Question 1

Find the Jacobian for the change of variables given below.   ,

Accepted Answer

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

Question 2

Sketch the region R of integration and then switch the order of integration for the following integral.

Accepted Answer

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

Question 3

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?

Accepted Answer

A)  14 
B)  197 
C)  28 
D)  150 
E)  27 
A)  14 
B)  197 
C)  28 
D)  150 
E)  27

Question 4

Evaluate the double integral below.

Accepted Answer

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

Question 5

Evaluate the following iterated integral. ​   ​

Accepted Answer

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

Question 6

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.

Accepted Answer

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

Question 7

Find the area of the surface for the portion of the paraboloid   in the first octant.

Accepted Answer

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

Question 8

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

Accepted Answer

A)  81 
B)  729 
C)  741 
D)  243 
E)  162 
A)  81 
B)  729 
C)  741 
D)  243 
E)  162

Question 9

Use spherical coordinates to find the volume of the solid inside the torus given by   .

Accepted Answer

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

Question 10

Use a double integral to find the volume of the indicated solid.

Accepted Answer

A)  32 
B)  16 
C)  60 
D)  15 
E)  24 
A)  32 
B)  16 
C)  60 
D)  15 
E)  24

Question 11

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.) ​   ​

Accepted Answer

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

Question 12

Evaluate   . Round your answer to two decimal places.

Accepted Answer

A)  12.64 
B)  26.34 
C)  105.35 
D)  88.49 
E)  29.50 
A)  12.64 
B)  26.34 
C)  105.35 
D)  88.49 
E)  29.50

Question 13

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)  0 
A)    
B)    
C)    
D)    
E)  0

Question 14

Evaluate the following iterated integral by converting to polar coordinates.

Accepted Answer

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

Question 15

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 16

Evaluate the following iterated integral by converting to polar coordinates.

Accepted Answer

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

Question 17

Use cylindrical coordinates to find the volume of the solid inside the sphere   and above the upper nappe of the cone   .

Accepted Answer

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

Question 18

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. ​

Accepted Answer

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

Question 19

Set up a double integral that gives the area of the surface of the graph of f over the region R.

Accepted Answer

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

Question 20

Find the mass of the lamina described by the inequalities   given that its density is  
(Hint: Some of the integrals are simpler in polar coordinates.)

Accepted Answer

A)  384 
B)  400 
C)  320 
D)  256 
E)  180 
A)  384 
B)  400 
C)  320 
D)  256 
E)  180

Find the Jacobian for the change of variables given below. ,

Sketch the region R of integration and then switch the order of integration for the following integral.

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?

Evaluate the double integral below.

Evaluate the following iterated integral.

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.

Find the area of the surface for the portion of the paraboloid in the first octant.

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

Use spherical coordinates to find the volume of the solid inside the torus given by .

Use a double integral to find the volume of the indicated solid.

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.)

Evaluate . 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 by converting to polar coordinates.

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

Evaluate the following iterated integral by converting to polar coordinates.

Use cylindrical coordinates to find the volume of the solid inside the sphere and above the upper nappe of the cone .

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.

Set up a double integral that gives the area of the surface of the graph of f over the region R.

Find the mass of the lamina described by the inequalities given that its density is (Hint: Some of the integrals are simpler in polar coordinates.)

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

Find the Jacobian for the change of variables given below. ,

Sketch the region R of integration and then switch the order of integration for the following integral.

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?

Evaluate the double integral below.

Evaluate the following iterated integral. ​ ​

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.

Find the area of the surface for the portion of the paraboloid in the first octant.

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

Use spherical coordinates to find the volume of the solid inside the torus given by .

Use a double integral to find the volume of the indicated solid.

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.) ​ ​

Evaluate . 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 by converting to polar coordinates.

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

Evaluate the following iterated integral by converting to polar coordinates.

Use cylindrical coordinates to find the volume of the solid inside the sphere and above the upper nappe of the cone .

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. ​

Set up a double integral that gives the area of the surface of the graph of f over the region R.

Find the mass of the lamina described by the inequalities given that its density is (Hint: Some of the integrals are simpler in polar coordinates.)

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

Evaluate the following iterated integral.

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

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.)

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

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.