Calculo 2 De dos variables_9na Edición - Ron Larson

728 CAPÍTULO 10 Cónicas, ecuaciones paramétricas y coordenadas polares
Longitud de arco
En los ejercicios 45 a 48, dar una integral que
represente la longitud de arco de la curva en el intervalo dado.
No evaluar la integral.
Ecuaciones paramétricas
Intervalo
45.
46.
47.
48.
Longitud de arco
En los ejercicios 49 a 56, hallar la longitud de
arco de la curva en el intervalo dado.
Ecuaciones paramétricas
Intervalo
53.
54.
55.
56.
Longitud de arco
En los ejercicios 57 a 60, hallar la longitud de
arco de la curva en el intervalo
57. Perímetro de una hipocicloide:
58. Circunferencia de un círculo:
59. Arco de una cicloide:
60. Evolvente o involuta de un círculo:
61. Trayectoria de un proyectil La trayectoria de un proyectil se
describe por medio de las ecuaciones paramétricas
y
donde x y y se miden en pies.
a) Utilizar una herramienta de graficación para trazar la trayectoria
del proyectil.
b) Utilizar una herramienta de graficación para estimar el
alcance del proyectil.
c) Utilizar las funciones de integración de una herramienta de
graficación para aproximar la longitud de arco de la trayectoria.
Comparar este resultado con el alcance del proyectil.
62. Trayectoria de un proyectil Si el proyectil del ejercicio 61 se
lanza formando un ángulo
con la horizontal, sus ecuaciones
paramétricas son
y
Usar una herramienta de graficación para hallar el ángulo que
maximiza el alcance del proyectil. ¿Qué ángulo maximiza la
longitud de arco de la trayectoria?
63. Hoja (o folio) de Descartes Considerar las ecuaciones paramétricas
y
a) Usar una herramienta de graficación para trazar la curva
descrita por las ecuaciones paramétricas.
b) Usar una herramienta de graficación para hallar los puntos de
tangencia horizontal a la curva.
c) Usar las funciones de integración de una herramienta de
graficación para aproximar la longitud de arco del lazo cerrado.
(Sugerencia: Usar la simetría e integrar sobre el intervalo
64. Hechicera o bruja de Agnesi Considerar las ecuaciones paramétricas
y
a) Emplear una herramienta de graficación para trazar la curva
descrita por las ecuaciones paramétricas.
b) Utilizar una herramienta de graficación para hallar los puntos
de tangencia horizontal a la curva.
c) Usar las funciones de integración de una herramienta de
graficación para aproximar la longitud de arco en el intervalo
65. Redacción
a) Usar una herramienta de graficación para representar cada
conjunto de ecuaciones paramétricas.
b) Comparar las gráficas de los dos conjuntos de ecuaciones
paramétricas del inciso a). Si la curva representa el movimiento
de una partícula y t es tiempo, ¿qué puede inferirse
acerca de las velocidades promedio de la partícula en las
trayectorias representadas por los dos conjuntos de ecuaciones
paramétricas?
c) Sin trazar la curva, determinar el tiempo que requiere la
partícula para recorrer las mismas trayectorias que en los
incisos a) y b) si la trayectoria está descrita por
y
66. Redacción
a) Cada conjunto de ecuaciones paramétricas representa el movimiento
de una partícula. Usar una herramienta de graficación
para representar cada conjunto.
Primera partícula
Segunda partícula
b) Determinar el número de puntos de intersección.
c) ¿Estarán las partículas en algún momento en el mismo lugar
al mismo tiempo? Si es así, identificar esos puntos.
d) Explicar qué ocurre si el movimiento de la segunda partícula
se representa por
0 t 2.
y 2 4 cos t,
x 2 3 sin t,
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 32
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
y 3 cos t
y 4 sin t
x 4 sin t
x 3 cos t
y 1 cos 1 2 t.
x 1 2 t sin1 2 t 0 ≤ t ≤
0 ≤ t ≤ 2
y 1 cos2t
y 1 cos t
x 2t sin2t
x t sin t
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 3 2
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 3 2
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
y 4 sin 2 ,
x 4 cot
rcises 45–48, write an integral that repreof
the curve on the given interval. Do not
l.
rcises 49–56, find the arc length of the curve
l.
rcises 57–60, find the arc length of the curve
imeter:
nce:
le:
ile
The path of a projectile is modeled by the
ions
and
e measured in feet.
ng utility to graph the path of the projectile.
ing utility to approximate the range of the
gration capabilities of a graphing utility to
the arc length of the path. Compare this result
e of the projectile.
ectile
If the projectile in Exercise 61 is
angle
with the horizontal, its parametric
and
utility to find the angle that maximizes the
ectile. What angle maximizes the arc length of
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
y 90 sin 30 t 16t 2
t
x cos sin , y sin cos
a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
].
