Post on 28-Apr-2015
822 CAPÍTULO 11 Vectores y la geometría del espacio
11.7 Coordenadas cilíndricas y esféricas
n Usar coordenadas cilíndricas para representar superficies en el espacio.n Usar coordenadas esféricas para representar superficies en el espacio.
Coordenadas cilíndricas
Ya se ha visto que algunas gráficas bidimensionales son más fáciles de representar en coor-denadas polares que en coordenadas rectangulares. Algo semejante ocurre con las superfi-cies en el espacio. En esta sección se estudiarán dos sistemas alternativos de coordenadasespaciales. El primero, el sistema de coordenadas cilíndricas, es una extensión de lascoordenadas polares del plano al espacio tridimensional.
Para convertir coordenadas rectangulares en coordenadas cilíndricas (o viceversa), hay queusar las siguientes fórmulas, basadas en las coordenadas polares, como se ilustra en la figu-ra 11.66.
Cilíndricas a rectangulares:
Rectangulares a cilíndricas:
Al punto (0, 0, 0) se le llama el polo. Como la representación de un punto en el sistema decoordenadas polares no es única, la representación en el sistema de las coordenadas cilín-dricas tampoco es única.
EJEMPLO 1 Conversión de coordenadas cilíndricas a coordenadas rectangulares
Convertir el punto a coordenadas rectangulares.
Solución Usando las ecuaciones de conversión de cilíndricas a rectangulares se obtiene
Por tanto, en coordenadas rectangulares, el punto es co-mo se muestra en la figura 11.67.
sx, y, zd 5 s22!3, 2, 3d,z 5 3.
y 5 4 sin 5p
65 4 11
22 5 2
x 5 4 cos 5p
65 4 12!3
2 2 5 22!3
sr, u, zd 5 14, 5p
6, 32
z 5 zy 5 r sin u,x 5 r cos u,
z 5 ztan u 5yx
,r2 5 x2 1 y2,
x
y
z
(x, y, z)(r, , z)θ
θθ
θ
θ
P
x
y
Coordenadasrectangulares:x = r cosy = r senz = z
tan =
r2 = x2 + y2
z = zxy
r
Coordenadas cilíndricas:
Figura 11.66
z
y
x
θ
θ
π(r, , z) = 4, , 356( )
r
z
P
1
−2
−3
−4
−11 2 3 4−1
1
2
3
4
(x, y, z) = (−2 3, 2, 3)
Figura 11.67
EL SISTEMA DE COORDENADAS CILÍNDRICAS
En un sistema de coordenadas cilíndricas, un punto P en el espacio se representapor medio de una terna ordenada
1. es una representación polar de la proyección de P en el plano xy.
2. es la distancia dirigida de a P.sr, udz
sr, ud
sr, u, zd.
sen
sen
Larson-11-07.qxd 3/12/09 17:44 Page 822
SECCIÓN 11.7 Coordenadas cilíndricas y esféricas 823
EJEMPLO 2 Conversión de coordenadas rectangularesa coordenadas cilíndricas
Convertir el punto a coordenadas cilíndricas.
Solución Usar las ecuaciones de conversión de rectangulares a cilíndricas.
Hay dos posibilidades para r y una cantidad infinita de posibilidades para Como semuestra en la figura 11.68, dos representaciones adecuadas del punto son
y en el cuadrante I.
y en el cuadrante III.
Las coordenadas cilíndricas son especialmente adecuadas para representar superficiescilíndricas y superficies de revolución en las que el eje z sea el eje de simetría, como semuestra en la figura 11.69.
Los planos verticales que contienen el eje z y los planos horizontales también tienen ecua-ciones simples de coordenadas cilíndricas, como se muestra en la figura 11.70.
ur < 0122, 4p
3, 22.
ur > 012, p
3, 22
u.
