Question 1

Use cylindrical coordinates to find the mass of the solid   where   .

Accepted Answer

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

Question 2

Combine the sum of the two iterated integrals into a single integral by converting to polar coordinates. Evaluate the resulting iterated integral.

Accepted Answer

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

Question 3

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

Accepted Answer

A)  81.00 
B)  88.83 
C)  508.94 
D)  254.47 
E)  799.44 
A)  81.00 
B)  88.83 
C)  508.94 
D)  254.47 
E)  799.44

Question 4

Find the area of the surface for the portion of the sphere   inside the cylinder   .

Accepted Answer

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

Question 5

Find the area of the surface given by   over the region R. ​     ​

Accepted Answer

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

Question 6

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.   ​

Accepted Answer

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

Question 7

Suppose the population density of a city is approximated by the model   where x and y are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. Round your answer to the nearest integer.

Accepted Answer

A)  417,128 
B)  417,001 
C)  833,901 
D)  833,903 
E)  416,951 
A)  417,128 
B)  417,001 
C)  833,901 
D)  833,903 
E)  416,951

Question 8

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.

Accepted Answer

A)  44.76 
B)  44.88 
C)  17.88 
D)  54.88 
E)  21.88 
A)  44.76 
B)  44.88 
C)  17.88 
D)  54.88 
E)  21.88

Question 9

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral.

Accepted Answer

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

Question 10

Evaluate the following iterated integral.

Accepted Answer

A)    
B)      
C)  160   
D)      
E)  160 
A)    
B)      
C)  160   
D)      
E)  160

Question 11

Find the area of the surface given by   over the region R. ​   ​
R: square with vertices   ​

Accepted Answer

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

Question 12

Evaluate the following integral.

Accepted Answer

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D)    
E)    
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E)

Question 13

Find the area of the surface of the portion of the plane   in the first octant.

Accepted Answer

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

Question 14

Find the area of the surface given by   over the region R. ​   ​
R: rectangle with vertices   ​

Accepted Answer

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

Question 15

Use a double integral in polar coordinates to find the volume of the solid in the first octant bounded by the graphs of the equations given below.

Accepted Answer

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

Question 16

Find the average value of   over the region R, where R is a triangle with vertices   . ​

Accepted Answer

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

Question 17

Find the Jacobian   for the following change of variables: ​   ​

Accepted Answer

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

Question 18

Given   use polar coordinates to set up and evaluate the double integral   .

Accepted Answer

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

Question 19

Find the center of mass of the solid bounded by   and   with density function   . ​

Accepted Answer

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

Question 20

Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformation.

Accepted Answer

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

Use cylindrical coordinates to find the mass of the solid where .

Combine the sum of the two iterated integrals into a single integral by converting to polar coordinates. Evaluate the resulting iterated integral.

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

Find the area of the surface for the portion of the sphere inside the cylinder .

Find the area of the surface given by over the region R.

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.

Suppose the population density of a city is approximated by the model where x and y are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. Round your answer to the nearest integer.

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.

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral.

Evaluate the following iterated integral.

Find the area of the surface given by over the region R. R: square with vertices

Evaluate the following integral.

Find the area of the surface of the portion of the plane in the first octant.

Find the area of the surface given by over the region R. R: rectangle with vertices

Use a double integral in polar coordinates to find the volume of the solid in the first octant bounded by the graphs of the equations given below.

Find the average value of over the region R, where R is a triangle with vertices .

Find the Jacobian for the following change of variables:

Given use polar coordinates to set up and evaluate the double integral .

Find the center of mass of the solid bounded by and with density function .

Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformation.

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 cylindrical coordinates to find the mass of the solid where .

Combine the sum of the two iterated integrals into a single integral by converting to polar coordinates. Evaluate the resulting iterated integral.

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

Find the area of the surface for the portion of the sphere inside the cylinder .

Find the area of the surface given by over the region R. ​ ​

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. ​

Suppose the population density of a city is approximated by the model where x and y are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. Round your answer to the nearest integer.

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.

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral.

Evaluate the following iterated integral.

Find the area of the surface given by over the region R. ​ ​ R: square with vertices ​

Evaluate the following integral.

Find the area of the surface of the portion of the plane in the first octant.

Find the area of the surface given by over the region R. ​ ​ R: rectangle with vertices ​

Use a double integral in polar coordinates to find the volume of the solid in the first octant bounded by the graphs of the equations given below.

Find the average value of over the region R, where R is a triangle with vertices . ​

Find the Jacobian for the following change of variables: ​ ​

Given use polar coordinates to set up and evaluate the double integral .

Find the center of mass of the solid bounded by and with density function . ​

Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformation.

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

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

Find the area of the surface given by over the region R.

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.

Find the area of the surface given by over the region R. R: square with vertices

Find the area of the surface given by over the region R. R: rectangle with vertices

Find the average value of over the region R, where R is a triangle with vertices .

Find the Jacobian for the following change of variables:

Find the center of mass of the solid bounded by and with density function .