Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 14.2 Integrales dobles y volumen 1003
Probabilidad
Una función de densidad de probabilidad conjunta
de las variables aleatorias continuas x y y es una función
ƒ(x, y) que satisface las propiedades siguientes.
a) para todo b)
c)
En los ejercicios 73 a 76, mostrar que la función es una función
de densidad de probabilidad conjunta y hallar la probabilidad
requerida.
73.
74.
75.
76.
77. Aproximación En una fábrica de cemento la base de un montón
de arena es rectangular con dimensiones aproximadas de 20
por 30 metros. Si la base se coloca en el plano xy con un vértice
en el origen, las coordenadas de la superficie del montón son
(5, 5, 3), (15, 5, 6), (25, 5, 4), (5, 15, 2), (15, 15, 7) y (25, 15, 3).
Aproximar el volumen de la arena en el montón.
78. Programación Considerar una función continua sobre
la región rectangular R con vértices (a, c), (b, c), (a, d) y (b, d)
donde y Dividir los intervalos y en
y
subintervalos, de modo que los subintervalos en una dirección
dada sean de igual longitud. Escribir un programa para que
una herramienta de graficación calcule la suma
donde
es el centro de un rectángulo representativo en
Aproximación
En los ejercicios 79 a 82, a) utilizar un sistema
algebraico por computadora y aproximar la integral iterada, y
b) utilizar el programa del ejercicio 78 para aproximar la integral
iterada con los valores dados de m y n.
79. 80.
81. 82.
Aproximación
En los ejercicios 83 y 84, determinar qué valor
aproxima mejor el volumen del sólido entre el plano xy y la función
sobre la región. (Hacer la elección con base en un dibujo del sólido
y sin realizar ningún cálculo.)
83.
cuadrado con vértices
a) b) 600 c) 50 d) 125 e) 1 000
84.
círculo acotado por
a) 50 b) 500 c) d) 5 e) 5 000
¿Verdadero o falso?
En los ejercicios 85 y 86, determinar si la
declaración es verdadera o falsa. Si es falsa, explicar por qué o
dar un ejemplo que demuestre que es falsa.
85. El volumen de la esfera está dado por la integral
86. Si para todo en R, y ƒ y g son continuas
en R, entonces
87. Sea Hallar el valor promedio de f en el intervalo
88. Hallar Sugerencia: Evaluar
89. Determinar la región R en el plano xy que maximiza el valor de
90. Determinar la región R en el plano xy que minimiza el valor de
91. Hallar (Sugerencia: Convertir la
integral en una integral doble.)
92. Utilizar un argumento geométrico para mostrar que
3
0
9y 2
0
9 x 2 y 2 dx dy 9
2 .
2 0 arctanx arctan x dx.
Probability
A joint density function of the continuous random
variables and is a function satisfying the following
properties.
(a) for all (b)
(c)
In Exercises 73–76, show that the function is a joint density
function and find the required probability.
73.
74.
75.
76.
77. Approximation The base of a pile of sand at a cement plant is
rectangular with approximate dimensions of 20 meters by
30 meters. If the base is placed on the plane with one vertex
at the origin, the coordinates on the surface of the pile are
and
Approximate the volume of sand in the pile.
78. Programming Consider a continuous function over
the rectangular region with vertices and
where and Partition the intervals and
into and subintervals, so that the subintervals in a
given direction are of equal length. Write a program for a
graphing utility to compute the sum
where
is the center of a representative rectangle in
Approximation
In Exercises 79–82, (a) use a computer algebra
system to approximate the iterated integral, and (b) use the
program in Exercise 78 to approximate the iterated integral for
the given values of
and
79. 80.
81. 82.
Approximation
In Exercises 83 and 84, determine which value
best approximates the volume of the solid between the
-plane
and the function over the region. (Make your selection on the
basis of a sketch of the solid and not by performing any
calculations.)
83.
square with vertices
(a) (b) 600 (c) 50 (d) 125 (e) 1000
84.
circle bounded by
(a) 50 (b) 500 (c) (d) 5 (e) 5000
True or False?
In Exercises 85 and 86, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
85. The volume of the sphere is given by the
integral
86. If for all in and both and are
continuous over
then
87. Let Find the average value of on the interval
88. Find Hint: Evaluate
89. Determine the region in the -plane that maximizes the
value of
90. Determine the region in the -plane that minimizes the
value of
91. Find (Hint: Convert the integral
to a double integral.)
92. Use a geometric argument to show that
3
0
9 y 2
0
9 x 2 y 2 dx dy
9
2 .
