32. Right circular cylinder: I z 2ma 2
3
CAPSTONE
r 2a CAS
sen sin , 32. ,
Right 0 ≤ zzcircular ≤ hh
cylinder: I z 2ma 2
48. Convert the CAPSTONE integral from rectangular coordinates to both
r 2a sin , 0 z h
48. 48. Convert (a) Convertir cylindrical the la integral integral and (b) from spherical desde rectangular coordenadas coordinates. coordinates rectangulares Without to both calcu-lating,
which integral
Utilizar Use a computer un sistema r algebra 2a algebraico sin system , 3 0por to evaluate zcomputadora h the triple y calcular integral. la (a) a) cylindrical coordenadas 48. Convert
and cilíndricas the
(b) spherical y integral b) coordinates. esféricas. from Sin rectangular
Without calcular, coordinates
calculating,
integral which parece (a) ser cylindrical
¿qué to both
CAS 32. Right circular cylinder: I z 2ma 2
CAPSTONE
3
I z 2ma 2 Use integral a computer triple. algebra system
CAPSTONE
to evaluate the triple integral.
appears
más appears sencilla and
to be
to be de (b)
the
the evaluar? spherical
simplest
simplest ¿Por coordinates.
to evaluate?
to evaluate? qué? Without calculating,
from which rectangular integral appears coordinates to be the to simplest both to evaluate?
Volume r 2a In sin Exercises , 0
Use a
z
computer
33–36, h
algebra system to evaluate the triple integral.
use spherical coordinates to find 48. Convert Why? the integral
h
48. Convert the integral from rectangular coordinates Why?
Volume Volumen the volume In En Exercises of los the ejercicios solid. 33–36, 33 use a 36, spherical utilizar coordinates coordenadas to esféricas
volume para calcular of the solid.
lating,
find
(a) acylindrical a 2 to x 2 both and Why? a 2 (b) x 2 spherical y 2 coordinates. Without calcu-
Use a computer Volume algebra In system Exercises to
(a)
evaluate 33–36, cylindrical use the triple spherical and (b)
integral.
spherical coordinates coordinates. to find
the
a
Without a
el volumen del sólido.
2 which x 2 calculating,
outside which z integral x 2 appears y 2 a
integral a 2 x 2 y 2 x 2 y 2 z 2 dz dy dx
system to evaluate
33. Solid
the triple
inside
integral.
x 2 y 2 z 2 appears to be the simplest to evaluate?
the volume of the solid.
0 0 0 a
9,
, andto be the simplest to evaluate?
2 x 2
ay 2 x 2 z 2 y
Why?
2
dz dy dx
Volume
Sólido
In
interior
Exercises
33. Solid inside x 2 x 2 y 2 33–36,
y 2 z 2 z 2
use
9,
spherical
exterior
coordinates
outside Why? z x 2 y 2 to
y
find
0 0 0
x 2 y 2 z 2 dz dy dx
arriba
36, use spherical coordinates
above the xy-
9,
, and
33.
del plano xy.
to
plane Solid find inside x 2 y 2 z 2 outside z x 2 y 2 a
0 0 0
the volume of the solid.
9,
, and a 2 x 2 a 2 x 2 y 2
above the xy-plane
a
34. Solid bounded above by x 2 y 2 az 2 x 2
z and a 2 x 2 below y 2
by 49. Hallar el “volumen” de la “esfera
x
above the xy-plane
2 y
en 2 cuatro
z 2 dz dy
dimensiones”
dx
34. Sólido limitado arriba por x 2 y 2 2 z y abajo xpor
2 y 2 49. z 2 dz Find dy dx the “volume” of the “four-dimensional sphere”
34. 33. Solid z bounded inside x 2 yx 2 2 above y 2 by z 2 x 2 youtside 2 z 2 zz
and x 2 below y 2 0 0 0
9,
, by and
z 2 9, outside z x 2 34.
y 2 0 0 0
,.
