Question 1

Find the average value of the function f over the given region.
-f(x, y) = 2x + 8y; R = {(x, y): 0 $\le$x $\le$ 1, 0$\le$ y $\le$ 1}

Accepted Answer

A)  10 
B)  5 
C)  6 
D)  9 
A)  10 
B)  5 
C)  6 
D)  9

Question 2

Find the average value of the function f over the given region.
-f(x, y) =   over the region bounded by   ,   ,   , and   .

Accepted Answer

A)  e - 1 
B)    
C)  2e - 1 
D)    
A)  e - 1 
B)    
C)  2e - 1 
D)

Question 3

Find the volume of the indicated region.
-the region bounded by the paraboloid   and the xy-plane

Accepted Answer

A)    $\pi$ 
B)    $\pi$ 
C)  500$\pi$ 
D)  5000$\pi$ 
A)    $\pi$ 
B)    $\pi$ 
C)  500$\pi$ 
D)  5000$\pi$

Question 4

Evaluate the cylindrical coordinate integral.
-

Accepted Answer

A)  756$\pi$ 
B)  504$\pi$ 
C)  2520$\pi$ 
D)  1008$\pi$ 
A)  756$\pi$ 
B)  504$\pi$ 
C)  2520$\pi$ 
D)  1008$\pi$

Question 5

Change the Cartesian integral to an equivalent polar integral, and then evaluate.
-

Accepted Answer

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

Question 6

Solve the problem.
-Rewrite the integral   in the order   .

Accepted Answer

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

Question 7

Solve the problem.
-Set up the triple integral for the volume of the sphere   in spherical coordinates.

Accepted Answer

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

Question 8

Provide an appropriate response.
-What does the graph of the equation   look like?

Accepted Answer

A)  The xy-plane 
B)  The y-axis 
C)  The z-axis 
D)  The yz-plane 
A)  The xy-plane 
B)  The y-axis 
C)  The z-axis 
D)  The yz-plane

Question 9

Change the Cartesian integral to an equivalent polar integral, and then evaluate.
-

Accepted Answer

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

Question 10

Find the area of the region specified in polar coordinates.
-one petal of the rose curve r = 9 sin 2$\theta$

Accepted Answer

A)    $\pi$ 
B)    $\pi$ 
C)    $\pi$ 
D)    $\pi$ 
A)    $\pi$ 
B)    $\pi$ 
C)    $\pi$ 
D)    $\pi$

Question 11

Evaluate the spherical coordinate integral.
-

Accepted Answer

A)    $\pi$ 
B)    $\pi$ 
C)      
D)      
A)    $\pi$ 
B)    $\pi$ 
C)      
D)

Question 12

Evaluate the cylindrical coordinate integral.
-

Accepted Answer

A)  40 
B)  60$\pi$ 
C)  20$\pi$ 
D)  60 
A)  40 
B)  60$\pi$ 
C)  20$\pi$ 
D)  60

Question 13

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)      
B)  8541   
C)  18,018   
D)      
A)      
B)  8541   
C)  18,018   
D)

Question 14

Provide an appropriate response.
-What does the graph of the equation   look like?

Accepted Answer

A)  A cylinder 
B)  A line 
C)  A plane 
D)  A cone 
A)  A cylinder 
B)  A line 
C)  A plane 
D)  A cone

Question 15

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)  3780 
B)  7560 
C)  2835 
D)  5670 
A)  3780 
B)  7560 
C)  2835 
D)  5670

Question 16

Find the volume of the indicated region.
-the region that lies under the plane   and over the triangle with vertices at   ,   , and

Accepted Answer

A)  8 
B)    
C)  4 
D)    
A)  8 
B)    
C)  4 
D)

Question 17

Find the center of mass of a thin plate of constant density covering the given region.
-The region bounded by the x-axis and the curve y = 7sin x, 0$\le$ x $\le$ $\pi$

Accepted Answer

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

Question 18

Find the center of mass of a thin plate covering the given region with the given density function.
-The region between the x-axis and the curve y =   , 1 $\le$x $\le$ 2, with density (x) =

Accepted Answer

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

Question 19

Find the average value of over the given region.  
-  +   over the cube in the first octant bounded by the coordinate planes and the planes       ,,