2
0 arctan x arctan x dx.
R x2 y 2 4 dA.
xy
R
R 9 x2 y 2 dA.
xy
R
2
1
e xy dy.
0
e x
e 2x
x
dx.
0, 1 .
f
f x
x
1 e t2 dt.
R fx, y dA
R gx, y dA.
R, g
f
R,
x, y
f x, y
g x, y
V 8
1
0
1
0
1 x 2 y 2 dx dy.
x 2 y 2 z 2 1
500
x 2 y 2 9
R:
fx, y x 2 y 2
200
0, 0 , 4, 0 , 4, 4 , 0, 4
R:
fx, y
4x
xy
m 6, n 4
m 4, n 8
4
1
2
1
x 3
y 3 dx dy
6
4
2
0
y cos
x dx dy
m 10, n 20
m 4, n 8
2
0
4
0
20e x3 8 dy dx
1
0
2
0
sen x y dy dx
n.
m
R.
x i , y j
n
i 1
m
j 1
fx i , y j A i b
a
d
c
fx, y dA
n
m
c, d
a, b
c < d.
a < b
b, d ,
a, d ,
b, c ,
a, c ,
R
fx, y
25, 15, 3 .
15, 15, 7 ,
5, 15, 2 ,
25, 5, 4 ,
15, 5, 6 ,
5, 5, 3 ,
xy-
P 0 x 1, x y 1
fx, y
e x y ,
0,
x 0, y 0
elsewhere
P 0 x 1, 4 y 6
fx, y
1
27 9 x y ,
0,
0 x 3, 3 y 6
elsewhere
P 0 x 1, 1 y 2
fx, y
1
4 xy,
0,
0 x 2, 0 y 2
elsewhere
P 0 x 2, 1 y 2
fx, y
1
10 ,
0,
0 x 5, 0 y 2
elsewhere
P[ x, y R]
R
fx, y dA
fx, y dA 1
x, y
f x, y ~ 0
fx, y
y
x
14.2 Double Integrals and Volume 1003
72. The following iterated integrals represent the solution to the
same problem. Which iterated integral is easier to evaluate?
Explain your reasoning.
4
0
2
x 2
sen y 2 dy dx
2
0
2y
0
sen y 2 dx dy
CAPSTONE
CAS
93. Evaluate where and are
positive.
94. Show that if there does not exist a real-valued function
such that for all in the closed interval
These problems were composed by the Committee on the Putnam Prize
Competition. © The Mathematical Association of America. All rights reserved.
ux 1
1
x uyuy x dy. 0 x 1,
x
u
> 1 2
b
a
a
0 b
0 emax b2 x 2 , a 2 y 2 dy dx,
PUTNAM EXAM CHALLENGE
Probability
A joint density function of the continuous random
variables and is a function satisfying the following
properties.
(a) for all (b)
(c)
In Exercises 73–76, show that the function is a joint density
function and find the required probability.
73.
74.
75.
76.
77. Approximation The base of a pile of sand at a cement plant is
rectangular with approximate dimensions of 20 meters by
30 meters. If the base is placed on the plane with one vertex
at the origin, the coordinates on the surface of the pile are
and
Approximate the volume of sand in the pile.
78. Programming Consider a continuous function over
the rectangular region with vertices and
where and Partition the intervals and
into and subintervals, so that the subintervals in a
given direction are of equal length. Write a program for a
graphing utility to compute the sum
where
is the center of a representative rectangle in
Approximation
In Exercises 79–82, (a) use a computer algebra
system to approximate the iterated integral, and (b) use the
program in Exercise 78 to approximate the iterated integral for
the given values of
and
79. 80.
81. 82.
Approximation
In Exercises 83 and 84, determine which value
best approximates the volume of the solid between the
-plane
and the function over the region. (Make your selection on the
basis of a sketch of the solid and not by performing any
calculations.)
83.
square with vertices
(a) (b) 600 (c) 50 (d) 125 (e) 1000
84.
circle bounded by
(a) 50 (b) 500 (c) (d) 5 (e) 5000
True or False?
In Exercises 85 and 86, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
85. The volume of the sphere is given by the
integral
86. If for all in and both and are
continuous over
then
87. Let Find the average value of on the interval
88. Find Hint: Evaluate
89. Determine the region in the -plane that maximizes the
value of
90. Determine the region in the -plane that minimizes the
value of
91. Find (Hint: Convert the integral
to a double integral.)
