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Deck 14: Multiple Integration
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Question
Use spherical coordinates to find the volume of the solid inside 
and outside 
, and above the xy-plane.
A)
B)
C)
D)
E)
and outside , and above the xy-plane. A)
B)
C)
D)
E)
Question
Given 
use polar coordinates to set up and evaluate the double integral 
.
A)
B)
C)
D)
E)
use polar coordinates to set up and evaluate the double integral .A)
B)
C)
D)
E)
Question
Find the mass of the lamina described by the inequalities 
and 
, given that its density is 
.
and , given that its density is . Question
Find the average value of 
over the region Q, where Q is a tetrahedron in the first octant with vertices 
. The average value of a continuous function 
over a solid region Q is 
, where V is the volume of the solid region Q.
A) 
B) 
C) 
D) 
E)
over the region Q, where Q is a tetrahedron in the first octant with vertices . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q. A)
B)
C)
D)
E)
Question
Find the mass of the lamina bounded by the graphs of the equations 
for the density 
.
A)
B)
C)
D)
E)
for the density . A)
B)
C)
D)
E)
Question
Find the centroid of the solid region bounded by the graphs of the equations. Use a computer algebra system to evaluate the triple integral. (Assume uniform density and find the center of mass.) 
A)
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the center of mass of the lamina bounded by the graphs of the equations 
for the density 
.
A)
B)
C)
D)
E)
for the density .A)
B)
C)
D)
E)
Question
Evaluate the iterated integral 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
Find the average value of 
over the region Q, where Q is a cube in the first octant bounded by the coordinate planes, and the planes 
. The average value of a continuous function 
over a solid region Q is 
, where V is the volume of the solid region Q.
A) 
B) 
C) 
D) 
E) 
over the region Q, where Q is a cube in the first octant bounded by the coordinate planes, and the planes . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q. A)
B)
C)
D)
E)
Question
Write a double integral that represents the surface area of 
over the region R: triangle with vertices 
. Use a computer algebra system to evaluate the double integral. Round your answer to two decimal places.
A)
B)
C)
D)
E)
over the region R: triangle with vertices . Use a computer algebra system to evaluate the double integral. Round your answer to two decimal places. A)
B)
C)
D)
E)
Question
Find a transformation 
that when applied to region 
, its image will be 
. 
A)
B)
C)
D)
E)
that when applied to region , its image will be . A)
B)
C)
D)
E)
Question
Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Identify the region of integration for the following integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
The area of a region R is given by the iterated integral 
. Switch the order of integration and show that both orders yield the same area. What is this area?
A)
B)
C)
D)
E)
. Switch the order of integration and show that both orders yield the same area. What is this area?A)
B)
C)
D)
E)
Question
Find the center of mass of the rectangular lamina with vertices 
for the density 
.
A) 
B) 
C) 
D) 
E) 
for the density .A)
B)
C)
D)
E)
Question
Find the mass and center of mass of the lamina bounded by the graphs of the equations given below for the given density. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Rewrite the iterated integral 
using the order dydxdz.
A) 
B) 
C) 
D) 
E) 
using the order dydxdz.
A)
B)
C)
D)
E)
Question
Use cylindrical coordinates to find the volume of the solid inside both 
and 
.
A)
B)
C)
D)
E)
and . A)
B)
C)
D)
E)
Question
Set up the double integral required to find the moment of inertia 
, about the line 
of the lamina bounded by the graphs of the equations 
and 
for the density 
. Use a computer algebra system to evaluate the double integral.
A)
B)
C)
D)
E)
, about the line of the lamina bounded by the graphs of the equations and for the density . Use a computer algebra system to evaluate the double integral.A)
B)
C)
D)
E)
Question
Use a double integral to find the volume of the indicated solid. 
A)
B)
C)
D)
E) none of these
A)
B)
C)
D)
E) none of these
Question
Evaluate the following iterated integral by converting to polar coordinates. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
If 
and 
then the Jacobian of x, y, and z with respect to u, v, and w is 
.
Find the Jacobian for the following change of variables:
A)
B)
C)
D)
E)
and then the Jacobian of x, y, and z with respect to u, v, and w is .
Find the Jacobian for the following change of variables:
A)
B)
C)
D)
E)
Question
Use a change of variables to find the volume of the solid region lying below the surface 
and above the plane region R: region bounded by the square with vertices 
.
A)
B)
C)
D)
E)
and above the plane region R: region bounded by the square with vertices .A)
B)
C)
D)
E)
Question
Find the area of the surface for the portion of the paraboloid 
in the first octant.