z 5 2
u 5 arctan s!3 d 1 np 5p
31 nptan u 5 !3
r 5 ±!1 1 3 5 ±2
sx, y, zd 5 s1, !3, 2d
( , , ) = (1, 3, 2)x y z
θ
θ π
π π( , , ) = 2, , 2r z
3
3 3( ) ( )
y
x
=
3
2
1 2 3
3
2
1
z = 2
r = 2
o −2, , 24
z
Figura 11.68
y
x
Planovertical:
= cθ
θ = c
z
r = 2 z
yx
z
x2 + y2 = 4zr = z
z
y
x
x2 + y2 = z2
r2 = z2 + 1
z
yx
x2 + y2 − z2 = 1
y
x
z
r = 3x2 + y2 = 9
CilindroFigura 11.69
Paraboloide Cono Hiperboloide
y
x
z Planohorizontal:z = c
Figura 11.70
Larson-11-07.qxd 3/12/09 17:44 Page 823
824 CAPÍTULO 11 Vectores y la geometría del espacio
EJEMPLO 3 Conversión de coordenadas rectangulares a coordenadas cilíndricas
Hallar una ecuación en coordenadas cilíndricas para la superficie representada por cadaecuación rectangular.
a)
b)
Solución
a) Según la sección anterior, se sabe que la gráfica de es un cono “de doshojas” con su eje a lo largo del eje z, como se muestra en la figura 11.71. Si se susti-tuye por la ecuación en coordenadas cilíndricas es
Ecuación rectangular.
Ecuación cilíndrica.
b) La gráfica de la superficie es un cilindro parabólico con rectas generatricesparalelas al eje z, como se muestra en la figura 11.72. Sustituyendo y2 por r2 sen2
q yx por r cos q, se obtiene la ecuación siguiente en coordenadas cilíndricas.
Ecuación rectangular.
Sustituir y por r sen q y x por r cos q.
Agrupar términos y factorizar.
Dividir cada lado entre r.
Despejar r.
Ecuación cilíndrica.
Hay que observar que esta ecuación comprende un punto en el que por lo cual nada se pierde al dividir cada lado entre el factor
La conversión de coordenadas rectangulares a coordenadas cilíndricas es más sencillaque la conversión de coordenadas cilíndricas a coordenadas rectangulares, como se mues-tra en el ejemplo 4.
EJEMPLO 4 Conversión de coordenadas cilíndricas a coordenadas rectangulares
Hallar una ecuación en coordenadas rectangulares de la superficie representada por laecuación cilíndrica
Solución
Ecuación cilíndrica.
Identidad trigonométrica.
Sustituya r cos q por x y r sen q por y.
Ecuación rectangular.
Es un hiperboloide de dos hojas cuyo eje se encuentra a lo largo del eje y, como se mues-tra en la figura 11.73.
y2 2 x2 2 z2 5 1
x2 2 y2 1 z2 5 21
r2 cos2 u 2 r2 sin2 u 1 z2 5 21
r2scos2 u 2 sin2 ud 1 z2 1 1 5 0
r2 cos 2u 1 z2 1 1 5 0
r2 cos 2u 1 z2 1 1 5 0.
r.r 5 0,
r 5 csc u cot u
r 5cos usin2 u
r sin2 u 2 cos u 5 0
rsr sin2 u 2 cos ud 5 0
r2 sin2 u 5 r cos u
y2 5 x
y2 5 x
r2 5 4z2.
x2 1 y2 5 4z2
r2,x2 1 y2
x2 1 y2 5 4z2
y2 5 x
x2 1 y2 5 4z2
y
z
4 6
3
46
x2 + y2 = 4z2
Rectangular:
r2 = 4z2
Cilíndrica:
x
Figura 11.71
Cilíndrica:r = csc cotθ θ
Rectangular:y2 = x
y
x
z
2
2
4
1
z
23
23
3
−3
−2
−1
Rectangular:y2 − x2 − z2 = 1
Cilíndrica:r2 cos 2 + z2 + 1 = 0θ
yx
Figura 11.73
Figura 11.72
sen2
sen2
sen2
sen2
sen2
sen2
Larson-11-07.qxd 3/12/09 17:44 Page 824
SECCIÓN 11.7 Coordenadas cilíndricas y esféricas 825
Coordenadas esféricas
En el sistema de coordenadas esféricas, cada punto se representa por una terna ordena-da: la primera coordenada es una distancia, la segunda y la tercera coordenadas son ángu-los. Este sistema es similar al sistema de latitud-longitud que se usa para identificar pun-tos en la superficie de la Tierra. Por ejemplo, en la figura 11.74 se muestra el punto en lasuperficie de la Tierra cuya latitud es 40° Norte (respecto al ecuador) y cuya longitud es80° Oeste (respecto al meridiano cero). Si se supone que la Tierra es esférica y tiene unradio de 4 000 millas, este punto sería
(4 000, 280°, 50°).