92. Use a geometric argument to show that
3
0
9 y 2
0
9 x 2 y 2 dx dy
9
2 .
2
0 arctan x arctan x dx.
R x2 y 2 4 dA.
xy
R
R 9 x2 y 2 dA.
xy
R
2
1
e xy dy.
0
e x
e 2x
x
dx.
0, 1 .
f
f x
x
1 e t2 dt.
R fx, y dA
R gx, y dA.
R, g
f
R,
x, y
f x, y
g x, y
V 8
1
0
1
0
1 x 2 y 2 dx dy.
x 2 y 2 z 2 1
500
x 2 y 2 9
R:
fx, y x 2 y 2
200
0, 0 , 4, 0 , 4, 4 , 0, 4
R:
fx, y
4x
xy
m 6, n 4
m 4, n 8
4
1
2
1
x 3
y 3 dx dy
6
4
2
0
y cos
x dx dy
m 10, n 20
m 4, n 8
2
0
4
0
20e x3 8 dy dx
1
0
2
0
sen x y dy dx
n.
m
R.
x i , y j
n
i 1
m
j 1
fx i , y j A i b
a
d
c
fx, y dA
n
m
c, d
a, b
c < d.
a < b
b, d ,
a, d ,
b, c ,
a, c ,
R
fx, y
25, 15, 3 .
15, 15, 7 ,
5, 15, 2 ,
25, 5, 4 ,
15, 5, 6 ,
5, 5, 3 ,
xy-
P 0 x 1, x y 1
fx, y
e x y ,
0,
x 0, y 0
elsewhere
P 0 x 1, 4 y 6
fx, y
1
27 9 x y ,
0,
0 x 3, 3 y 6
elsewhere
P 0 x 1, 1 y 2
fx, y
1
4 xy,
0,
0 x 2, 0 y 2
elsewhere
P 0 x 2, 1 y 2
fx, y
1
10 ,
0,
0 x 5, 0 y 2
elsewhere
P[ x, y R]
R
fx, y dA
fx, y dA 1
x, y
f x, y ~ 0
fx, y
y
x
14.2 Double Integrals and Volume 1003
72. The following iterated integrals represent the solution to the
same problem. Which iterated integral is easier to evaluate?
Explain your reasoning.
4
0
2
x 2
sen y 2 dy dx
2
0
2y
0
sen y 2 dx dy
CAPSTONE
CAS
93. Evaluate where and are
positive.
94. Show that if there does not exist a real-valued function
such that for all in the closed interval
These problems were composed by the Committee on the Putnam Prize
Competition. © The Mathematical Association of America. All rights reserved.
ux 1
1
x uyuy x dy. 0 x 1,
x
u
> 1 2
b
a
a
0 b
0 emax b2 x 2 , a 2 y 2 dy dx,
PUTNAM EXAM CHALLENGE
2
1
e xy dy.
0
e x e 2x
x
dx.
0, 1.
f x x 1 e t2 dt.
Probability
A joint density function of the continuous random
variables and is a function satisfying the following
properties.
(a) for all (b)
(c)
In Exercises 73–76, show that the function is a joint density
function and find the required probability.
73.
74.
75.
76.
77. Approximation The base of a pile of sand at a cement plant is
rectangular with approximate dimensions of 20 meters by
30 meters. If the base is placed on the plane with one vertex
at the origin, the coordinates on the surface of the pile are
and
Approximate the volume of sand in the pile.
78. Programming Consider a continuous function over
the rectangular region with vertices and
where and Partition the intervals and
into and subintervals, so that the subintervals in a
given direction are of equal length. Write a program for a
graphing utility to compute the sum
where
is the center of a representative rectangle in
Approximation
In Exercises 79–82, (a) use a computer algebra
system to approximate the iterated integral, and (b) use the
program in Exercise 78 to approximate the iterated integral for
the given values of
and
79. 80.
81. 82.
Approximation
In Exercises 83 and 84, determine which value
best approximates the volume of the solid between the
-plane
and the function over the region. (Make your selection on the
basis of a sketch of the solid and not by performing any
calculations.)
83.
square with vertices
(a) (b) 600 (c) 50 (d) 125 (e) 1000
84.
circle bounded by
(a) 50 (b) 500 (c) (d) 5 (e) 5000
True or False?
In Exercises 85 and 86, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
85. The volume of the sphere is given by the
integral
86. If for all in and both and are
continuous over
then
87. Let Find the average value of on the interval
88. Find Hint: Evaluate
89. Determine the region in the -plane that maximizes the
value of
90. Determine the region in the -plane that minimizes the
value of
91. Find (Hint: Convert the integral
to a double integral.)
