Question 1

Use the indicated change of variables to evaluate the following double integral.     ​

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

Use a triple integral to find the volume of the solid shown below. ​   ​   ​

Accepted Answer

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

Question 4

Find a such that the volume inside the hemisphere   and outside the cylinder   is one-half the volume of the hemisphere. Round your answer to four decimal places. ​

Accepted Answer

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

Question 5

Use the indicated change of variables to evaluate the following double integral.   ,

Accepted Answer

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

Question 6

Find   for the indicated solid with density function   .   ​

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

Evaluate   .

Accepted Answer

A)  6 
B)  14 
C)  12 
D)  18 
E)  8 
A)  6 
B)  14 
C)  12 
D)  18 
E)  8

Question 9

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

Accepted Answer

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

Question 10

Find the area of the portion of the surface   that lies above the triangular region with vertices   , and   .

Accepted Answer

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

Question 11

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?

Accepted Answer

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

Question 12

Use an iterated integral to find the area of the region bounded by   .

Accepted Answer

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 13

Find a transformation   that when applied to region   , its image will be   . ​   ​

Accepted Answer

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

Question 14

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

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

Question 15

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

Accepted Answer

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

Question 16

Suppose the temperature in degrees Celsius on the surface of a metal plate is   where x and y are measured in centimeters. Estimate the average temperature if x varies between 0 and 3 centimeters and y varies between 0 and 2 centimeters.

Accepted Answer

A)    degrees Celsius 
B)    degrees Celsius 
C)    degrees Celsius 
D)    degrees Celsius 
E)    degrees Celsius 
A)    degrees Celsius 
B)    degrees Celsius 
C)    degrees Celsius 
D)    degrees Celsius 
E)    degrees Celsius

Question 17

Use spherical coordinates to find the volume of the solid inside   and outside   , and above the xy-plane. ​

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Find the average value of   over the region R where: ​   ​
R: rectangle with vertices   ​

Accepted Answer

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

Question 20

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 7, and having uniform density k.

Accepted Answer

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

Use the indicated change of variables to evaluate the following double integral.

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

Use a triple integral to find the volume of the solid shown below.

Find a such that the volume inside the hemisphere and outside the cylinder is one-half the volume of the hemisphere. Round your answer to four decimal places.

Use the indicated change of variables to evaluate the following double integral. ,

Find for the indicated solid with density function .

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

Evaluate .

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

Find the area of the portion of the surface that lies above the triangular region with vertices , and .

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?

Use an iterated integral to find the area of the region bounded by .

Find a transformation that when applied to region , its image will be .

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

Suppose the temperature in degrees Celsius on the surface of a metal plate is where x and y are measured in centimeters. Estimate the average temperature if x varies between 0 and 3 centimeters and y varies between 0 and 2 centimeters.

Use spherical coordinates to find the volume of the solid inside and outside , and above the xy-plane.

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

Find the average value of over the region R where: R: rectangle with vertices

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 7, and having uniform density k.

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 the indicated change of variables to evaluate the following double integral. ​

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

Use a triple integral to find the volume of the solid shown below. ​ ​ ​

Find a such that the volume inside the hemisphere and outside the cylinder is one-half the volume of the hemisphere. Round your answer to four decimal places. ​

Use the indicated change of variables to evaluate the following double integral. ,

Find for the indicated solid with density function . ​

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

Evaluate .

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

Find the area of the portion of the surface that lies above the triangular region with vertices , and .

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?

Use an iterated integral to find the area of the region bounded by .

Find a transformation that when applied to region , its image will be . ​ ​

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 mass of the lamina bounded by the graphs of the equations for the density . ​

Suppose the temperature in degrees Celsius on the surface of a metal plate is where x and y are measured in centimeters. Estimate the average temperature if x varies between 0 and 3 centimeters and y varies between 0 and 2 centimeters.

Use spherical coordinates to find the volume of the solid inside and outside , and above the xy-plane. ​

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

Find the average value of over the region R where: ​ ​ R: rectangle with vertices ​

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 7, and having uniform density k.

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

Use the indicated change of variables to evaluate the following double integral.

Use a triple integral to find the volume of the solid shown below.

Find a such that the volume inside the hemisphere and outside the cylinder is one-half the volume of the hemisphere. Round your answer to four decimal places.

Find for the indicated solid with density function .

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

Find a transformation that when applied to region , its image will be .

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

Use spherical coordinates to find the volume of the solid inside and outside , and above the xy-plane.

Find the average value of over the region R where: R: rectangle with vertices