Radio 80° en el sentido 50° hacia abajo de las manecillas del Polo Nortedel reloj, desde el
meridiano cero
La relación entre coordenadas rectangulares y esféricas se ilustra en la figura 11.75.Para convertir de un sistema al otro, usar lo siguiente.
Esféricas a rectangulares:
Rectangulares a esféricas:
Para cambiar entre los sistemas de coordenadas cilíndricas y esféricas, usar lo siguiente.
Esféricas a cilíndricas :
Cilíndricas a esféricas :xr ≥ 0c
sr ≥ 0d
x
y
80° O40° N
Ecuador
Meridianocero
z
Figura 11.74
x
y
( , , )(x, y, z)
θ φρ
θ
φ
ρ
P
x
y
r
O
φρr x2 + y2= sen =
z
z
z 5 r cos fy 5 r sin f sin u,x 5 r sin f cos u,
z 5 r cos fu 5 u,r2 5 r2 sin2 f,
f 5 arccos1 z!x2 1 y2 1 z22tan u 5
yx
,r2 5 x2 1 y2 1 z2,
f 5 arccos1 z!r2 1 z22u 5 u,r 5 !r2 1 z2,
Coordenadas esféricasFigura 11.75
EL SISTEMA DE COORDENADAS ESFÉRICAS
En un sistema de coordenadas esféricas, un punto P en el espacio se representapor medio de una terna ordenada
1. es la distancia entre P y el origen,
2. es el mismo ángulo utilizado en coordenadas cilíndricas para
3. es el ángulo entre el eje z positivo y el segmento de recta
Hay que observar que la primera y tercera coordenadas, r y f, son no negativas.r es la letra minúscula ro, y f es la letra griega minúscula fi.
0 ≤ f ≤ p.OP\
,f
r ≥ 0.u
r ≥ 0.r
sr, u, fd.
sen sen sen
sen2
Larson-11-07.qxd 3/12/09 17:44 Page 825
826 CAPÍTULO 11 Vectores y la geometría del espacio
El sistema de coordenadas esféricas es útil principalmente para superficies en el espa-cio que tiene un punto o centro de simetría. Por ejemplo, la figura 11.76 muestra tressuperficies con ecuaciones esféricas sencillas.
EJEMPLO 5 Conversión de coordenadas rectangulares a coordenadas esféricas
Hallar una ecuación en coordenadas esféricas para la superficie representada por cada unade las ecuaciones rectangulares.
a) Cono:
b) Esfera:
Solución
a) Haciendo las sustituciones apropiadas de x, y y z en la ecuación dada se obtiene lo si-guiente.
.
.
La ecuación representa el semicono superior, y la ecuación repre-senta el semicono inferior.
b) Como y la ecuación dada tiene la forma esférica si-guiente.
Descartando por el momento la posibilidad de que se obtiene la ecuación esfé-rica
o
Hay que observar que el conjunto solución de esta ecuación comprende un punto en elcual de manera que no se pierde nada al eliminar el factor La esfera repre-sentada por la ecuación se muestra en la figura 11.77.r 5 4 cos f
r.r 5 0,
r 5 4 cos f.r 2 4 cos f 5 0
r 5 0,
rsr 2 4 cos fd 5 0r2 2 4r cos f 5 0
z 5 r cos f,r2 5 x2 1 y2 1 z2
f 5 3py4f 5 py4
f 5 py4 or f 5 3py4tan2 f 5 1
r ≥ 0sin2 f
cos2 f5 1
r2 sin2 f 5 r2 cos2 f
r2 sin2 f scos2 u 1 sin2 ud 5 r2 cos2 f
r2 sin2 f cos2 u 1 r2 sin2 f sin2 u 5 r2 cos2 f
x2 1 y2 5 z2
x2 1 y2 1 z2 2 4z 5 0
x2 1 y2 5 z2
Esfera:= cρ
y
x
c
z
yx θ = c
Semiplano vertical:= cθ
z
Rectangular:x2 + y2 + z2 − 4z = 0 ρ φ
Esférica:= 4 cos
y
x
z
−2
2
4
11
2
Figura 11.77
y
x
Semicono:= cφ 0 < c < π
2( )
φ = c
z
Figura 11.76
sen2 sen2 sen2
sen2 sen2
sen2
sen2
o
≥
Larson-11-07.qxd 3/12/09 17:44 Page 826
SECCIÓN 11.7 Coordenadas cilíndricas y esféricas 827
En los ejercicios 1 a 6, convertir las coordenadas cilíndricas delpunto en coordenadas rectangulares.