Solid
and
bounded above by x 2 y 2 z 2 z and below 49. Find the “volume” of the “four-dimensional sphere”
z x 2 y 2 x 2 by
above the xy-plane
y 2 49. z 2 Find w 2 the a“volume” 2 of the “four-dimensional sphere”
CAS 35. The torus given zby x
35. El toro dado por 2 4 sin y 2 (Use a computer algebra system x
. (Utilizar un sistema algebraico
2 y 2 z 2 w 2 a 2
CAS CAS 35. 34. The Solid to torus evaluate bounded given the by triple above integral.) 4 sin by sen x 2 (Use y 2 a computer z 2 z algebra and below systemby
by evaluando evaluating x 2 y 2 z 2 w 2 a 2
by x 2 y 2 z 2 por zcomputadora CAS 35. The
and below y byevaluar torus given la integral by triple.) 4 sin (Use a computer algebra 49. system by Find evaluating the “volume” of the “four-dimensional sphere”
to evaluate the triple integral.) 49. Find the “volume” of the “four-dimensional sphere”
a
36. El sólido comprendido entre las esferas y 16
x
a x y
a x y z
36.
z
The solid
x 2 y
between 2 a a
the spheres and
2 x 2 a 2 x 2 y 2 a 2 x 2 y 2 z
x 2 y 2 z 2 a 2
by evaluating
2
to evaluate the triple integral.)
x 36. solid between the spheres x y z a 2
2 a y 2 a x (Use a computer x 2 yalgebra 2 z 2 system b 2 , b > a, e interior 2 y
al 2 z
cono
2 w
z 2 2 a
x 2 2 and
2 xz 2 2 aw 2 2 x a 2 y 2 2 a 2 x 2 y 2 z 2 dw dz dy dx.
CAS 35. The x 2 torus y 2 given z 2 by b 2 , b > 4 sin a, and (Use inside a x 2 computer the ycone
2 algebra z 2 a 2 2
a a
4 sin
and inside the cone y 2
16 0 0 0
0 dw dz dy dx.
x 2 y 2 36.
z 2 The
b 2 solid between the spheres
z 0 0 0
0
by evaluating
2 x 2 y 2
and
2 x 2 a 2 x 2 y 2 a 2 x 2 y 2 z
x
, b > a,
2 system
y 2 z 2 a 2 by evaluating
2
to evaluate the triple 16
dw dz dy dx.
x
egral.) Mass In Exercises 2 integral.)
y
37 and 2 z
38, 2 b
use spherical 2 , b > a, and inside the cone z
coordinates to find
2 x50. 2 Use Utilizar
y 2 aspherical las a coordenadas coordinates 0 0
esféricas to show 0
para that 0
36. The solid between the spheres and
2 x 2 a 2 x 2 y 2 a 2 x 2 y 2 z
x
mostrar que
Mass Masa En los ejercicios 37 38, utilizar a coordenadas a esféricas
the spheres
and
2 x 2 a 2 2 y 2 a 2 x 2 y 2 z
x 2
para hallar la masa de la esfera x16
de densidad
especificada.
2 In
y 2 Exercises
z 2 a 2 37 and 38, use spherical 2 y
coordinates 2 z 2 a
to 2
2
find 50. Use spherical coordinates to show that
the mass of the sphere x 2 1 y 2 1 z 2 a 2 with the given density. 16
dw dz dy dx.
x 2 y 2 z 2 a 2
dw dz dy dx.
x x2 y 2 z
the mass 2 y
of the 2 Mass z 2
a, and inside the cone z 2 sphere
2 b 2 In , b
x 2 x
y 2
2 Exercises > a, and inside
1 y 2 1 z 2 37 and the
a 2 38, cone use zspherical with the given 2 x
density.
2 coordinates y 2 to find 50. Use spherical coordinates
x 2 y 2 z 2 e x2 y 2 z
to show that
0 0 0
0
2 dx dy dz 22.
.
0 0 0
0
x 2 y 2 z 2 e x2 y 2 z
37. The density the at mass any point of the is sphere proportional x 2 1 to y 2 the 1 distance z 2 a 2 with between the given density.
2 dx dy dz .
37. The density at any point is proportional
38, use spherical 37. La coordinates densidad en to find 50. Use spherical to the distance coordinates between
x
the point and the origin.