Accepted Answer

A)  9 
B)  18 
C)  3 
D)  27 
A)  9 
B)  18 
C)  3 
D)  27

Question 20

Use spherical coordinates to find the volume of the indicated region.
-the region enclosed by the sphere   and the cylinder

Accepted Answer

A)    (3 $\pi$ - 4) 
B)    (3 $\pi$ - 4) 
C)    (3 $\pi$ - 4) 
D)    (3 $\pi$ - 4) 
A)    (3 $\pi$ - 4) 
B)    (3 $\pi$ - 4) 
C)    (3 $\pi$ - 4) 
D)    (3 $\pi$ - 4)

Find the average value of the function f over the given region. -f(x, y) = 2x + 8y; R = {(x, y): 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1}

Find the average value of the function f over the given region. -f(x, y) = over the region bounded by , , , and .

Find the volume of the indicated region. -the region bounded by the paraboloid and the xy-plane

Evaluate the cylindrical coordinate integral. -

Change the Cartesian integral to an equivalent polar integral, and then evaluate. -

Solve the problem. -Rewrite the integral in the order .

Solve the problem. -Set up the triple integral for the volume of the sphere in spherical coordinates.

Provide an appropriate response. -What does the graph of the equation look like?

Change the Cartesian integral to an equivalent polar integral, and then evaluate. -

Find the area of the region specified in polar coordinates. -one petal of the rose curve r = 9 sin 2 $\theta$

Evaluate the spherical coordinate integral. -

Evaluate the cylindrical coordinate integral. -

Change the order of integration and evaluate the integral. -

Provide an appropriate response. -What does the graph of the equation look like?

Change the order of integration and evaluate the integral. -

Find the volume of the indicated region. -the region that lies under the plane and over the triangle with vertices at , , and

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the x-axis and the curve y = 7sin x, 0 $\le$ x $\le$ $\pi$

Find the average value of over the given region. - + over the cube in the first octant bounded by the coordinate planes and the planes ,,

Use spherical coordinates to find the volume of the indicated region. -the region enclosed by the sphere and the cylinder

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Vector Calculus

Filters

Exam 16: Multiple Integration

Find the average value of the function f over the given region. -f(x, y) = 2x + 8y; R = {(x, y): 0 ≤\le≤ x ≤\le≤ 1, 0 ≤\le≤ y ≤\le≤ 1}

Find the average value of the function f over the given region. -f(x, y) = over the region bounded by , , , and .

Find the volume of the indicated region. -the region bounded by the paraboloid and the xy-plane

Evaluate the cylindrical coordinate integral. -

Change the Cartesian integral to an equivalent polar integral, and then evaluate. -

Solve the problem. -Rewrite the integral in the order .

Solve the problem. -Set up the triple integral for the volume of the sphere in spherical coordinates.

Provide an appropriate response. -What does the graph of the equation look like?

Change the Cartesian integral to an equivalent polar integral, and then evaluate. -

Find the area of the region specified in polar coordinates. -one petal of the rose curve r = 9 sin 2 θ\thetaθ

Evaluate the spherical coordinate integral. -

Evaluate the cylindrical coordinate integral. -

Change the order of integration and evaluate the integral. -

Provide an appropriate response. -What does the graph of the equation look like?

Change the order of integration and evaluate the integral. -

Find the volume of the indicated region. -the region that lies under the plane and over the triangle with vertices at , , and

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the x-axis and the curve y = 7sin x, 0 ≤\le≤ x ≤\le≤ π\piπ

Find the center of mass of a thin plate covering the given region with the given density function. -The region between the x-axis and the curve y = , 1 ≤\le≤ x ≤\le≤ 2, with density (x) =

Find the average value of over the given region. - + over the cube in the first octant bounded by the coordinate planes and the planes ,,

Use spherical coordinates to find the volume of the indicated region. -the region enclosed by the sphere and the cylinder

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Vector Calculus

Filters

Find the average value of the function f over the given region. -f(x, y) = 2x + 8y; R = {(x, y): 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1}

Find the area of the region specified in polar coordinates. -one petal of the rose curve r = 9 sin 2 $\theta$

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the x-axis and the curve y = 7sin x, 0 $\le$ x $\le$ $\pi$