En los ejercicios 7 a 12, convertir las coordenadas rectangularesdel punto en coordenadas cilíndricas.
En los ejercicios 13 a 20, hallar una ecuación en coordenadascilíndricas de la ecuación dada en coordenadas rectangulares.
En los ejercicios 21 a 28, hallar una ecuación en coordenadasrectangulares de la ecuación dada en coordenadas cilíndricas ydibujar su gráfica.
En los ejercicios 29 a 34, convertir las coordenadas rectangularesdel punto en coordenadas esféricas.
En los ejercicios 35 a 40, convertir las coordenadas esféricas delpunto en coordenadas rectangulares.
En los ejercicios 41 a 48, hallar una ecuación en coordenadasesféricas de la ecuación dada en coordenadas rectangulares.
En los ejercicios 49 a 56, encontrar una ecuación en coordenadasrectangulares de la ecuación dada en coordenadas esféricas ydibujar su gráfica.
En los ejercicios 57 a 64, convertir las coordenadas cilíndricasdel punto en coordenadas esféricas.
En los ejercicios 65 a 72, convertir las coordenadas esféricas delpunto en coordenadas cilíndricas.
En los ejercicios 73 a 88, usar un sistema algebraico por compu-tadora o una herramienta de graficación para convertir las coor-denadas del punto de un sistema a otro, entre los sistemas decoordenadas rectangulares, cilíndricas y esféricas.
11.7 Ejercicios
1. 2.
3. 4.
5. 6. 0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
7. 8.
9. 10.
11. 12.
I E i 13 20 fi d i i li d i l di
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
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11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827
11.7 Cylindrical and Spherical Coordinates 827
11.7 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
In Exercises 1–6, convert the point from cylindrical coordinatesto rectangular coordinates.
1. 2.
3. 4.
5. 6.
In Exercises 7–12, convert the point from rectangular coordinatesto cylindrical coordinates.
7. 8.
9. 10.
11. 12.
In Exercises 13–20, find an equation in cylindrical coordinatesfor the equation given in rectangular coordinates.
13. 14.
15. 16.
17. 18.
19. 20.
In Exercises 21–28, find an equation in rectangular coordinatesfor the equation given in cylindrical coordinates, and sketch itsgraph.
21. 22.
23. 24.
25. 26.
27. 28.
In Exercises 29–34, convert the point from rectangular coordinatesto spherical coordinates.
29. 30.
31. 32.
33. 34.
In Exercises 35– 40, convert the point from spherical coordinatesto rectangular coordinates.
35. 36.
37. 38.
39. 40.
In Exercises 41–48, find an equation in spherical coordinatesfor the equation given in rectangular coordinates.
41. 42.
43. 44.
45. 46.
47. 48.
In Exercises 49–56, find an equation in rectangular coordinatesfor the equation given in spherical coordinates, and sketch itsgraph.
49. 50.
51. 52.
53. 54.
55. 56.
In Exercises 57–64, convert the point from cylindrical coordinatesto spherical coordinates.
57. 58.
59. 60.
61. 62.
63. 64.
In Exercises 65–72, convert the point from spherical coordinatesto cylindrical coordinates.
65. 66.
67. 68.
69. 70.
71. 72.
