Calculo 2 De dos variables_9na Edición - Ron Larson

1002 CAPÍTULO 14 Integración múltiple
37. primer octante
38. primer octante
39. primer octante
40.
En los ejercicios 41 a 46, establecer una integral doble para
encontrar el volumen de una región sólida limitada por las gráficas
de las ecuaciones. No evaluar la integral.
41. 42.
43.
44.
En los ejercicios 47 a 50, utilizar un sistema algebraico por computadora
y hallar el volumen del sólido limitado o acotado por las
gráficas de las ecuaciones.
47.
48. primer octante
49.
50.
51. Si es una función continua tal que en una
región R de área 1, demostrar que
52. Hallar el volumen del sólido que se encuentra en el primer
octante, acotado por los planos coordenados y el plano
donde
y
En los ejercicios 53 a 58, trazar la región de integración. Después
evaluar la integral iterada y, si es necesario, cambiar el orden de
integración.
53. 54.
57.
Valor promedio
En los ejercicios 59 a 64, encontrar el valor
promedio de f (x, y) sobre la región R.
59.
rectángulo con vértices
60. f(x, y) 2xy
rectángulo con vértices (0, 0), (5, 0), (5, 3), (0, 3)
61.
cuadrado con vértices
62.
R: triángulo con vértices (0, 0), (1, 0), (1, 1)
63.
R: triángulo con vértices (0, 0), (0, 1), (1, 1)
64.
R: rectángulo con vértices (0, 0), (, 0),(, ), (0, )
65. Producción promedio La función de producción Cobb-Douglas
para un fabricante de automóviles es
donde x es el número de unidades de trabajo y y es el número de
unidades de capital. Estimar el nivel promedio de producción si
el número x de unidades de trabajo varía entre 200 y 250 y el
número y de unidades de capital varía entre 300 y 325.
66. Temperatura promedio La temperatura en grados Celsius
sobre la superficie de una placa metálica es T(x, y) 20 4x 2
y 2 , donde x y y están medidas en centímetros. Estimar la temperatura
promedio si x varía entre 0 y 2 centímetros y y varía
entre 0 y 4 centímetros.
f x, y 100x 0.6 y 0.4
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
f x, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
fx, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
f x, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
f x, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
fx, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
fx, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
f x, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
f x, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
f x, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
f x, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
0, 0, 2, 0, 2, 2, 0, 2
R:
f x, y x 2 y 2
R:
0, 0, 4, 0, 4, 2, 0, 2
R:
f x, y x
1
0
arccos y
0
sin x1 sin 2 x dx dy
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 ≤ R f x, y dA ≤ 1.
0 ≤ f x, y ≤ 1
f
z ln1 x y, z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
fx, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
fx, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
z sin 2 x, z 0, 0 ≤ x ≤ , 0 ≤ y ≤ 5
z x 2 y 2 , x 2 y 2 4, z 0
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
f x, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
f x, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
fx, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
fx, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
f x, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
f x, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
z 1
1 y 2, x 0, x 2, y ≥ 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
Desarrollo de conceptos
67. Enunciar la definición de integral doble. Dar la interpretación
geométrica de una integral doble si el integrando es
una función no negativa sobre la región de integración.
68. Sea R una región en el plano xy cuya área es B. Si ƒ(x, y) k
para todo punto (x, y) en R, ¿cuál es el valor de
Explicar.
69. Sea R un condado en la parte norte de Estados Unidos, y sea
ƒ(x, y) la precipitación anual de nieve en el punto (x, y) de R.
Interpretar cada uno de los siguientes.
a) b)
70. Identificar la expresión que es inválida. Explicar el razonamiento.
71. Sea la región plana R un círculo unitario y el máximo valor
de f sobre R sea 6. ¿Es el valor más grande posible de
igual a 6? ¿Por qué sí o por qué no? Si es
no, ¿cuál es el valor más grande posible?
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
f x, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
fx, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
f x, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
Rf x, y dA
RdA
Rf x, y dA
R f x, y dA?
sen
sen
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
f x, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
fx, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R fx, y dA? R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
CAS
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
f x, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
fx, y
x
R.
f x, y
2
0
2
12x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
f x, y dy dx
2
0
3
x
f x, y dy dx
2
0
y
0
f x, y dy dx
2
0
3
0
f x, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R f x, y dA?
R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS
sen
37. first octant
38. first octant
39. first octant
40.