92. Use a geometric argument to show that
3
0
9 y 2
0
9 x 2 y 2 dx dy
9
2 .
2
0 arctan x arctan x dx.
R x2 y 2 4 dA.
xy
R
R 9 x2 y 2 dA.
xy
R
2
1
e xy dy.
0
e x
e 2x
x
dx.
0, 1 .
f
f x
x
1 e t2 dt.
R f x, y dA
R g x, y dA.
R, g
f
R,
x, y
f x, y
g x, y
V 8
1
0
1
0
1 x 2 y 2 dx dy.
x 2 y 2 z 2 1
500
x 2 y 2 9
R:
fx, y x 2 y 2
200
0, 0 , 4, 0 , 4, 4 , 0, 4
R:
fx, y
4x
xy
m 6, n 4
m 4, n 8
4
1
2
1
x 3
y 3 dx dy
6
4
2
0
y cos
x dx dy
m 10, n 20
m 4, n 8
2
0
4
0
20e x3 8 dy dx
1
0
2
0
sen x y dy dx
n.
m
R.
x i , y j
n
i 1
m
j 1
fx i , y j A i b
a
d
c
fx, y dA
n
m
c, d
a, b
c < d.
a < b
b, d ,
a, d ,
b, c ,
a, c ,
R
fx, y
25, 15, 3 .
15, 15, 7 ,
5, 15, 2 ,
25, 5, 4 ,
15, 5, 6 ,
5, 5, 3 ,
xy-
P 0 x 1, x y 1
fx, y
e x y ,
0,
x 0, y 0
elsewhere
P 0 x 1, 4 y 6
fx, y
1
27 9 x y ,
0,
0 x 3, 3 y 6
elsewhere
P 0 x 1, 1 y 2
fx, y
1
4 xy,
0,
0 x 2, 0 y 2
elsewhere
P 0 x 2, 1 y 2
fx, y
1
10 ,
0,
0 x 5, 0 y 2
elsewhere
P[ x, y R]
R
fx, y dA
fx, y dA 1
x, y
f x, y ~ 0
fx, y
y
x
14.2 Double Integrals and Volume 1003
72. The following iterated integrals represent the solution to the
same problem. Which iterated integral is easier to evaluate?
Explain your reasoning.
4
0
2
x 2
sen y 2 dy dx
2
0
2y
0
sen y 2 dx dy
CAPSTONE
CAS
93. Evaluate where and are
positive.
94. Show that if there does not exist a real-valued function
such that for all in the closed interval
These problems were composed by the Committee on the Putnam Prize
Competition. © The Mathematical Association of America. All rights reserved.
ux 1
1
x uyuy x dy. 0 x 1,
x
u
> 1 2
b
a
a
0 b
0 emax b2 x 2 , a 2 y 2 dy dx,
PUTNAM EXAM CHALLENGE
x, y
f x, y ≤ gx, y
V 8
1
0
1
0
1 x 2 y 2 dx dy.
x 2 y 2 z 2 1
500
x 2 y 2 9
R:
f x, y x 2 y 2
200
0, 0, 4, 0, 4, 4, 0, 4
R:
f x, y 4x
m 6, n 4
m 4, n 8
4
1
2
1
x 3 y 3 dx dy
6
4
2
0
y cos x dx dy
m 10, n 20
m 4, n 8
2
0
4
0
20e x3 8 dy dx
1
0
2
0
sin x y dy dx
R.
x i , y j
n
i1 m
j1
f x i , y j A i
b
a
d
c
f x, y dA
n
m
c, d
a, b
c < d.
a < b
f x, y
P0 ≤ x ≤ 1, x ≤ y ≤ 1
f x, y e xy ,
0,
x ≥ 0, y ≥ 0
elsewhere
P0 ≤ x ≤ 1, 4 ≤ y ≤ 6
f x, y 1
279 x y,
0,
0 ≤ x ≤ 3, 3 ≤ y ≤ 6
elsewhere
P0 ≤ x ≤ 1, 1 ≤ y ≤ 2
f x, y 1
4 xy,
0,
0 ≤ x ≤ 2, 0 ≤ y ≤ 2
elsewhere
P0 ≤ x ≤ 2, 1 ≤ y ≤ 2
f x, y 1
10 ,
0,
0 ≤ x ≤ 5, 0 ≤ y ≤ 2
elsewhere
P[x, y R] R f x, y dA
f x, y dA 1
x, y
f x, y ≥ 0
en cualquier otro punto
en cualquier otro punto
en cualquier otro punto
en cualquier otro punto
Para discusión
72. Las siguientes integrales iteradas representan la solución al
mismo problema. ¿Cuál integral iterada es más fácil de evaluar?