In Exercises 73–88, use a computer algebra system or graphingutility to convert the point from one system to another amongthe rectangular, cylindrical, and spherical coordinate systems.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88. 8, 6,
3, 3 4, 3
8.25, 1.3, 4
3.5, 2.5, 6
2, 11 6, 3
5, 3 4, 5
0, 5, 4
5 2, 4 3, 3 2
3 2, 3 2, 3
3, 2, 2
7.5, 0.25, 1
20, 2 3, 4
10, 0.75, 6
5, 9, 8
6, 2, 3
4, 6, 3
Esféricas Cilíndricas Rectangulares
7, 4, 3 48, 7 6, 6
5, 5 6, 6, 6, 3
18, 3, 336, , 2
4, 18, 210, 6, 2
4, 2, 312, , 5
4, 3, 44, 6, 6
2, 2 3, 24, 2, 4
3, 4, 04, 4, 0
4 csc sec csc
2 sec 4 cos
26
34
5
x2 y2 z2 9z 0x2 y2 2z2
x 13x2 y2 16
x2 y2 3z2 0x2 y2 z2 49
z 6y 2
6, , 25, 4, 3 4
9, 4, 12, 4, 0
12, 3 4, 94, 6, 4
1, 2, 13, 1, 2 3
2, 2, 4 22, 2 3, 4
4, 0, 04, 0, 0
r 2 cos r 2 sen
z r2 cos2 r2 z2 5
r 12z6
z 2r 3
x2 y2 z2 3z 0y2 10 z2
x2 y2 8xy x2
z x2 y2 11x2 y2 z2 17
x 9z 4
2 3, 2, 61, 3, 4
3, 3, 72, 2, 4
2 2, 2 2, 40, 5, 1
0.5, 4 3, 84, 7 6, 3
6, 4, 23, 4, 1
2, , 47, 0, 5
1053714_1107.qxp 10/27/08 10:41 AM Page 827Larson-11-07.qxd 3/12/09 17:44 Page 827
828 CAPÍTULO 11 Vectores y la geometría del espacio
En los ejercicios 89 a 94, asociar la ecuación (dada en términosde coordenadas cilíndricas o esféricas) con su gráfica. [Los grá-ficos se marcan a), b), c), d), e) y f).]
a) b)
c) d)
e) f)
89. 90.
91. 92.
93. 94.
En los ejercicios 99 a 106, convertir la ecuación rectangular auna ecuación a) en coordenadas cilíndricas y b) en coordenadasesféricas.
En los ejercicios 107 a 110, dibujar el sólido que tiene la descrip-ción dada en coordenadas cilíndricas.
En los ejercicios 111 a 114, dibujar el sólido que tiene la descrip-ción dada en coordenadas esféricas.
Para pensar En los ejercicios 115 a 120, hallar las desigualdadesque describen al sólido, y especificar el sistema de coordenadasutilizado. Posicionar al sólido en el sistema de coordenadas en elque las desigualdades sean tan sencillas como sea posible.
115. Un cubo con cada arista de 10 centímetros de largo.
116. Una capa cilíndrica de 8 metros de longitud, 0.75 metros dediámetro interior y un diámetro exterior de 1.25 metros.
117. Una capa esférica con radios interior y exterior de 4 pulgadasy 6 pulgadas, respectivamente.
118. El sólido que queda después de perforar un orificio de 1 pul-gada de diámetro a través del centro de una esfera de 6 pul-gadas de diámetro.
119. El sólido dentro tanto de como de
.
120. El sólido entre las esferas x2 1 y2 1 z2 = 4 y x2 1 y2 1 z2 = 9, ydentro del cono .
¿Verdadero o falso? En los ejercicios 121 a 124, determinar si ladeclaración es verdadera o falsa. Si es falsa, explicar por qué odar un ejemplo que pruebe que es falsa.
121. En coordenadas cilíndricas, la ecuación r = z es un cilindro.
122. Las ecuaciones y representan lamisma superficie.
123. Las coordenadas cilíndricas de un punto (x, y, z) son únicas.
124. Las coordenadas esféricas de un punto (x, y, z) son únicas.
125. Identificar la curva de intersección de las superficies (en coor-denadas cilíndricas) z 5 sen q y r 5 1.
126. Identificar la curva de intersección de las superficies (en coor-denadas esféricas) y r 5 4.r 5 2 sec f
x2 1 y2 1 z2 5 4r 5 2
z2 5 x2 1 y2
sx 232d2
1 y2 594
x2 1 y2 1 z2 5 9
r 5 4 sec fr2 5 z
f 5p
4r 5 5
u 5p
4r 5 5
yx
3
2
−212
z
yπ4x
2
1
2−2
2
z
yx
55
5
z
y
x55
5
z
y
x
4
−4
4
2
z
y
x
π4
1 2 3
3
−3 −2
3 2
z
Desarrollo de conceptos95. Dar las ecuaciones para la conversión de coordenadas rec-
tangulares a coordenadas cilíndricas y viceversa.
