Calculo 2 De dos variables_9na Edición - Ron Larson

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76. For Para each cada polar ecuación equation, polar, sketch dibujar its su graph, gráfica, determine determinar theel
intervalo that que traces traza the la gráfica graph only sólo once, una vez and y encontrar find the area el área of
76. For each polar equation, sketch its graph, determine the
the de region la región bounded acotada by por the la graph gráfica using utilizando a geometric una formula
interval that traces the graph only once, find the area fórmula of
and geométrica integration.
the region bounded e integración. by the graph using a geometric formula
and (a) rintegration.
10 cos
(b) r b) 5 r sin 5 sen
(a) r 10 cos (b) r 5 sin
77. Área de la superficie de un toro Hallar el área de la superficie
del toro Area generado of a Torus por revolución Find the surface de la circunferencia area of the torus r 2
77. Surface
generated alrededor by de revolving la recta r the 5 circle sec . given by r 2 about the line
77. Surface Area of a Torus Find the surface area of the torus
r 5 sec .
78. generated Área de la by superficie revolving de the un circle torogiven Hallar by rel área 2 about de la the superficie
del 5 sec toro Area . generado of a Torus por revolución Find the surface de la circunferencia area of the torus r a
line
78. rSurface generated torno a by la revolving recta r the b sec circle , donde given 0 by < r a < a about b. the line
78. Surface Area of a Torus Find the surface area of the torus
r b sec , where 0 < a < b.
79. generated Aproximación by revolving de un área the circle Considerar given by la rcircunferencia a about the rline
8
79. rApproximating cos b . sec , where Area 0 < Consider a < b. the circle r 8 cos .
79. Approximating (a) Hallar Find the el area área Area of del the círculo. Consider circle. the circle r 8 cos .
(a) (b) Find Completar Complete the area the la of tabla table the dando giving circle. las the áreas areas
A de of the los sectors sectores of circulares
circle entre between y los and valores the values de dados of in en the la tabla. table.
the
(b) Complete the table giving the areas A of the sectors of the
circle between and the values of in the table.
0.2 0.4 0.6 0.8 1.0 1.2 1.4
A
(c) Emplear Use the table la tabla in part del inciso (b) to approximate b) para aproximar the values los valores of for de
1
para los cuales el sector circular contiene 4, 1 3
which the sector of the circle composes 4, 1 2, and y 4
del of the área
(c) Use the table in part (b) to approximate the 2,
values of for
total de area la of circunferencia.
the circle.
1 3
which the sector of the circle composes 4, 1 2, and 4 of the
(d) total Usar Use area una a graphing herramienta of the circle. utility de graficación to approximate, para aproximar, to two decimal con una
precisión places, the de angles dos cifras for decimales, which the los sector ángulos of the para circle los
(d) Use a graphing 1 3
1 3
cuales composes el sector 4, 1 utility to approximate,
circular contiene 4, 1 to two decimal
and 4 of the total area 2, of y 4the del circle. área total de la
places, the angles 2,
for which the sector of the circle
(e)
circunferencia.
Do the results 1 of part 3
composes (d) depend on the radius of the circle?
4, 1 2, and 4 of the total area of the circle.
e) ¿Dependen Explain. los resultados del inciso d) del radio del círcu-lo?
(e) Do the results of part (d) depend on the radius of the circle?
80. Approximate
Explicar la
Area
respuesta.
Explain. Consider the circle r 3 sin .
80.
(a)
Área
Find
aproximada
the area of the
Dado
circle.
el círculo r 3 sen sin .
80. Approximate Area Consider the circle r 3 sin .
(b)
a) Hallar
Complete
el área
the
de
table
la circunferencia
giving the areas
correspondiente.
(a) Find the area of the circle. A of the sectors of the
b) Completar circle between la tabla dando the las values áreas of de in los the sectores table. circulares
comprendidos entre y los valores de dados en
(b) Complete the table giving the areas A of the sectors of the
circle between and the values of in the table.
la tabla. 0.2 0.4 0.6 0.8 1.0 1.2 1.4
A
A
0 0
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0
0
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4
A
(c) Use the table in part (b) to approximate the values of for
1 1 1
which the sector of the circle composes and of the
(c) Utilizar Use the la table tabla in del part inciso (b) to b) approximate para aproximar the 8,
values 4,
los valores of 2 for de
total area of the circle.
1
1 1 1
para los cuales el sector circular representa 8 , 4 , 1
which the sector of the circle composes 8, 4, and y 2 2del of área the
(d) total Use de area a graphing la of circunferencia.
the circle. utility to approximate, to two decimal
d ) Usar places, una the herramienta angles de for graficación which the para sector aproximar, of the con circle
(d) Use a graphing una
1 1 utility 1 to approximate, to two decimal
precisión composes de 8, dos 4, and cifras 2 of decimales, the total area los of ángulos the circle.
places, the angles for which para los que
1 1 1
81. What el conic sector section circular does representa the following 8 , 4 ,
the sector of the circle
1 1 1
composes
y 2polar del área equation total represent? del círculo.
8, 4, and 2 of the total area of the circle.
81. 81. What r¿Qué a conic sin sección section cónica b cos does representa the following la siguiente polar equation ecuación represent? polar?
r sin cos
82. rArea a sin sen
Find the b cos area of the circle given by r sin cos .
82. Check Área Hallar your result el área by del converting círculo dado the por polar r sen sin equation cos to.
82. Area Find the area of the circle given by r sin cos .
rectangular Comprobar form, el resultado then using transformando the formula for la the ecuación area of polar a circle. a la
Check your result by converting the polar equation to
83. Spiral forma of rectangular Archimedes y usando The después curve represented la fórmula by para the el equation área del
rectangular form, then using the formula for the area of a circle.
rcírculo.
a, where a is a constant, is called the spiral of
83. Spiral of Archimedes The curve represented by the equation
83. Archimedes. Espiral Arquímedes La curva representada por la ecuación
r a, where a is a constant, is called the spiral of
r a, donde a es una constante, se llama espiral de Arquímedes.