A)
B)
C)
D)
E)
in the first octant.A)
B)
C)
D)
E)
Question
Use cylindrical coordinates to find the volume of the solid bounded above by 
and below by 
.
A)
B)
C)
D)
E)
and below by .A)
B)
C)
D)
E)
Question
Use a double integral to find the area enclosed by the graph of 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find the area of the surface for the portion of the sphere 
inside the cylinder 
.
A)
B)
C)
D)
E)
inside the cylinder .A)
B)
C)
D)
E)
Question
Use the indicated change of variables to evaluate the following double integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Evaluate the following integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Evaluate the following iterated integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find 
of the center of mass of the solid of given density 
bounded by the graphs of the equations 
.
A)
B)
C)
D)
E)
of the center of mass of the solid of given density bounded by the graphs of the equations . A)
B)
C)
D)
E)
Question
Evaluate the following iterated integral. 
A) 43
B) 42
C) 36
D) 33
E) 38
A) 43
B) 42
C) 36
D) 33
E) 38
Question
Find the average value of 
over the region R, where R is a triangle with vertices 
.
A)
B)
C)
D)
E)
over the region R, where R is a triangle with vertices . A)
B)
C)
D)
E)
Question
Use spherical coordinates to find the mass of the sphere 
with the given density. The density at any point is proportional to the distance of the point from the z-axis.
A)
B)
C)
D)
E)
with the given density. The density at any point is proportional to the distance of the point from the z-axis.A)
B)
C)
D)
E)
Question
Evaluate the double integral below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations given below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the area of the portion of the surface 
that lies above the region 
. Round your answer to two decimal places.
A) 29.20
B) 58.39
C) 439.36
D) 36.18
E) 1.64
that lies above the region . Round your answer to two decimal places.A) 29.20
B) 58.39
C) 439.36
D) 36.18
E) 1.64
Question
Evaluate the iterated integral 
by converting to polar coordinates.
A)
B)
C)
D)
E)
by converting to polar coordinates.A)
B)
C)
D)
E)
Question
Evaluate the following improper integral. 
A)
B)
C)
D)
E) The integral does not converge.
A)
B)
C)
D)
E) The integral does not converge.
Question
Use cylindrical coordinates to find the volume of the solid inside the sphere 
and above the upper nappe of the cone 
.
A)
B)
C)
D)
E)
and above the upper nappe of the cone .A)
B)
C)
D)
E)
Question
Evaluate the following iterated integral. 
A) -514
B) -499
C) -473
D) -463
E) -399
A) -514
B) -499
C) -473
D) -463
E) -399
Question
Evaluate 
.
A) 6
B) 14
C) 12
D) 18
E) 8
.A) 6
B) 14
C) 12
D) 18
E) 8
Question
Find the center of mass of the lamina bounded by the graphs of the equations 
for the density 
.
A)
B)
C)
D)
E)
for the density . A)
B)
C)
D)
E)
Question
Evaluate the following iterated integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use a double integral to find the volume of the indicated solid. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Set up and evaluate a double integral required to find the moment of inertia, I, about the given line, of the lamina bounded by the graphs of the following equations. Use a computer algebra system to evaluate the double integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use an iterated integral to find the area of the region bounded by 
.
A)
B)
C)
D)
E) The integral is improper and does not converge.
.A)
B)
C)
D)
E) The integral is improper and does not converge.
Question
Use a double integral to find the area enclosed by the graph of 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Evaluate the double integral below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
The area of a region R is given by the iterated integral 
. Switch the order of integration and show that both orders yield the same area. What is this area?
A) 101
B) 20
C) 30
D) 10
E) 5
. Switch the order of integration and show that both orders yield the same area. What is this area?A) 101
B) 20
C) 30
D) 10
E) 5
Question
Evaluate the double integral below. 
A)
B) 0
C)
D)
E)
A)
B) 0
C)
D)
E)
Question
Use spherical coordinates to find the volume of the solid inside the torus given by 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
Use a change of variables to find the volume of the solid region lying below the surface 
and above the plane region R: region bounded by the parallelogram with vertices 
. Round your answer to two decimal places.
A)
B)
C)
D)
E)
and above the plane region R: region bounded by the parallelogram with vertices . Round your answer to two decimal places.A)
B)
C)
D)
E)
Question
Evaluate the iterated integral below. Note that it is necessary to switch the order of integration. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use a double integral to find the area of the region inside the circle 
and outside the cardioid 
. Round your answer to two decimal places.
