Question 1

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.

Accepted Answer

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Question 2

Find the center of mass of the lamina bounded by the graphs of the equations   for the density   .

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Question 3

Set up a triple integral for the volume of the solid bounded above by the cylinder   and below by the paraboloid   .

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Question 4

Use a double integral to find the area enclosed by the graph of   .

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Question 5

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:
​   ​

Accepted Answer

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Question 6

Find the center of mass of the rectangular lamina with vertices   and   for the density   . ​

Accepted Answer

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Question 7

Use spherical coordinates to find the z coordinate of the center of mass of the solid lying between two concentric hemispheres of radii 4 and 5, and having uniform density k.

Accepted Answer

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Question 8

Use the following change of variables to evaluate the double integral   . Round your answer to four decimal places.
​   ​   ​

Accepted Answer

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Question 9

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations given below.

Accepted Answer

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Question 10

Evaluate the following integral.

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Question 11

Evaluate the following iterated integral.

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Question 12

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration.

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Question 13

Evaluate the double integral below.

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Question 14

Use cylindrical coordinates to find the volume of the cone   where   and   . ​   ​

Accepted Answer

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Question 15

Use spherical coordinates to find the volume of the solid between the spheres   and   and inside the cone   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 16

Find the mass of the triangular lamina with vertices   for the density   . ​

Accepted Answer

A)  170k 
B)  347k 
C)  338k 
D)  169k 
E)  343k 
A)  170k 
B)  347k 
C)  338k 
D)  169k 
E)  343k

Question 17

Write a double integral that represents the surface area of   over the region R:   . Use a computer algebra system to evaluate the double integral. Round your answer to four decimal places. ​

Accepted Answer

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Question 18

A company produces a spherical object of radius 30 centimeters. A hole of radius 6 centimeters is drilled through the center of the object. Find the volume of the object.

Accepted Answer

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B)    
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Question 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.

Accepted Answer

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B)    
C)    
D)    
E)    
A)    
B)    
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Question 20

Find the area of the portion of the surface   that lies above the region   . Round your answer to two decimal places.

Accepted Answer

A)  0.67 
B)  3.87 
C)  30.30 
D)  10.64 
E)  30.84 
A)  0.67 
B)  3.87 
C)  30.30 
D)  10.64 
E)  30.84

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.

Find the center of mass of the lamina bounded by the graphs of the equations for the density .

Set up a triple integral for the volume of the solid bounded above by the cylinder and below by the paraboloid .

Use a double integral to find the area enclosed by the graph of .

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:

Find the center of mass of the rectangular lamina with vertices and for the density .

Use spherical coordinates to find the z coordinate of the center of mass of the solid lying between two concentric hemispheres of radii 4 and 5, and having uniform density k.

Use the following change of variables to evaluate the double integral . Round your answer to four decimal places.

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations given below.

Evaluate the following integral.

Evaluate the following iterated integral.

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration.

Evaluate the double integral below.

Use cylindrical coordinates to find the volume of the cone where and .

Use spherical coordinates to find the volume of the solid between the spheres and and inside the cone .

Find the mass of the triangular lamina with vertices for the density .

Write a double integral that represents the surface area of over the region R: . Use a computer algebra system to evaluate the double integral. Round your answer to four decimal places.

A company produces a spherical object of radius 30 centimeters. A hole of radius 6 centimeters is drilled through the center of the object. Find the volume of the object.

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.

Find the area of the portion of the surface that lies above the region . Round your answer to two decimal places.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Vector Anal

Filters

Exam 14: Multiple Integration

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.

Find the center of mass of the lamina bounded by the graphs of the equations for the density .

Set up a triple integral for the volume of the solid bounded above by the cylinder and below by the paraboloid .

Use a double integral to find the area enclosed by the graph of .

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: ​ ​

Find the center of mass of the rectangular lamina with vertices and for the density . ​

Use spherical coordinates to find the z coordinate of the center of mass of the solid lying between two concentric hemispheres of radii 4 and 5, and having uniform density k.

Use the following change of variables to evaluate the double integral . Round your answer to four decimal places. ​ ​ ​

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations given below.

Evaluate the following integral.

Evaluate the following iterated integral.

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration.

Evaluate the double integral below.

Use cylindrical coordinates to find the volume of the cone where and . ​ ​

Use spherical coordinates to find the volume of the solid between the spheres and and inside the cone . ​

Find the mass of the triangular lamina with vertices for the density . ​

Write a double integral that represents the surface area of over the region R: . Use a computer algebra system to evaluate the double integral. Round your answer to four decimal places. ​

A company produces a spherical object of radius 30 centimeters. A hole of radius 6 centimeters is drilled through the center of the object. Find the volume of the object.

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.

Find the area of the portion of the surface that lies above the region . Round your answer to two decimal places.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Vector Anal

Filters

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:

Find the center of mass of the rectangular lamina with vertices and for the density .

Use the following change of variables to evaluate the double integral . Round your answer to four decimal places.

Use cylindrical coordinates to find the volume of the cone where and .

Use spherical coordinates to find the volume of the solid between the spheres and and inside the cone .

Find the mass of the triangular lamina with vertices for the density .

Write a double integral that represents the surface area of over the region R: . Use a computer algebra system to evaluate the double integral. Round your answer to four decimal places.