96. Explicar por qué en las coordenadas esféricas la gráfica deq = c es un semiplano y no un plano entero.
97. Dar las ecuaciones para la conversión de coordenadas rec-tangulares a coordenadas esféricas y viceversa.
Para discusión98. a) Dadas las constantes a, b y c, describir las gráficas de las ecua-
ciones y en coordenadas cilíndricas.
b) Dadas las constantes a, b y c, describir las gráficas de lasecuaciones y en coordenadas esfé-ricas.
f 5 cu 5 b,r 5 a,
z 5 cu 5 b,r 5 a,
In Exercises 89–94, match the equation (written in terms ofcylindrical or spherical coordinates) with its graph. [The graphsare labeled (a), (b), (c), (d), (e), and (f).]
(a) (b)
(c) (d)
(e) (f)
89. 90.
91. 92.
93. 94.
In Exercises 99–106, convert the rectangular equation to anequation in (a) cylindrical coordinates and (b) spherical coordinates.
99. 100.
101. 102.
103. 104.
105. 106.
In Exercises 107–110, sketch the solid that has the givendescription in cylindrical coordinates.
107.
108.
109.
110.
In Exercises 111–114, sketch the solid that has the givendescription in spherical coordinates.
111.
112.
113.
114.
Think About It In Exercises 115–120, find inequalities thatdescribe the solid, and state the coordinate system used.Position the solid on the coordinate system such that theinequalities are as simple as possible.
115. A cube with each edge 10 centimeters long
116. A cylindrical shell 8 meters long with an inside diameter of0.75 meter and an outside diameter of 1.25 meters
117. A spherical shell with inside and outside radii of 4 inches and6 inches, respectively
118. The solid that remains after a hole 1 inch in diameter is drilledthrough the center of a sphere 6 inches in diameter
119. The solid inside both and
120. The solid between the spheres andand inside the cone
True or False? In Exercises 121–124, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.
121. In cylindrical coordinates, the equation is a cylinder.
122. The equations and represent thesame surface.
123. The cylindrical coordinates of a point are unique.
124. The spherical coordinates of a point are unique.
125. Identify the curve of intersection of the surfaces (in cylindricalcoordinates) and
126. Identify the curve of intersection of the surfaces (in sphericalcoordinates) and r 5 4.r 5 2 sec f
r 5 1.z 5 sin u
sx, y, zdsx, y, zd
x2 1 y2 1 z2 5 4r 5 2
r 5 z
z2 5 x2 1 y2x2 1 y2 1 z2 5 9,x2 1 y2 1 z2 5 4
sx 232d2
1 y2 594
x2 1 y2 1 z2 5 9
1 # r # 30 # f # py2,0 # u # p,
0 # r # 20 # f # py2,0 # u # py2,
0 # u # 2p, py4 # f # py2, 0 # r # 1
0 # u # 2p, 0 # f # py6, 0 # r # a sec f
0 # u # 2p, 2 # r # 4, z2 # 2r2 1 6r 2 8
0 # u # 2p, 0 # r # a, r # z # a
2py2 # u # py2, 0 # r # 3, 0 # z # r cos u
0 # u # py2, 0 # r # 2, 0 # z # 4
y 5 4x2 2 y2 5 9
x2 1 y2 5 36x2 1 y2 5 4y
x2 1 y2 5 zx2 1 y2 1 z2 2 2z 5 0
4sx2 1 y2d 5 z2x2 1 y2 1 z2 5 25
r 5 4 sec fr2 5 z
f 5p
4r 5 5
u 5p
4r 5 5
yx
3
2
−212
z
y
x
2
1
2−2
2
z
π4
y55
5
z
y55
5
z
y
x
4
−4
4
2
z
y
x
π4
1 2 3
3
−3 −2
3 2
z
828 Chapter 11 Vectors and the Geometry of Space
95. Give the equations for the coordinate conversion fromrectangular to cylindrical coordinates and vice versa.
96. Explain why in spherical coordinates the graph of isa half-plane and not an entire plane.
97. Give the equations for the coordinate conversion fromrectangular to spherical coordinates and vice versa.
u 5 c
WRITING ABOUT CONCEPTS
98. (a) For constants and describe the graphs of theequations and in cylindrical coordinates.
