Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 14.1 Integrales iteradas y área en el plano 991
2 x
4 4x
CAS En 14.1 los ejercicios 14.1
14.1
Iterated Iterated
Iterated 73 a Integrals 76, utilizar Integrals
Integrals
and
and un Area and sistema Area
Area
in
in
the algebraico in
the
Plane the Plane
Plane por computadora
y evaluar la integral iterada.
991 991
991
59. dy dx
0 dy dx
0
20
4 x2
6 6x
2 2x
60.
2 x 2 x
4 4 x
CAS
dy dx
4 4 x
73.
In Exercises
x
0
3 3y
73–76, 2 dy dx
use a computer algebra system to evaluate
CAS
59.
In Exercises 73–76, use computer algebra system to evaluate
59. 0 dy dx
CAS In Exercises 73–76, use a computer algebra system to evaluate
59. 0 dy
dy
dxdy dx
dx 40
dy
dy
dxdy dx
dx
the x
the
iterated the iterated
iterated
1
61. 62.
74. 0
2 integral. integral.
02
01
0 0 2 0 2 0
9 3
integral.
2y
4 x 24
x 2
6 6 x
2 2x
dy dx
6 6 x
dy dx
2 2x
2x
60. 60.
sinx y dx dy
60. 0x2
dy
dy
dxdy dx
dx
dy
dy
dxdy dx
dx 0
73. sen
x
73.
73.
x
y
3
4
63. 64.
75. 0
x3y 3y 3 2 dy 3y
dy 2 dxdy dx
0 0
4 0
dx
0 x 2 2 1
9 3
2 y 1 2y
2y 2
61. dy dx
2 4y
62.
61. 62. dy dx dy dx
dy dx
74.
0 x
sin
1y
x
1
y dx dy
0 x 2
74. sin dx dy
0 y
a
ax
1 3
2
03 0 0
4 0
1 y
2
0 x 2
2 1
9 3
1 2y
61. dy dx dx dy
62. dy dx
dy
74. sin x y dx dy
0 yx 2 2
0
0x0
x
0 y
4 y
1 3 y
76.
65. Para pensar Dar un argumento geométrico para la igualdad.
Verificar 2
0 0 x 1 y 1
x
0
3
2 4 y 2
4 y
y
2 4 y 2
2
2
y 2 2
dy 63. 64.
2
75.
dx dy
63.
63.
dx dy
64.
75.
0
dx dy
dx dy
64.
dx dy
75.
dx dy
dx dy
dx
dx
dy
dy
0
0 x 1 y 1
0 y 0 y 2
la igualdad analíticamente.
2
2 0 2 0
a a ax
a x
76.
76. En los ejercicios x
77 y
y dy 78, 2 dy dx
5 50x
65. Think About It Give a geometric argument for the equality.
dx a) dibujar la región de integración,
0 0
65. Think About It Give geometric argument for the equality.
b) cambiar el orden de integración y c) usar un sistema algebraico
In
Verify 0
2
CAS
76. x 2 y 2 dy dx
65. About
x
Verify the 2 It
y 2 dy
Give
dx
a geometric argument for the equality.
0 0
Verify x the
the
equality equality
equality
analytically. analytically.
analytically.
In Exercises por Exercises computadora 77 and
77 and 78, (a) y 78, mostrar (a) sketch
sketch the que the
region ambos region
of órdenes of integration,
5
CAS CAS In Exercises 77 and 78, (a) sketch the region of integration,
integration, dan el
5
y 50 x 50 x 2
CAS
50 2
(b) mismo switch valor.
x
dy 2 y
dx
2 52
dy dx
(b) switch the order of integration, and (c) use a computer
x the order of integration, and (c) use computer
0 x
0
2 y 2 dx dy
2 50y 2
x 2 y 2 dy dx x 2 y 2 (b) switch the order of integration, and (c) use a computer
dx dy
0
algebra system to show that both orders yield the same value.