A) 44.76
B) 44.88
C) 17.88
D) 54.88
E) 21.88
and outside the cardioid . Round your answer to two decimal places.A) 44.76
B) 44.88
C) 17.88
D) 54.88
E) 21.88
Question
Use cylindrical coordinates to find the volume of the cone 
where 
and 
. 
A)
B)
C)
D)
E)
where and . A)
B)
C)
D)
E)
Question
Use a double integral to find the area of the shaded region as shown in the figure below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations 
if one-tenth of the volume of the solid is removed. Round your answer to four decimal places.
A) 1.2983
B) 36.5966
C) 3.2983
D) 5.5966
E) 37.2983
if one-tenth of the volume of the solid is removed. Round your answer to four decimal places.A) 1.2983
B) 36.5966
C) 3.2983
D) 5.5966
E) 37.2983
Question
Find the mass of the lamina bounded by the graphs of the equations 
for the density 
.
A)
B)
C)
D)
E)
for the density . A)
B)
C)
D)
E)
Question
Find the mass of the triangular lamina with vertices 
for the density 
.
A) 11,337,408k
B) 22,674,826k
C) 11,337,398k
D) 11,337,413k
E) 22,674,816k
for the density . A) 11,337,408k
B) 22,674,826k
C) 11,337,398k
D) 11,337,413k
E) 22,674,816k
Question
Find the area of the portion of the surface 
that lies above the region 
. Round your answer to two decimal places.
A) 0.67
B) 3.87
C) 30.30
D) 10.64
E) 30.84
that lies above the region . Round your answer to two decimal places.A) 0.67
B) 3.87
C) 30.30
D) 10.64
E) 30.84
Question
Evaluate the iterated integral below. Note that it is necessary to switch the order of integration. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Sketch the region R of integration and then switch the order of integration for the following integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the area of the surface given by 
over the region R. 
A)
B)
C)
D)
E)
over the region R. A)
B)
C)
D)
E)
Question
Find the Jacobian 
for the following change of variables: 
, 
A)
B)
C)
D)
E)
for the following change of variables: , A)
B)
C)
D)
E)
Question
The area of a region R is given by the iterated integrals 
. Switch the order of integration and show that both orders yield the same area. What is this area?
A) 14
B) 197
C) 28
D) 150
E) 27
. Switch the order of integration and show that both orders yield the same area. What is this area?A) 14
B) 197
C) 28
D) 150
E) 27
Question
Use an iterated integral to find the area of the region bounded by 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
Question
Set up a double integral that gives the area of the surface on the graph of 
over the region 
.
A)
B)
C)
D)
E)
over the region .A)
B)
C)
D)
E)
Question
Evaluate the integral 
by switching the order of integration. Round your answer to two decimal places.
A) 5.16
B) 48.66
C) 15.38
D) 13.38
E) 56.14
by switching the order of integration. Round your answer to two decimal places.A) 5.16
B) 48.66
C) 15.38
D) 13.38
E) 56.14
Question
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations 
and 
in the first octant.
A) 273,375
B) 30,375
C) 182,250
D) 91,125
E) 60,750
and in the first octant.A) 273,375
B) 30,375
C) 182,250
D) 91,125
E) 60,750
Question
Find the area of the surface of the portion of the plane 
in the first octant.
A)
B)
C)
D)
E)
in the first octant.A)
B)
C)
D)
E)
Question
Use a double integral to find the volume of the indicated solid. 
A) 16
B) 9
C) 4
D) 20
E) 6
A) 16
B) 9
C) 4
D) 20
E) 6
Question
Evaluate the following integral. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Set up a triple integral for the volume of the solid bounded by the coordinate planes and the plane given below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use cylindrical coordinates to find the mass of the solid 
where 
.
A)
B)
C)
D)
E)
where .A)
B)
C)
D)
E)
Question
Consider the region 
in the xy-plane bounded by the ellipse 
and the transformation 
and 
. Find 
.
A)
B) 0
C)
D)
E)
in the xy-plane bounded by the ellipse and the transformation and . Find .A)
B) 0
C)
D)
E)
Question
Use an iterated integral to find the area of the region bounded by the graphs of the equations 
and 
.
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
Question
Use polar coordinates to describe the region as shown in the figure below: 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the area of the portion of the surface 
that lies above the region 
. Round your answer to two decimal places.