2 y 2 z 2 e x2 y 2 z
Mass In Exercises 37 and 38, use spherical coordinates to find 50. Use spherical coordinates to show that
2 dx dy dz 2 .
the to show that
the
mass
point
of the 37.
and
sphere The
the origin. cualquier xdensity 2 1 y 2 punto at 1 any z 2 es point proporcional a 2 with is proportional the given a la density. distancia to the distance between
y 2 1 z 2 a 2 38. with entre
The the density
el given punto density. at
y the any
el origen. point and is proportional the origin. to the distance of the PUTNAM EXAM x CHALLENGE
x 2 y 2 z 2 e x2 y 2 z
38. point from the z-axis.
of the PUTNAM EXAM CHALLENGE
2
51. Preparación 2 y
dx dy Find dz the 2 volume . del 2 z
of the examen 2 e x2 y 2 z 2 dx dy dz 2 .
37. The density at any point is proportional to the distance between
region of Putnam points x, y, z such that
t is proportional 38. to point La densidad 38.
distance from the between en z-cualquier The density
axis. punto at any es proporcional point is proportional a la distancia to the deldistance of the PUTNAM EXAM CHALLENGE
the point and the origin.
51. Find x 2 the y 2 volume z 2 of 8the 2 region 36 x 2 of ypoints 2 . x, y, z such that
.
Center punto of al Mass eje z. point from the z-axis.
51. Encontrar
In Exercises 39 and 40, use spherical coordinates
x 2 y 2 51. el volumen
z 2 Find
8 2 the de volume la región
36 x 2 y 2 of the de region puntos of (x, points y, z) x, eny, z such that
38. The density at any point is proportional to the distance of the PUTNAM EXAM CHALLENGE .
This problem forma was tal que composed
Center of Mass In Exercises 39
t is proportional to the
to find
distance
the center
of the
of mass PUTNAM of and the 40, solid use EXAM of spherical uniform CHALLENGE
coordinates
Centro to de find masa the center En los of ejercicios mass of the 39 solid y 40, of utilizar uniform coordenadas density.
x
density.
2 by the y 2 Committee z 2 8on 2 the Putnam 36 x 2 Prize y 2 Competition. .
point from Center the z-axis.
of Mass In Exercises 39 and 40, use spherical This coordinates
51. © The problem Find Mathematical was composed volume Association by of the the Committee of region America. on of All the points rights Putnam reserved. x, Prize y, zCompetition.
such that
51. Find the volume of the region of points
©
Este
The Mathematical
problema fue
x, y, z such
Association This preparado problem por
that
of was America.
el Committee composed All rights by on the reserved. Committee Putnam Prize on Competition.
the Putnam Prize Competition.
esféricas 39. Hemispherical para hallar solid to find
el of centro radius the center
de r of mass of the solid of uniform density. © The x
masa del sólido de densidad
2 Mathematical y 2 © z 2 Association The Mathematical 8 2 of 36 America. xAssociation 2 Todos y 2 . of los America. derechos All reservados. rights reserved.
39.
Center x
ises 39 and 40, uniforme. Hemispherical
of Mass
solid
In Exercises
of radius
39
r
and 2 40, yuse 2 spherical z 2 8 2 coordinates
to find the 39. center Hemispherical of mass of solid the of solid radius of uniform r density.
36 x 2 y 2 .
40. use Solid spherical lying between coordimass
of the solid
two concentric hemispheres of radii r and
This problem was composed by the Committee on the Putnam Prize Competition.
This problem was composed by the Committee on the
©
Putnam
The Mathematical
Prize Competition.
Association of America. All rights reserved.
40. Solid
39. of Sólido
R, uniform where lying between
hemisférico
r density. < R two concentric hemispheres of radii r and
40. Solid de lying radio between r© The Mathematical two concentric Association hemispheres of America. of All radii rights reserved.