0
x
5 0
algebra
algebra
system
2 system
to
42y to
show
show
that
that
both
both
orders
orders
yield
yield
the
the
same
same
value.
value.
5 y 5 y
5 2 5 50 2 y 50 y
y
50 77.
2y
x
0
2 2
2
2 4 2 2yy xy 2 dx dy
x
dx 2 yy dy
2 = dx dy 50 − x 2
x
dx 2 y
dy
2 2 4 2y
x dx dy
77. y
77.
2 4x
78. 0
3
x 2 y
xy xy
dx 2 dx dy
( 2 y
0, 5 02
)
2 dx dy
x 2 y 2 dx dy
77. x 2 y xy 2 dx dy
0 0 0
5 50
0
dy
0 y 3 2 4
3 0 y 3 xy
2
x 2 dy dx
2 4x x 2 4
5 y y
y = 50 y 2 1 50 − x 2
2 4 x
(5, 5)
2 4
xy
y = 50 − x 78.
78.
2 xy dy dx
dy dx
( 0, 5 2
0,
)
2
xy
78.
0 4 x x y 2 dy dx
( 0, 5 2 )
1
0 4 x x 2 2 y 2 1
y = x
2 2
5 5
(5, 5)
CAS
(5, En los ejercicios a 82, usar un sistema algebraico por computadora
(5,
5)
5)
CAS CAS In In Exercises 79–82, use a computer system to approximate
the iterated integral.
CAS In
Exercises
Exercises
79–82,
y aproximar
79–82,
use
use
a
la computer
computer
algebra
iterada.
algebra
system
system
to
to
approximate
x
approximate
5
the
the
iterated
iterated
integral.
y integral.
= x y = x
22
44x 2x 79. 0
2
2 4 2
xy x 2
x x
dy dx
79. 79.
79.
e
0
xy xy dy e
dy
dx
xy dy dx
5 5
dx
0 0
0 0
2
Para discusión
2 22
2 2
80.
80.
80.
16 16 16 0 16
x 3 3 x y 3 3 dy
dy ydx
dx
3 dy dx
66. Para x 0 x
CAPSTONE pensar Completar las integrales iteradas en forma tal
0 x
CAPSTONE
2 1 1cos 2
1
cos
cos
que cada una represente el área de la región R (ver la figura).
Entonces demostrar que ambas integrales tienen la
66.
66.
Think 66. Think
Think
About About
About
It
It
Complete It Complete
Complete
the
the
iterated the iterated
iterated
integrals integrals
integrals
so
so
that so that
that 81.
81.
81.
6r 6r 2 cos 6r 2 cos dr dr ddr d
6r 2 cos dr d
0 0 0
each each
each
one
one
represents one represents
represents
the
the
area the
area
of area
of
the of
the
region the region
region
R
(see R
(see
figure). (see figure).
0 0
misma área.
figure).
2 1 sin
2
2 1 1sin
sin
Then Then
Then
show show
show
that
that
both that
both
integrals both integrals
integrals
yield yield
yield
the
the
same the same
same
area. area.
area.
82.
82.
82.
15 15r r 15 dr
dr dr d
dr d
0
0
0
0
0
a)
a)
Área a) Área
b)
Área
dx
dx
dydx dy
dy
b)
b)
Área Área
Área
dy
dy
dxdy dx
dx
y
WRITING ABOUT CONCEPTS
y
WRITING ABOUT CONCEPTS
83. Desarrollo 83. Explain what is meant by an iterated integral. How is it
(4, 2)
83.
Explain
Explain
what de
what
is
is
meant conceptos
meant
by
by
an
an
iterated
iterated
integral.
integral.
How
How
is
is
it
(4, it
2 y = x (4,
2)
2 y 2)
evaluated?
=
x
83.
evaluated?
evaluated? Explicar qué se quiere decir con una integral iterada. ¿Có-
84. Describe regions that are vertically simple and regions that
y = x mo 84. se Describe evalúa? regions that are vertically simple and regions that
1
y = x 84. Describe regions that are vertically simple and regions that
1
are horizontally simple.