Explicar el razonamiento.
Probability
A joint density function of the continuous random
variables and is a function satisfying the following
properties.
(a) for all (b)
(c)
In Exercises 73–76, show that the function is a joint density
function and find the required probability.
73.
74.
75.
76.
77. Approximation The base of a pile of sand at a cement plant is
rectangular with approximate dimensions of 20 meters by
30 meters. If the base is placed on the plane with one vertex
at the origin, the coordinates on the surface of the pile are
and
Approximate the volume of sand in the pile.
78. Programming Consider a continuous function over
the rectangular region with vertices and
where and Partition the intervals and
into and subintervals, so that the subintervals in a
given direction are of equal length. Write a program for a
graphing utility to compute the sum
where
is the center of a representative rectangle in
Approximation
In Exercises 79–82, (a) use a computer algebra
system to approximate the iterated integral, and (b) use the
program in Exercise 78 to approximate the iterated integral for
the given values of
and
79. 80.
81. 82.
Approximation
In Exercises 83 and 84, determine which value
best approximates the volume of the solid between the
-plane
and the function over the region. (Make your selection on the
basis of a sketch of the solid and not by performing any
calculations.)
83.
square with vertices
(a) (b) 600 (c) 50 (d) 125 (e) 1000
84.
circle bounded by
(a) 50 (b) 500 (c) (d) 5 (e) 5000
True or False?
In Exercises 85 and 86, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
85. The volume of the sphere is given by the
integral
86. If for all in and both and are
continuous over
then
87. Let Find the average value of on the interval
88. Find Hint: Evaluate
89. Determine the region in the -plane that maximizes the
value of
90. Determine the region in the -plane that minimizes the
value of
91. Find (Hint: Convert the integral
to a double integral.)
92. Use a geometric argument to show that
3
0
9 y 2
0
9 x 2 y 2 dx dy
9
2 .
2
0 arctan x arctan x dx.
R x2 y 2 4 dA.
xy
R
R 9 x2 y 2 dA.
xy
R
2
1
e xy dy.
0
e x
e 2x
x
dx.
0, 1 .
f
f x
x
1 e t2 dt.
R fx, y dA
R gx, y dA.
R, g
f
R,
x, y
f x, y
g x, y
V 8
1
0
1
0
1 x 2 y 2 dx dy.
x 2 y 2 z 2 1
500
x 2 y 2 9
R:
fx, y x 2 y 2
200
0, 0 , 4, 0 , 4, 4 , 0, 4
R:
fx, y
4x
xy
m 6, n 4
m 4, n 8
4
1
2
1
x 3
y 3 dx dy
6
4
2
0
y cos
x dx dy
m 10, n 20
m 4, n 8
2
0
4
0
20e x3 8 dy dx
1
0
2
0
sen x y dy dx
n.
m
R.
x i , y j
n
i 1
m
j 1
fx i , y j A i b
a
d
c
fx, y dA
n
m
c, d
a, b
c < d.
a < b
b, d ,
a, d ,
b, c ,
a, c ,
R
fx, y
25, 15, 3 .
15, 15, 7 ,
5, 15, 2 ,
25, 5, 4 ,
15, 5, 6 ,
5, 5, 3 ,
xy-
P 0 x 1, x y 1
fx, y
e x y ,
0,
x 0, y 0
elsewhere
P 0 x 1, 4 y 6
fx, y
1
27 9 x y ,
0,
0 x 3, 3 y 6
elsewhere
P 0 x 1, 1 y 2
fx, y
1
4 xy,
0,
0 x 2, 0 y 2
elsewhere
P 0 x 2, 1 y 2
fx, y
1
10 ,
0,
0 x 5, 0 y 2
elsewhere
P[ x, y R]
R
fx, y dA
fx, y dA 1
x, y
f x, y ~ 0
fx, y
y
x
14.2 Double Integrals and Volume 1003
72. The following iterated integrals represent the solution to the
same problem. Which iterated integral is easier to evaluate?
Explain your reasoning.
4
0
2
x 2
sen y 2 dy dx
2
0
2y
0
sen y 2 dx dy
CAPSTONE
CAS
93. Evaluate where and are
positive.