A) 81.00
B) 88.83
C) 508.94
D) 254.47
E) 799.44
that lies above the region . Round your answer to two decimal places. A) 81.00
B) 88.83
C) 508.94
D) 254.47
E) 799.44
Question
Find the Jacobian for the change of variables given below. 
, 
A)
B)
C)
D)
E)
, A)
B)
C)
D)
E)
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Deck 14: Multiple Integration
1
Use spherical coordinates to find the volume of the solid inside and outside , and above the xy-plane.
A)
B)
C)
D)
E)
and outside , and above the xy-plane. A)
B)
C)
D)
E)
C
2
Given use polar coordinates to set up and evaluate the double integral .
A)
B)
C)
D)
E)
use polar coordinates to set up and evaluate the double integral .A)
B)
C)
D)
E)
E
3
Find the mass of the lamina described by the inequalities and , given that its density is .
and , given that its density is .A
4
Find the average value of over the region Q, where Q is a tetrahedron in the first octant with vertices . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q.
A)
B)
C)
D)
E)
over the region Q, where Q is a tetrahedron in the first octant with vertices . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q. A)
B)
C)
D)
E)
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5
Find the mass of the lamina bounded by the graphs of the equations for the density .
A)
B)
C)
D)
E)
for the density . A)
B)
C)
D)
E)
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6
Find the centroid of the solid region bounded by the graphs of the equations. Use a computer algebra system to evaluate the triple integral. (Assume uniform density and find the center of mass.)
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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7
Find the center of mass of the lamina bounded by the graphs of the equations for the density .
A)
B)
C)
D)
E)
for the density .A)
B)
C)
D)
E)
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8
Evaluate the iterated integral .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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9
Find the average value of over the region Q, where Q is a cube in the first octant bounded by the coordinate planes, and the planes . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q.
A)
B)
C)
D)
E)
over the region Q, where Q is a cube in the first octant bounded by the coordinate planes, and the planes . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q. A)
B)
C)
D)
E)
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10
Write a double integral that represents the surface area of over the region R: triangle with vertices . Use a computer algebra system to evaluate the double integral. Round your answer to two decimal places.
A)
B)
C)
D)
E)
over the region R: triangle with vertices . Use a computer algebra system to evaluate the double integral. Round your answer to two decimal places. A)
B)
C)
D)
E)
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11
Find a transformation that when applied to region , its image will be .
A)
B)
C)
D)
E)
that when applied to region , its image will be . A)
B)
C)
D)
E)
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12
Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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13
Identify the region of integration for the following integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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14
The area of a region R is given by the iterated integral . Switch the order of integration and show that both orders yield the same area. What is this area?
A)
B)
C)
D)
E)
. Switch the order of integration and show that both orders yield the same area. What is this area?A)
B)
C)
D)
E)
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15
Find the center of mass of the rectangular lamina with vertices for the density .
A)
B)
C)
D)
E)
for the density .A)
B)
C)
D)
E)
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16
Find the mass and center of mass of the lamina bounded by the graphs of the equations given below for the given density.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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17
Rewrite the iterated integral using the order dydxdz.
A)
B)
C)
D)
E)
using the order dydxdz.
A)
B)
C)
D)
E)
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18
Use cylindrical coordinates to find the volume of the solid inside both and .
A)
B)
C)
D)
E)
and . A)
B)
C)
D)
E)
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19
Set up the double integral required to find the moment of inertia , about the line of the lamina bounded by the graphs of the equations and for the density . Use a computer algebra system to evaluate the double integral.
A)
B)
C)
D)
E)
, about the line of the lamina bounded by the graphs of the equations and for the density . Use a computer algebra system to evaluate the double integral.A)
B)
C)
D)
E)
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20
Use a double integral to find the volume of the indicated solid.
A)
B)
C)
D)
E) none of these
A)
B)
C)
D)
E) none of these
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21
Evaluate the following iterated integral by converting to polar coordinates.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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22
If and then the Jacobian of x, y, and z with respect to u, v, and w is .
Find the Jacobian for the following change of variables:
A)
B)
C)
D)
E)
and then the Jacobian of x, y, and z with respect to u, v, and w is .
Find the Jacobian for the following change of variables:
A)
B)
C)
D)
E)
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23
Use a change of variables to find the volume of the solid region lying below the surface and above the plane region R: region bounded by the square with vertices .
A)
B)
C)
D)
E)
and above the plane region R: region bounded by the square with vertices .A)
B)
C)
D)
E)
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24
Find the area of the surface for the portion of the paraboloid in the first octant.