Archimedes.
SECCIÓN 10.5 Área y longitud de arco en coordenadas polares 749
10.5 Area and Arc Length in Polar Coordinates 749
10.5 Area and Arc Length in Polar Coordinates 749
(a)
a)
Use
Emplear
a graphing
una herramienta
utility to
de
graph
graficación
r ,
para
where
trazar la gráfica
de donde ≥
0.
What happens
r ,
to the graph
0.
of
¿Qué
r a
ocurre
as a increases?
con la gráfica
What
de
(a) Use r a a graphing a medida que utility a aumenta? to graph ¿Qué r pasa , where si ≤ 0? 0.
happens if 0?
b)
What
Determinar
happens
los
to
puntos
the graph
de la
of r
espiral
a
r
as
a increases?
a a > 0,
What
≥
(b)
0,
happens Determine
en los que
if the
la points
curva
0? on the spiral r a a > 0, 0,
where the curve crosses
cruza
the
el
polar
eje polar.
axis.
(b)
c)
Determine
Hallar la longitud
the points
de
on
r
the
sobre
spiral
el
r
intervalo
a a >
0
0,
≤
≤ 0,
(c) Find the length of over the interval
2.
where the curve crosses r the polar axis. 0 2.
d) Hallar el área bajo la curva r para
≤ ≤
(d) area under the curve for
2.
(c) Find the length of r over the r interval
0 0
2.
2.
84.
84.
Logarithmic
Espiral logarítmica La curva descrita por la ecuación r
(d)
ae
Find b Spiral The represented by the equation
r ae
, b, donde
the area
where
a y
under
a
b
and
son
the
b
constantes,
curve r
are constants,
se denomina
for 0
is called a
espiral 2.
logarithmic
logarítmica.
La figura Spiralsiguiente The curve muestra represented la gráfica by the de equation r e6
84. Logarithmic
spiral. The figure shows the graph of r
,
e6
,
r ae ≤ where ≤ 2. a and
Find
Hallar b are
the
el
area
área constants,
of
de
the
la
shaded
zona is called sombreada. a logarithmic
b, 2.
region.
spiral. The π figure shows the graph of r e6
,
π
22.
Find the area of the shaded region.
2
2
2
2
π
2
85. La mayor de las circunferencias mostradas en la figura siguiente
larger es circle la gráfica in the de figure r 1. is Hallar the graph la ecuación of r 1.
85. The polar Find para thela
polar circunferencia equation of menor the smaller de manera circle que such las that áreas the sombreadas shaded regions
85. The larger circle in the figure is the graph of r 1. Find the sean
are iguales. equal.
polar equation of the smaller circle such that the shaded regions
are equal. π π
2 2
π
2
0
0
86. 86. Folium Hoja (o of folio) Descartes de Descartes A curve Una called curva the folium llamada of hoja Descartes (o folio)
can de be Descartes represented puede by representarse the parametric por equations medio de las ecuaciones
86. Folium paramétricas of Descartes A curve called the folium of Descartes
x can x be 3t represented 3t
by the parametric
and y
3t2 equations
y y 1 3t2
1 t 3. t 3.
x 1 t 3
(a) 3t and y
3t2
Convert 1 t 3 the parametric equations 1 t 3. to polar form.
a) Convertir las ecuaciones paramétricas a la forma polar.
(a) (b)
b) Convert Sketch the
Dibujar the graph
la gráfica parametric of the
de la equations polar equation
ecuación to polar polar from
del form. part (a).
inciso a).
(b) (c)
c) Sketch Use a graphing
Emplear the una graph utility
herramienta of the to polar approximate
de equation the
graficación from area
para part enclosed
aproximar (a). by
el
the loop of the curve.
(c) Use área a comprendida graphing utility en el to lazo approximate de la curva. the area enclosed by
True or the False? loop of In the Exercises curve. 87 and 88, determine whether the
¿Verdadero o falso? En los ejercicios 87 y 88, determinar si la
statement is true or false. If it is false, explain why or give an
True afirmación or False? es verdadera In Exercises o falsa. 87 and Si 88, es falsa, determine explicar whether por qué the o
example that shows it is false.
statement dar un ejemplo is true que or false. demuestre If it is que false, es falsa. explain why or give an
example 87. If f that > 0shows for all it is and false. g < 0 for all , then the graphs of
87. Si f > 0 para todo y g < 0 para todo , entonces las
r f and r g do not intersect.
87. If gráficas f > 0de for r all fand y rg g < 0no for se all cortan. , then the graphs of
88. If f g for and 32, then the graphs of
88. r Si f f and
g r para g do not intersect. y 32, entonces las gráficas de
r f and r g have at least four points of intersection.
88. If rf f g y r for g tienen cuando menos 32, cuatro then the puntos graphs de intersección.
f
of
89.
r
Use the formula
and r
for
g
the
have
arc length
at least
of
four
a curve
points
in parametric
of intersection.
form
89. to Usar derive la fórmula the formula para for la longitud the arc length de arco of de a polar una curva curve. en forma
89. Use
paramétrica
the formula
para
for
obtener
the arc length
la fórmula
of a curve
de la
in
longitud
parametric
de arco
form
de
to
una
derive
curva
the
polar.
formula for the arc length of a polar curve.
1 t 3 0
0
1 2 3 0
1 2 3
0
1 2 3
0, 2,
0, 2,
0, 2,