94. Show that if there does not exist a real-valued function
such that for all in the closed interval
These problems were composed by the Committee on the Putnam Prize
Competition. © The Mathematical Association of America. All rights reserved.
ux 1
1
x uyuy x dy. 0 x 1,
x
u
> 1 2
b
a
a
0 b
0 emax b2 x 2 , a 2 y 2 dy dx,
PUTNAM EXAM CHALLENGE
CAS
Preparación del examen Putnam
93. Evaluar donde a y b son positivos.
94. Probar que si no existe una función real u tal que, para
todo x en el intervalo cerrado ,
Estos problemas fueron preparados por el Committee on the Putnam Prize
Competition. © The Mathematical Association of America. Todos los derechos reservados.
Probability
A joint density function of the continuous random
variables and is a function satisfying the following
properties.
(a) for all (b)
(c)
In Exercises 73–76, show that the function is a joint density
function and find the required probability.
73.
74.
75.
76.
77. Approximation The base of a pile of sand at a cement plant is
rectangular with approximate dimensions of 20 meters by
30 meters. If the base is placed on the plane with one vertex
at the origin, the coordinates on the surface of the pile are
and
Approximate the volume of sand in the pile.
78. Programming Consider a continuous function over
the rectangular region with vertices and
where and Partition the intervals and
into and subintervals, so that the subintervals in a
given direction are of equal length. Write a program for a
graphing utility to compute the sum
where
is the center of a representative rectangle in
Approximation
In Exercises 79–82, (a) use a computer algebra
system to approximate the iterated integral, and (b) use the
program in Exercise 78 to approximate the iterated integral for
the given values of
and
79. 80.
81. 82.
Approximation
In Exercises 83 and 84, determine which value
best approximates the volume of the solid between the
-plane
and the function over the region. (Make your selection on the
basis of a sketch of the solid and not by performing any
calculations.)
83.
square with vertices
(a) (b) 600 (c) 50 (d) 125 (e) 1000
84.
circle bounded by
(a) 50 (b) 500 (c) (d) 5 (e) 5000
True or False?
In Exercises 85 and 86, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
85. The volume of the sphere is given by the
integral
86. If for all in and both and are
continuous over
then
87. Let Find the average value of on the interval
88. Find Hint: Evaluate
89. Determine the region in the -plane that maximizes the
value of
90. Determine the region in the -plane that minimizes the
value of
91. Find (Hint: Convert the integral
to a double integral.)
92. Use a geometric argument to show that
3
0
9 y 2
0
9 x 2 y 2 dx dy
9
2 .
2
0 arctan x arctan x dx.
R x2 y 2 4 dA.
xy
R
R 9 x 2 y 2 dA.
xy
R
2
1
e xy dy.
0
e x
e 2x
x
dx.
0, 1 .
f
f x
x
1 e t2 dt.
R fx, y dA
R gx, y dA.
R, g
f
R,
x, y
f x, y
g x, y
V 8
1
0
1
0
1 x 2 y 2 dx dy.
x 2 y 2 z 2 1
500
x 2 y 2 9
R:
fx, y x 2 y 2
200
0, 0 , 4, 0 , 4, 4 , 0, 4
R:
fx, y
4x
xy
m 6, n 4
m 4, n 8
4
1
2
1
x 3
y 3 dx dy
6
4
2
0
y cos
x dx dy
m 10, n 20
m 4, n 8
2
0
4
0
20e x3 8 dy dx
1
0
2
0
sen x y dy dx
n.
m
R.
x i , y j
n
i 1
m
j 1
fx i , y j A i b
a
d
c
fx, y dA
n
m
c, d
a, b
c < d.
a < b
b, d ,
a, d ,
b, c ,
a, c ,
R
fx, y
25, 15, 3 .
15, 15, 7 ,
5, 15, 2 ,
25, 5, 4 ,
15, 5, 6 ,
5, 5, 3 ,
xy-
P 0 x 1, x y 1
fx, y
e x y ,
0,
x 0, y 0
elsewhere
P 0 x 1, 4 y 6
fx, y
1
27 9 x y ,
0,
0 x 3, 3 y 6
elsewhere
P 0 x 1, 1 y 2
fx, y
1
4 xy,
0,
0 x 2, 0 y 2
elsewhere
P 0 x 2, 1 y 2
fx, y
1
10 ,
0,
0 x 5, 0 y 2
elsewhere
P[ x, y R]
R
fx, y dA
fx, y dA 1
x, y
f x, y ~ 0
fx, y
y
x
14.2 Double Integrals and Volume 1003
72. The following iterated integrals represent the solution to the
same problem. Which iterated integral is easier to evaluate?