In Exercises 41–46, set up a double integral to find the volume
of the solid region bounded by the graphs of the equations. Do
not evaluate the integral.
41. 42.
43.
44.
45.
46.
In Exercises 47–50, use a computer algebra system to find the
volume of the solid bounded by the graphs of the equations.
47.
48. first octant
49.
50.
51. If is a continuous function such that over a
region
of area 1, prove that
52. Find the volume of the solid in the first octant bounded by the
coordinate planes and the plane
where
and
In Exercises 53–58, sketch the region of integration. Then
evaluate the iterated integral, switching the order of integration
if necessary.
53. 54.
55. 56.
57.
58.
Average Value
In Exercises 59– 64, find the average value of
over the region
59.
rectangle with vertices
60.
rectangle with vertices
61.
square with vertices
62.
triangle with vertices
63.
triangle with vertices
64.
rectangle with vertices
65. Average Production The Cobb-Douglas production function
for an automobile manufacturer is
where
is the number of units of labor and is the number of units of
capital. Estimate the average production level if the number of
units of labor varies between 200 and 250 and the number of
units of capital varies between 300 and 325.
66. Average Temperature The temperature in degrees Celsius on
the surface of a metal plate is
where
and
are measured in centimeters. Estimate the average
temperature if varies between 0 and 2 centimeters and varies
between 0 and 4 centimeters.
y
x
y
x
Tx, y 20 4x 2 y 2 ,
y
x
y
x
fx, y 100x 0.6 y 0.4 ,
0, 0 , , 0 , , , 0,
R:
fx, y sen x y
0, 0 , 0, 1 , 1, 1
R:
fx, y e x y 0, 0 , 1, 0 , 1, 1
R:
fx, y
1
x
y
0, 0 , 2, 0 , 2, 2 , 0, 2
R:
f x, y x 2 y 2 0, 0 , 5, 0 , 5, 3 , 0, 3
R:
fx, y
2xy
0, 0 , 4, 0 , 4, 2 , 0, 2
R:
fx, y
x
R.
f x, y
2
0
2
1 2 x 2
y cos y dy dx
1
0
arccos y
0
sin x 1 sin 2 x dx dy
3
0
1
y 3
1
1 x 4 dx dy
2
2
4 x 2
4 x 2 4 y 2 dy dx
ln 10
0
10
e x 1
ln y dy dx
1
0
12
y 2
e x2 dx dy
c > 0.
b > 0,
a > 0,
xa yb zc 1,
0 R f x, y dA 1.
R
0 fx, y 1
f
z ln 1 x y , z 0, y 0, x 0, x 4 y
z
2
1 x 2 y 2, z 0, y 0, x 0, y 0.5x 1
x 2 9 y, z 2 9 y,
z 9 x 2 y 2 , z 0
z x 2 y 2 , z 18 x 2 y 2
z x 2 2y 2 , z 4y
z sin 2 x, z 0, 0 x , 0 y 5
z x 2 y 2 , x 2 y 2 4, z 0
z = 2x
y
x
4
2
−2
−2
1
2
1
z
z = x 2 + y 2
z = 4 − x 2 − y 2
z = 4 − 2x
y
x
4
2
2
z
z
1
1 y 2, x 0, x 2, y 0
z x y, x 2 y 2 4,
y 4 x 2 , z 4 x 2 ,
x 2 z 2 1, y 2 z 2 1,
1002 Chapter 14 Multiple Integration
CAS
67. State the definition of a double integral. If the integrand is
a nonnegative function over the region of integration, give
the geometric interpretation of a double integral.
68. Let be a region in the plane whose area is If
for every point in what is the value of
Explain.
69. Let represent a county in the northern part of the United
States, and let
represent the total annual snowfall at
the point in Interpret each of the following.
(a)
(b)
70. Identify the expression that is invalid. Explain your
reasoning.
a) b)
c) d)
71. Let the plane region be a unit circle and let the maximum
value of on be 6. Is the greatest possible value of
equal to 6? Why or why not? If not, what
is the greatest possible value?
R
fx, y dy dx R
f
R
2
0
x
0
fx, y dy dx
2
0
3
x
fx, y dy dx
2
0
y
0
fx, y dy dx
2
0
3
0
fx, y dy dx
R
fx, y dA
R
dA
R
fx, y dA
R.
x, y
fx, y
R
R f x, y dA?
R,
x, y
f x, y
k
B.
xy-
R
WRITING ABOUT CONCEPTS