A)
B)
C)
D)
E)
in the first octant.A)
B)
C)
D)
E)
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25
Use cylindrical coordinates to find the volume of the solid bounded above by and below by .
A)
B)
C)
D)
E)
and below by .A)
B)
C)
D)
E)
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26
Use a double integral to find the area enclosed by the graph of .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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27
Find the area of the surface for the portion of the sphere inside the cylinder .
A)
B)
C)
D)
E)
inside the cylinder .A)
B)
C)
D)
E)
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28
Use the indicated change of variables to evaluate the following double integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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29
Evaluate the following integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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30
Evaluate the following iterated integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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31
Find of the center of mass of the solid of given density bounded by the graphs of the equations .
A)
B)
C)
D)
E)
of the center of mass of the solid of given density bounded by the graphs of the equations . A)
B)
C)
D)
E)
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32
Evaluate the following iterated integral.
A) 43
B) 42
C) 36
D) 33
E) 38
A) 43
B) 42
C) 36
D) 33
E) 38
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33
Find the average value of over the region R, where R is a triangle with vertices .
A)
B)
C)
D)
E)
over the region R, where R is a triangle with vertices . A)
B)
C)
D)
E)
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34
Use spherical coordinates to find the mass of the sphere with the given density. The density at any point is proportional to the distance of the point from the z-axis.
A)
B)
C)
D)
E)
with the given density. The density at any point is proportional to the distance of the point from the z-axis.A)
B)
C)
D)
E)
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35
Evaluate the double integral below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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36
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations given below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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37
Find the area of the portion of the surface that lies above the region . Round your answer to two decimal places.
A) 29.20
B) 58.39
C) 439.36
D) 36.18
E) 1.64
that lies above the region . Round your answer to two decimal places.A) 29.20
B) 58.39
C) 439.36
D) 36.18
E) 1.64
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38
Evaluate the iterated integral by converting to polar coordinates.
A)
B)
C)
D)
E)
by converting to polar coordinates.A)
B)
C)
D)
E)
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39
Evaluate the following improper integral.
A)
B)
C)
D)
E) The integral does not converge.
A)
B)
C)
D)
E) The integral does not converge.
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40
Use cylindrical coordinates to find the volume of the solid inside the sphere and above the upper nappe of the cone .
A)
B)
C)
D)
E)
and above the upper nappe of the cone .A)
B)
C)
D)
E)
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41
Evaluate the following iterated integral.
A) -514
B) -499
C) -473
D) -463
E) -399
A) -514
B) -499
C) -473
D) -463
E) -399
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42
Evaluate .
A) 6
B) 14
C) 12
D) 18
E) 8
.A) 6
B) 14
C) 12
D) 18
E) 8
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43
Find the center of mass of the lamina bounded by the graphs of the equations for the density .
A)
B)
C)
D)
E)
for the density . A)
B)
C)
D)
E)
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44
Evaluate the following iterated integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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45
Use a double integral to find the volume of the indicated solid.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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46
Set up and evaluate a double integral required to find the moment of inertia, I, about the given line, of the lamina bounded by the graphs of the following equations. Use a computer algebra system to evaluate the double integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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47
Use an iterated integral to find the area of the region bounded by .
A)
B)
C)
D)
E) The integral is improper and does not converge.
.A)
B)
C)
D)
E) The integral is improper and does not converge.
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48
Use a double integral to find the area enclosed by the graph of .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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49
Evaluate the double integral below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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50
The area of a region R is given by the iterated integral . Switch the order of integration and show that both orders yield the same area. What is this area?
A) 101
B) 20
C) 30
D) 10
E) 5
. Switch the order of integration and show that both orders yield the same area. What is this area?A) 101
B) 20
C) 30
D) 10
E) 5
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51
Evaluate the double integral below.
A)
B) 0
C)
D)
E)
A)
B) 0
C)
D)
E)
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52
Use spherical coordinates to find the volume of the solid inside the torus given by .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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53
Use a change of variables to find the volume of the solid region lying below the surface and above the plane region R: region bounded by the parallelogram with vertices . Round your answer to two decimal places.
A)
B)
C)
D)
E)
and above the plane region R: region bounded by the parallelogram with vertices . Round your answer to two decimal places.A)
B)
C)
D)
E)
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54
Evaluate the iterated integral below. Note that it is necessary to switch the order of integration.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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55
Use a double integral to find the area of the region inside the circle and outside the cardioid . Round your answer to two decimal places.