Explain your reasoning.
4
0
2
x 2
sen y 2 dy dx
2
0
2y
0
sen y 2 dx dy
CAPSTONE
S
93. Evaluate where and are
positive.
94. Show that if there does not exist a real-valued function
such that for all in the closed interval
These problems were composed by the Committee on the Putnam Prize
Competition. © The Mathematical Association of America. All rights reserved.
ux 1
1
x u y u y x dy. 0 x 1,
x
u
> 1 2
b
a
a
0 b
0 emax b2 x 2 , a 2 y 2 dy dx,
PUTNAM EXAM CHALLENGE
Probability
A joint density function of the continuous random
variables and is a function satisfying the following
properties.
(a) for all (b)
(c)
In Exercises 73–76, show that the function is a joint density
function and find the required probability.
73.
74.
75.
76.
77. Approximation The base of a pile of sand at a cement plant is
rectangular with approximate dimensions of 20 meters by
30 meters. If the base is placed on the plane with one vertex
at the origin, the coordinates on the surface of the pile are
and
Approximate the volume of sand in the pile.
78. Programming Consider a continuous function over
the rectangular region with vertices and
where and Partition the intervals and
into and subintervals, so that the subintervals in a
given direction are of equal length. Write a program for a
graphing utility to compute the sum
where
is the center of a representative rectangle in
Approximation
In Exercises 79–82, (a) use a computer algebra
system to approximate the iterated integral, and (b) use the
program in Exercise 78 to approximate the iterated integral for
the given values of
and
79.
81.
Approximation In Exercises 83
best approximates the volume of
and the function over the region
basis of a sketch of the solid
calculations.)
83.
square with vertices
(a) (b) 600 (c) 50
84.
circle bounded by
(a) 50 (b) 500 (c)
True or False?
In Exercises 85 a
statement is true or false. If it is
example that shows it is false.
85. The volume of the sphere
integral
86. If for all
continuous over
then
87. Let Find the a
88. Find
89. Determine the region in th
value of
90. Determine the region in th
value of
91. Find
to a double integral.)
92. Use a geometric argument to s
3
0
9 y 2
0
9 x 2 y 2 dx d
2
0 arctan x arctan x
R x2 y 2 4 dA.
R
R 9 x2 y 2 dA.
R
0
e x
e 2x
x
dx.
0, 1 .
f x
x
1 e t2 dt.
R fx
R, x,
f x, y
g x, y
V 8
1
0
1
0
1 x 2 y 2 dx
x 2
500
x 2 y 2
R:
fx, y x 2 y 2
200
0, 0 ,
R:
fx, y
4x
m 4, n 8
6
4
2
0
y cos
x dx dy
m 4, n 8
1
0
2
0
sen x y dy dx
n.
m
R.
x i , y j
n
i 1
m
j 1
fx i , y j A i b
a
d
c
fx, y dA
n
m
c, d
a, b
c < d.
a < b
b, d ,
a, d ,
b, c ,
a, c ,
R
fx, y
25, 15, 3 .
15, 15, 7 ,
5, 15, 2 ,
25, 5, 4 ,
15, 5, 6 ,
5, 5, 3 ,
xy-
P 0 x 1, x y 1
fx, y
e x y ,
0,
x 0, y 0
elsewhere
P 0 x 1, 4 y 6
fx, y
1
27 9 x y ,
0,
0 x 3, 3 y 6
elsewhere
P 0 x 1, 1 y 2
fx, y
1
4 xy,
0,
0 x 2, 0 y 2
elsewhere
P 0 x 2, 1 y 2
fx, y
1
10 ,
0,
0 x 5, 0 y 2
elsewhere
P[ x, y R]
R
fx, y dA
fx, y dA 1
x, y
f x, y ~ 0
fx, y
y
x
14.2 Double In
72. The following iterated integrals represent the solution to the
same problem. Which iterated integral is easier to evaluate?
Explain your reasoning.
4
0
2
x 2
sen y 2 dy dx
2
0
2y
0
sen y 2 dx dy
CAPSTONE
CAS
93. Evaluate
positive.
94. Show that if there doe
tion such that for all in t
These problems were composed by th
Competition. © The Mathematical Associa
u x 1
1
x uyuy
x
x
u
> 1 2
a
0 b
0 emax b2 x 2 , a 2 y 2
PUTNAM EXAM CHALLEN
0 ≤ x ≤ 1
> 1 2
Probability
A joint density function of the continuous random
variables and is a function satisfying the following
properties.