R, where r < R
Sr
and
39. Hemispherical solid of radius r
E C T I O N P R O J E C T
radius r 40. Moment Sólido of comprendido R, where
Inertia In entre r <
Exercises dos R hemisferios 41 and concéntricos 42, use spherical de radios
of r y Inertia R, donde In r < Exercises R 41 and 42, use spherical
S EPROYECTO C T I O N P R O J E C T
SDE E TRABAJO
40. Solid lying between two concentric hemispheres of radii r and
C T I O N P R O J E C T
Moment
o concentric hemispheres
coordinates
of
to
radii
find
r
the
and
moment of inertia about the z-axis of the Wrinkled and Bumpy Spheres
R, where r Moment < R of Inertia In Exercises 41 and 42, use
coordinates solid of uniform to find density. the moment of inertia about the z-axis of the Wrinkled
spherical
and Bumpy Spheres
coordinates to find the moment of inertia about the z-axis
solid Momento of uniform de inercia density. En los ejercicios S E C T41 Iy O42, N utilizar P R Ocoorde-
nadas 41. Solid
J E C T
In Esferas S E C T I
of parts the (a) deformadas
O
and Wrinkled N P R O J
(b), find the and E C T
volume Bumpy of the Spheres wrinkled sphere or
of Inertia
In parts (a) and (b), find the volume of the wrinkled sphere or
xercises 41 and 42, esféricas bounded solid
use spherical para by of In
hallar the uniform Exercises
hemisphere density. 41 and 42, use spherical
el momento de cos inercia , 4con respecto 2, bumpy sphere. These solids are used as models for tumors.
41.
coordinates
ment of inertia al about eje Solid and z del bounded
to find
the
the sólido cone by
the
z-axis the
moment
of the densidad hemisphere
of inertia
4
Wrinkled uniforme. cos
about
,
the
4
z-axis of the
2, bumpy
Wrinkled
En los sphere. incisos These
and In parts
a) y b), solids
Bumpy (a) and
hallar are el used
Spheres (b), find the volume of the wrinkled sphere or
volumen as models las for esferas tumors. deformadas.
solid 41. Solid bounded by the hemisphere and Bumpy Spheres(a) Wrinkled sphere
(b) Bumpy sphere
and
of
the
uniform cos , 4 2,
cone
density.
bumpy sphere. These solids are used as models for tumors.
4
(a) Estos Wrinkled sólidos sphere se usan como modelos (b) de Bumpy tumores. sphere
41.
42.
Sólido
Solid lying
limitado
between and o acotado the two cone concentric
por
In parts
el hemisferio
hemispheres 4 of radii
(a) and (b), find the
4
r and In parts (a)
volume
1
and
0.2
(b),
sin 8
find
sin
the volume of the
1
wrinkled
0.2 sin 8
sphere
sin 4
or
(a) Wrinkled sphere
(b) Bumpy sphere
42. 41. Solid lying between two concentric hemispheres of radii r and of a) the Esfera wrinkled sphere or
R,
where bounded
2,
r <
y el
Rby the hemisphere cos , 4 2, bumpy sphere.
cono
1 deformada 0.2 These sin 8 solids sin are used as models b) 1Esfera for 0.2 deformada tumors.
42. Solid lying
sin 8 sin 4
misphere cos R, where , 4 r < R 2, bumpy
between
sphere.
two concentric
These solids
hemispheres
are used as
of
models
radii r0and
for tumors.
2 , 0
0 2 , 0
and the cone 4
1 0.2 sin 8 sin
1 0.2 sin 8 sin 4
42. Sólido comprendido R, where entre r <
(a)
dos R
(a)
Wrinkled
hemisferios concéntricos de radios
r y R, donde r < R
0 ≤ 1 ≤ 0.2 2,
0Wrinkled 2 sphere , 0sen sen (b) 0Bumpy sphere 2 , 0sen sen
WRITING ABOUT CONCEPTS
z
z
42. Solid lying between two concentric hemispheres of radii r and(b) Bumpy sphere 0 2 , 0
0 2 , 0
o concentric hemispheres WRITING of ABOUT radii r andCONCEPTS
z
sin 0 ≤8 sin ≤
0 ≤1 ≤0.2 2, z
sin 0 8 ≤ sin ≤4
43. R, Give where the r < WRITING equations R for ABOUT conversion 1CONCEPTS
from 0.2 sin rectangular 8 sin to 1 0.2 sin 8 sin 4
z
z
0 2 , 0
0 2 , 0
43. Give cylindrical the equations coordinates for and conversion vice from rectangular to
43. Give the equations 0
versa.
for 2 conversion , 0 from rectangular 0 to 2 , 0
WRITING
44.
cylindrical
Give the
ABOUT coordinates
equations
CONCEPTS and
for
vice
conversion
versa.
z
z
cylindrical coordinates from and rectangular to
NCEPTS
zvice versa.
z
44. 43. Desarrollo Give spherical the coordinates equations de conceptos
for and conversion vice versa.
from rectangular to to
44. Give the equations for conversion from rectangular to
for conversion from spherical rectangular to and vice versa.
s and vice versa. 43.