(b) For constants and describe the graphs of theequations and in spherical coordinates.
f 5 cu 5 b,r 5 a,c,b,a,
z 5 cu 5 b,r 5 a,c,b,a,
CAPSTONE
1053714_1107.qxp 10/27/08 10:41 AM Page 828
In Exercises 89–94, match the equation (written in terms ofcylindrical or spherical coordinates) with its graph. [The graphsare labeled (a), (b), (c), (d), (e), and (f).]
(a) (b)
(c) (d)
(e) (f)
89. 90.
91. 92.
93. 94.
In Exercises 99–106, convert the rectangular equation to anequation in (a) cylindrical coordinates and (b) spherical coordinates.
99. 100.
101. 102.
103. 104.
105. 106.
In Exercises 107–110, sketch the solid that has the givendescription in cylindrical coordinates.
107.
108.
109.
110.
In Exercises 111–114, sketch the solid that has the givendescription in spherical coordinates.
111.
112.
113.
114.
Think About It In Exercises 115–120, find inequalities thatdescribe the solid, and state the coordinate system used.Position the solid on the coordinate system such that theinequalities are as simple as possible.
115. A cube with each edge 10 centimeters long
116. A cylindrical shell 8 meters long with an inside diameter of0.75 meter and an outside diameter of 1.25 meters
117. A spherical shell with inside and outside radii of 4 inches and6 inches, respectively
118. The solid that remains after a hole 1 inch in diameter is drilledthrough the center of a sphere 6 inches in diameter
119. The solid inside both and
120. The solid between the spheres andand inside the cone
True or False? In Exercises 121–124, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.
121. In cylindrical coordinates, the equation is a cylinder.
122. The equations and represent thesame surface.
123. The cylindrical coordinates of a point are unique.
124. The spherical coordinates of a point are unique.
125. Identify the curve of intersection of the surfaces (in cylindricalcoordinates) and
126. Identify the curve of intersection of the surfaces (in sphericalcoordinates) and r 5 4.r 5 2 sec f
r 5 1.z 5 sin u
sx, y, zdsx, y, zd
x2 1 y2 1 z2 5 4r 5 2
r 5 z
z2 5 x2 1 y2x2 1 y2 1 z2 5 9,x2 1 y2 1 z2 5 4
sx 232d2
1 y2 594
x2 1 y2 1 z2 5 9
1 # r # 30 # f # py2,0 # u # p,
0 # r # 20 # f # py2,0 # u # py2,
0 # u # 2p, py4 # f # py2, 0 # r # 1
0 # u # 2p, 0 # f # py6, 0 # r # a sec f
0 # u # 2p, 2 # r # 4, z2 # 2r2 1 6r 2 8
0 # u # 2p, 0 # r # a, r # z # a
2py2 # u # py2, 0 # r # 3, 0 # z # r cos u
0 # u # py2, 0 # r # 2, 0 # z # 4
y 5 4x2 2 y2 5 9
x2 1 y2 5 36x2 1 y2 5 4y
x2 1 y2 5 zx2 1 y2 1 z2 2 2z 5 0
4sx2 1 y2d 5 z2x2 1 y2 1 z2 5 25
r 5 4 sec fr2 5 z
f 5p
4r 5 5
u 5p
4r 5 5
yx
3
2
−212
z
y
x
2
1
2−2
2
z
π4
y55
5
z
y55
5
z
y
x
4
−4
4
2
z
y
x
π4
1 2 3
3
−3 −2
3 2
z
828 Chapter 11 Vectors and the Geometry of Space
95. Give the equations for the coordinate conversion fromrectangular to cylindrical coordinates and vice versa.
96. Explain why in spherical coordinates the graph of isa half-plane and not an entire plane.
97. Give the equations for the coordinate conversion fromrectangular to spherical coordinates and vice versa.
u 5 c
WRITING ABOUT CONCEPTS
98. (a) For constants and describe the graphs of theequations and in cylindrical coordinates.