1 t 2
1
6t 3 0 t 1
t 1
0 t
1
2
ln 1 t 2 0 t
2
e t sen t
1 t 0
4t 3 3
1 t 4
t 3 0 t 2
1 t 3
7 2t
Interval
tions
0 t
y t cos t
2 t 2
2t 1
1 t 5
t 3
1 t 3
2t 3 2
Interval
tions
0 Conics, Parametric Equations, and Polar Coordinates
y 4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y 90 sin 30t 16t 2
x 90 cos 30t
x cos
sin , y sin
cos
x a sin , y a1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2].
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 32
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
x t, y t 5
10 1
6t 3
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 3 2
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
x t, y 3t 1
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 3 2
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
x arcsin t, y ln1 t 2
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 3 2
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
x e t cos t, y e t sin t
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 3 2
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
sen
arcsen
sen
sen
sen
sen
sen
sen
sen
sen
sen
sen
sen
sen 2
sen t,
sen
Arc Length
In Exercises 45–48, write an integral that represents
the arc length of the curve on the given interval. Do not
evaluate the integral.
45.
46.
47.
48.
Arc Length
In Exercises 49–56, find the arc length of the curve
on the given interval.
49.
50.
51.
52.
53.
54.
55.
56.
Arc Length
In Exercises 57–60, find the arc length of the curve
on the interval
57. Hypocycloid perimeter:
58. Circle circumference:
59. Cycloid arch:
60. Involute of a circle:
61. Path of a Projectile The path of a projectile is modeled by the
parametric equations
and
where and are measured in feet.
(a) Use a graphing utility to graph the path of the projectile.
(b) Use a graphing utility to approximate the range of the
projectile.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the path. Compare this result
with the range of the projectile.
62. Path of a Projectile If the projectile in Exercise 61 is
launched at an angle
with the horizontal, its parametric
equations are
and
Use a graphing utility to find the angle that maximizes the
range of the projectile. What angle maximizes the arc length of
the trajectory?
63. Folium of Descartes Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility to
approximate the arc length of the closed loop.
Hint: Use
symmetry and integrate over the interval
64. Witch of Agnesi Consider the parametric equations
and
(a) Use a graphing utility to graph the curve represented by the
parametric equations.
(b) Use a graphing utility to find the points of horizontal
tangency to the curve.
(c) Use the integration capabilities of a graphing utility
to approximate the arc length over the interval
65. Writing
(a) Use a graphing utility to graph each set of parametric
equations.
(b) Compare the graphs of the two sets of parametric equations
in part (a). If the curve represents the motion of a particle
and is time, what can you infer about the average speeds
of the particle on the paths represented by the two sets of
parametric equations?
(c) Without graphing the curve, determine the time required for
a particle to traverse the same path as in parts (a) and (b) if
the path is modeled by
and
66. Writing
(a) Each set of parametric equations represents the motion of a
particle. Use a graphing utility to graph each set.
(b) Determine the number of points of intersection.
(c) Will the particles ever be at the same place at the same
time? If so, identify the point(s).
(d) Explain what happens if the motion of the second particle
is represented by
0 t 2 .
y 2 4 cos t,
x 2 3 sin t,
0 t 2
0 t 2
y
3 cos t
y
4 sin t
x
4 sin t
x
3 cos t
Second Particle
First Particle
y 1 cos 1 2t .
x
1
2t
sin 1 2t
t
0 t
0 t 2
y 1 cos 2t
y 1 cos t
x 2t sin 2t
x t sin t
4 2.
2 2 .
y 4 sin2 ,
x
4 cot
0 t 1.
y
4t 2
1 t 3.
x
4t
1 t 3
y 90 sin t 16t 2 .
x 90 cos t
y
x
y 90 sin 30 t 16t 2
x
90 cos 30 t
x cos sin , y sin cos
x a sin , y a 1 cos
x a cos , y a sin
x a cos 3 , y a sin 3
[0, 2 ].
1 t 2
x t, y
t 5
10
1
6t 3 0 t 1
x t, y 3t 1
0 t
1
2
x arcsen t, y ln 1 t 2 0 t
2
x e t cos t, y e t sen t
1 t 0
x t 2 1, y 4t 3 3
1 t 4
x 6t 2 , y 2t 3 0 t 2
x t 2 , y 2t
1 t 3
x 3t 5, y 7 2t
Interval
Parametric Equations
0 t
y t cos t
x t sen t,
2 t 2
y 2t 1
x e t 2,
1 t 5
y 4t 3
x ln t,
1 t 3
y 2t 3 2
x 3t t 2 ,
Interval
Parametric Equations
728 Chapter 10 Conics, Parametric Equations, and Polar Coordinates