A) 44.76
B) 44.88
C) 17.88
D) 54.88
E) 21.88
and outside the cardioid . Round your answer to two decimal places.A) 44.76
B) 44.88
C) 17.88
D) 54.88
E) 21.88
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56
Use cylindrical coordinates to find the volume of the cone where and .
A)
B)
C)
D)
E)
where and . A)
B)
C)
D)
E)
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57
Use a double integral to find the area of the shaded region as shown in the figure below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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58
Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations if one-tenth of the volume of the solid is removed. Round your answer to four decimal places.
A) 1.2983
B) 36.5966
C) 3.2983
D) 5.5966
E) 37.2983
if one-tenth of the volume of the solid is removed. Round your answer to four decimal places.A) 1.2983
B) 36.5966
C) 3.2983
D) 5.5966
E) 37.2983
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59
Find the mass of the lamina bounded by the graphs of the equations for the density .
A)
B)
C)
D)
E)
for the density . A)
B)
C)
D)
E)
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60
Find the mass of the triangular lamina with vertices for the density .
A) 11,337,408k
B) 22,674,826k
C) 11,337,398k
D) 11,337,413k
E) 22,674,816k
for the density . A) 11,337,408k
B) 22,674,826k
C) 11,337,398k
D) 11,337,413k
E) 22,674,816k
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61
Find the area of the portion of the surface that lies above the region . Round your answer to two decimal places.
A) 0.67
B) 3.87
C) 30.30
D) 10.64
E) 30.84
that lies above the region . Round your answer to two decimal places.A) 0.67
B) 3.87
C) 30.30
D) 10.64
E) 30.84
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62
Evaluate the iterated integral below. Note that it is necessary to switch the order of integration.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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63
Sketch the region R of integration and then switch the order of integration for the following integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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64
Find the area of the surface given by over the region R.
A)
B)
C)
D)
E)
over the region R. A)
B)
C)
D)
E)
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65
Find the Jacobian for the following change of variables: ,
A)
B)
C)
D)
E)
for the following change of variables: , A)
B)
C)
D)
E)
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66
The area of a region R is given by the iterated integrals . Switch the order of integration and show that both orders yield the same area. What is this area?
A) 14
B) 197
C) 28
D) 150
E) 27
. Switch the order of integration and show that both orders yield the same area. What is this area?A) 14
B) 197
C) 28
D) 150
E) 27
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67
Use an iterated integral to find the area of the region bounded by .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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68
Set up a double integral that gives the area of the surface on the graph of over the region .
A)
B)
C)
D)
E)
over the region .A)
B)
C)
D)
E)
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69
Evaluate the integral by switching the order of integration. Round your answer to two decimal places.
A) 5.16
B) 48.66
C) 15.38
D) 13.38
E) 56.14
by switching the order of integration. Round your answer to two decimal places.A) 5.16
B) 48.66
C) 15.38
D) 13.38
E) 56.14
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70
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations and in the first octant.
A) 273,375
B) 30,375
C) 182,250
D) 91,125
E) 60,750
and in the first octant.A) 273,375
B) 30,375
C) 182,250
D) 91,125
E) 60,750
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71
Find the area of the surface of the portion of the plane in the first octant.
A)
B)
C)
D)
E)
in the first octant.A)
B)
C)
D)
E)
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72
Use a double integral to find the volume of the indicated solid.
A) 16
B) 9
C) 4
D) 20
E) 6
A) 16
B) 9
C) 4
D) 20
E) 6
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73
Evaluate the following integral.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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74
Set up a triple integral for the volume of the solid bounded by the coordinate planes and the plane given below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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75
Use cylindrical coordinates to find the mass of the solid where .
A)
B)
C)
D)
E)
where .A)
B)
C)
D)
E)
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76
Consider the region in the xy-plane bounded by the ellipse and the transformation and . Find .
A)
B) 0
C)
D)
E)
in the xy-plane bounded by the ellipse and the transformation and . Find .A)
B) 0
C)
D)
E)
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77
Use an iterated integral to find the area of the region bounded by the graphs of the equations and .
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
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78
Use polar coordinates to describe the region as shown in the figure below:
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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79
Find the area of the portion of the surface that lies above the region . Round your answer to two decimal places.
A) 81.00
B) 88.83
C) 508.94
D) 254.47
E) 799.44
that lies above the region . Round your answer to two decimal places. A) 81.00
B) 88.83
C) 508.94
D) 254.47
E) 799.44
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80
Find the Jacobian for the change of variables given below. ,
A)
B)
C)
D)
E)
, A)
B)
C)
D)
E)
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