(a) for all (b)
(c)
In Exercises 73–76, show that the function is a joint density
function and find the required probability.
73.
74.
75.
76.
77. Approximation The base of a pile of sand at a cement plant is
rectangular with approximate dimensions of 20 meters by
30 meters. If the base is placed on the plane with one vertex
at the origin, the coordinates on the surface of the pile are
and
Approximate the volume of sand in the pile.
78. Programming Consider a continuous function over
the rectangular region with vertices and
where and Partition the intervals and
into and subintervals, so that the subintervals in a
given direction are of equal length. Write a program for a
graphing utility to compute the sum
where
is the center of a representative rectangle in
Approximation
In Exercises 79–82, (a) use a computer algebra
system to approximate the iterated integral, and (b) use the
program in Exercise 78 to approximate the iterated integral for
the given values of
and
79. 80.
81. 82.
Approximation
In Exercises 83 and 84, determine which value
best approximates the volume of the solid between the
-plane
and the function over the region. (Make your selection on the
basis of a sketch of the solid and not by performing any
calculations.)
83.
square with vertices
(a) (b) 600 (c) 50 (d) 125 (e) 1000
84.
circle bounded by
(a) 50 (b) 500 (c) (d) 5 (e) 5000
True or False?
In Exercises 85 and 86, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
85. The volume of the sphere is given by the
integral
86. If for all in and both and are
continuous over
then
87. Let Find the average value of on the interval
88. Find Hint: Evaluate
89. Determine the region in the -plane that maximizes the
value of
90. Determine the region in the -plane that minimizes the
value of
91. Find (Hint: Convert the integral
to a double integral.)
92. Use a geometric argument to show that
3
0
9 y 2
0
9 x 2 y 2 dx dy
9
2 .
2
0 arctan x arctan x dx.
R x2 y 2 4 dA.
xy
R
R 9 x2 y 2 dA.
xy
R
2
1
e xy dy.
0
e x
e 2x
x
dx.
0, 1 .
f
f x
x
1 e t2 dt.
R fx, y dA
R gx, y dA.
R, g
f
R,
x, y
f x, y
g x, y
V 8
1
0
1
0
1 x 2 y 2 dx dy.
x 2 y 2 z 2 1
500
x 2 y 2 9
R:
fx, y x 2 y 2
200
0, 0 , 4, 0 , 4, 4 , 0, 4
R:
fx, y
4x
xy
m 6, n 4
m 4, n 8
4
1
2
1
x 3
y 3 dx dy
6
4
2
0
y cos
x dx dy
m 10, n 20
m 4, n 8
2
0
4
0
20e x3 8 dy dx
1
0
2
0
sen x y dy dx
n.
m
R.
x i , y j
n
i 1
m
j 1
fx i , y j A i b
a
d
c
fx, y dA
n
m
c, d
a, b
c < d.
a < b
b, d ,
a, d ,
b, c ,
a, c ,
R
fx, y
25, 15, 3 .
15, 15, 7 ,
5, 15, 2 ,
25, 5, 4 ,
15, 5, 6 ,
5, 5, 3 ,
xy-
P 0 x 1, x y 1
fx, y
e x y ,
0,
x 0, y 0
elsewhere
P 0 x 1, 4 y 6
fx, y
1
27 9 x y ,
0,
0 x 3, 3 y 6
elsewhere
P 0 x 1, 1 y 2
fx, y
1
4 xy,
0,
0 x 2, 0 y 2
elsewhere
P 0 x 2, 1 y 2
fx, y
1
10 ,
0,
0 x 5, 0 y 2
elsewhere
P[ x, y R]
R
fx, y dA
fx, y dA 1
x, y
f x, y ~ 0
f x, y
y
x
14.2 Double Integrals and Volume 1003
72. The following iterated integrals represent the solution to the
same problem. Which iterated integral is easier to evaluate?
Explain your reasoning.
4
0
2
x 2
sen y 2 dy dx
2
0
2y
0
sen y 2 dx dy
CAPSTONE
CAS
93. Evaluate where and are
positive.
94. Show that if there does not exist a real-valued function
such that for all in the closed interval
These problems were composed by the Committee on the Putnam Prize
Competition. © The Mathematical Association of America. All rights reserved.
ux 1
1
x uyuy x dy. 0 x 1,
x
u
> 1 2
b
a
a
0 b
0 emax b2 x 2 , a 2 y 2 dy dx,
PUTNAM EXAM CHALLENGE
e máx
sen