R
2
are horizontally simple.
84.
are
Describir
horizontally
regiones
simple.
R
2
que sean verticalmente simples y regiones
a geometric
85. 85. Give a geometric description of the region of integration if
85.
Give
Give geometric que sean horizontalmente description
description
of
of
the
the simples. region
region
of
of
integration
integration
if
if
x
the inside and outside limits of integration are constants.
the inside and outside limits of integration are constants.
1 2 3 4
85.
the
Dar
inside
una descripción
and outside
geométrica
limits of integration
de la región
are
de
constants.
integración
1 2 3 4
86. 86. Explain why it is sometimes an advantage to change the
86.
Explain
Explain si los límites why
why
it interiores it
is
is
sometimes
sometimes y exteriores an
an
advantage
advantage de integración to
to
change
change son constantes.
of
the
the
order order of integration.
order of
integration.
integration.
86. Explicar por qué algunas veces es una ventaja cambiar el
En In
In los Exercises In
Exercises ejercicios Exercises 67–72,
67–72, 67–72, a 72, sketch
sketch trazar sketch the
the la región region the region
region de of
of integración. integration. of integration.
integration. Después Then Then
Then
orden de integración.
evaluar evaluate evaluate
evaluate la the
the integral iterated the iterated
iterated iterada. integral. integral.
integral. (Observar (Note (Note
(Note
that
that que it that
it
is
is es necessary it
necessary necesario is necessary to
to cambiar switch to switch
switch True True or False? In Exercises 87 and 88, determine whether the
True
or
or
False?
False?
In
In
Exercises
Exercises
87
87
and
and
88,
88,
determine
determine
whether
whether the
el the
the orden order the order
order of
of integración.)
integration.) of integration.)
integration.)
statement statement is true or false. If it is false, explain why or give an
statement
is
is
true
true
or
or
false.
false.
If
If
it
it
is
is
false,
false,
explain
explain
why
why
or
or
give
give
an
an
2 2
4 2
example ¿Verdadero
example
that o
that
shows falso?
shows
it is En
it
false. los
is false.
3
ejercicios 87 y 88, determinar si la
67. x 1
67.
2
2
4 2
example that shows it is false.
67. x1 68.
y 68.
dy dx
3 3
x 1 y dy dx
declaración es verdadera o falsa. Si es falsa, explicar por qué o
dy 3
dy dx
dx
b d
d b
0
3 dy dx 68.
0 x
0
x 23 dy dx
y b d
d b
0
x
0 x 2 y 1 2
2 2
87. dar un 87. ejemplo fx, que y demuestre dy dx que fx, es falsa. y dx dy
1 2
2
87.
fx,
fx,
y dy
dy
dx
dx
fx,
fx,
y dx
dx
dy
dy
69. 69. 70. 70.
87. a
b d
fx, y dy dx
d b
0
y x
fx, y dx dy
2 4
c
c
2 4e a c
c a
69. 2 4e 70.
dy y 2 2
2
dy dx
dx
70.
2 e
dy y 2 a c
c a
4e y 2 dy dx
e y 2 dy dxdy dx
dx
1 x
1 y
0 2x
0 x
dy dx
1 x
1 y
0 2x
0 x
2x 1 1
2 4
88. 88. fx, y dy dx fx, y dx dy
1 1
2 4
88.
fx,
fx,
y dy
dy
dx
dx
fx,
fx,
y dx
dx
dy
dy
0 0
71.
a 0 0
71.
71.
71.
72.
1 x
1 y
sin x sin x dx dy
0
0y 88. fx, y dy dx
sin sin x 72.
sin x dx dy
dx
dx
dy
dy
2 0 0
0 0
sin x 2 dx dydx dy 72.
sen
72.
x sin
sin
x dx
dx
dy
dy
0 y 0 y
0 y 0 y 2 sen
2
2
2 f x, y dx dy
2
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