(b) For constants and describe the graphs of theequations and in spherical coordinates.
f 5 cu 5 b,r 5 a,c,b,a,
z 5 cu 5 b,r 5 a,c,b,a,
CAPSTONE
1053714_1107.qxp 10/27/08 10:41 AM Page 828
In Exercises 89–94, match the equation (written in terms ofcylindrical or spherical coordinates) with its graph. [The graphsare labeled (a), (b), (c), (d), (e), and (f).]
(a) (b)
(c) (d)
(e) (f)
89. 90.
91. 92.
93. 94.
In Exercises 99–106, convert the rectangular equation to anequation in (a) cylindrical coordinates and (b) spherical coordinates.
99. 100.
101. 102.
103. 104.
105. 106.
In Exercises 107–110, sketch the solid that has the givendescription in cylindrical coordinates.
107.
108.
109.
110.
In Exercises 111–114, sketch the solid that has the givendescription in spherical coordinates.
111.
112.
113.
114.
Think About It In Exercises 115–120, find inequalities thatdescribe the solid, and state the coordinate system used.Position the solid on the coordinate system such that theinequalities are as simple as possible.
115. A cube with each edge 10 centimeters long
116. A cylindrical shell 8 meters long with an inside diameter of0.75 meter and an outside diameter of 1.25 meters
117. A spherical shell with inside and outside radii of 4 inches and6 inches, respectively
118. The solid that remains after a hole 1 inch in diameter is drilledthrough the center of a sphere 6 inches in diameter
119. The solid inside both and
120. The solid between the spheres andand inside the cone
True or False? In Exercises 121–124, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.
121. In cylindrical coordinates, the equation is a cylinder.
122. The equations and represent thesame surface.
123. The cylindrical coordinates of a point are unique.
124. The spherical coordinates of a point are unique.
125. Identify the curve of intersection of the surfaces (in cylindricalcoordinates) and
126. Identify the curve of intersection of the surfaces (in sphericalcoordinates) and r 5 4.r 5 2 sec f
r 5 1.z 5 sin u
sx, y, zdsx, y, zd
x2 1 y2 1 z2 5 4r 5 2
r 5 z
z2 5 x2 1 y2x2 1 y2 1 z2 5 9,x2 1 y2 1 z2 5 4
sx 232d2
1 y2 594
x2 1 y2 1 z2 5 9
1 # r # 30 # f # py2,0 # u # p,
0 # r # 20 # f # py2,0 # u # py2,
0 # u # 2p, py4 # f # py2, 0 # r # 1
0 # u # 2p, 0 # f # py6, 0 # r # a sec f
0 # u # 2p, 2 # r # 4, z2 # 2r2 1 6r 2 8
0 # u # 2p, 0 # r # a, r # z # a
2py2 # u # py2, 0 # r # 3, 0 # z # r cos u
0 # u # py2, 0 # r # 2, 0 # z # 4
y 5 4x2 2 y2 5 9
x2 1 y2 5 36x2 1 y2 5 4y
x2 1 y2 5 zx2 1 y2 1 z2 2 2z 5 0
4sx2 1 y2d 5 z2x2 1 y2 1 z2 5 25
r 5 4 sec fr2 5 z
f 5p
4r 5 5
u 5p
4r 5 5
yx
3
2
−212
z
y
x
2
1
2−2
2
z
π4
y55
5
z
y55
5
z
y
x
4
−4
4
2
z
y
x
π4
1 2 3
3
−3 −2
3 2
z
828 Chapter 11 Vectors and the Geometry of Space
95. Give the equations for the coordinate conversion fromrectangular to cylindrical coordinates and vice versa.
96. Explain why in spherical coordinates the graph of isa half-plane and not an entire plane.
97. Give the equations for the coordinate conversion fromrectangular to spherical coordinates and vice versa.
u 5 c
WRITING ABOUT CONCEPTS
98. (a) For constants and describe the graphs of theequations and in cylindrical coordinates.
(b) For constants and describe the graphs of theequations and in spherical coordinates.
f 5 cu 5 b,r 5 a,c,b,a,
z 5 cu 5 b,r 5 a,c,b,a,
CAPSTONE
1053714_1107.qxp 10/27/08 10:41 AM Page 828Larson-11-07.qxd 3/12/09 17:44 Page 828