45.
cylindrical
Dar
Give
las
the
ecuaciones
iterated
coordinates
spherical form
de conversión
of
and
coordinates the
vice
triple
versa.
de
integral and coordenadas vice versa. fx,
Q
rectangulares
y, z dV
y
45. 44. Give the iterated of the triple integral fx, y, z dV
y
in cylindrical the equations
a coordenadas 45.
form. for conversion from rectangular to
Give cilíndricas the iterated y form viceversa. of the Q triple integral fx, y, z dV
y
for conversion from in cylindrical rectangular form. to
Q
x
x
and vice versa.
44. 46.
spherical
Dar Give las the
coordinates
ecuaciones iterated in form
and
cylindrical de conversión of the
vice
form. triple
versa.
de integral coordenadas fx, rectangulares
in spherical the
y, z dV
Q
x
Generado con Maple x
46. 45. Give
a coordenadas iterated form.
form
esféricas
of of the triple
y viceversa.
integral
fx, y, y, z z dV dV
x
y
Generado x con Maple y
46. Give the iterated form of the QQtriple integral fx, y, z dV
Generado con Maple
of the triple integral
spherical fx, y, form. z dV
y
Generado con Maple
Q
Generado con Maple
45. 47.
in
Dar Describe
cylindrical
Q la forma the iterada surface
form.
in spherical de whose la integral form. equation triple is a coordinate f x, y, z equal dV en to FOR FURTHER INFORMATION For more information on theseGenerado con Maple
x
x
47. 46. Describe Give forma a constant the cilíndrica. the iterated for surface each form whose of of the the equation coordinates triple integral is a coordinate Q(a) the fx, cylindrical equal y, z to dV FOR FURTHER INFORMATION For more information on these
47. Describe the surface x whose equation is a coordinate x equal types toof spheres, see the article “Heat Therapy Tumors” by Leah
Q
FOR Generado FURTHER con Maple INFORMATION For more information on these
of the triple integral a in constant fx, y, for z each dV of the coordinates in (a) the cylindrical
46. Dar
coordinate spherical types of spheres, see the article “Heat Therapy for Tumors” by Leah
Q la forma
system form.
Generado con Maple
iterada a constant
and
de
(b)
la for integral
the
each
spherical
triple of the
coordinate
coordinates Generado f x, y,
system.
con z Maple dV in (a) en the cylindrical Edelstein-Keshet in The UMAP Journal.
coordinate system and (b) the spherical coordinate Q system.
Generado
types
con Maple
of spheres, see the article “Heat Therapy for Tumors” by Leah
47. Describe forma esférica. the surface Edelstein-Keshet in The UMAP Journal.
coordinate whose system equation and is (b) a coordinate the spherical equal coordinate to system. FOR FURTHER Edelstein-Keshet INFORMATION in The For UMAP more information Journal. on these
hose equation is
47.
a coordinate
Describir
a constant equal
la
for
superficie
each to of
cuya
the coordinates
ecuación FOR FURTHER es
in
una
(a)
coordenada INFORMATION the cylindrical
igual For more types information on these
PARA
of
MAYOR
spheres, see
INFORMACIÓNMPara article “Heat Therapy
más
for
información
Tumors” by Leah
sobre
the coordinates in (a)
a
coordinate
una
the
constante
cylindrical system
en cada
and
una
(b) types the
de las
spherical of coordenadas spheres, coordinate see en the a) article el
system.
sistema “Heat Therapy Edelstein-Keshet for Tumors” by Leah
estos tipos de esferas,
in The
ver
UMAP
el artículo
Journal.
“Heat Therapy for Tumors” de
(b) the spherical coordinate
de coordenadas
system.
cilíndricas Edelstein-Keshet y b) sistema in de The coordenadas UMAP Journal.
Leah Edelstein-Keshet en The UMAP Journal.
esféricas.
4 sin
4
cos ,
1 0.2 sin 8 sin
1 0.2 sin 8 sin 4