Question 1

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

Accepted Answer

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

Question 2

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)      
B)      
C)  98   
D)      
A)      
B)      
C)  98   
D)

Question 3

Find the area of the region specified in polar coordinates.
-the region enclosed by the curve r = 6 cos $\theta$

Accepted Answer

A)  18 $\pi$ 
B)  9 $\pi$ 
C)  12 $\pi$ 
D)  36 $\pi$ 
A)  18 $\pi$ 
B)  9 $\pi$ 
C)  12 $\pi$ 
D)  36 $\pi$

Question 4

Find the average value of the function f over the given region.
-f(x, y) = 10x + 7y over the region bounded by the coordinate axes and the lines   and   .

Accepted Answer

A)  73 
B)    
C)    
D)    
A)  73 
B)    
C)    
D)

Question 5

Find the Jacobian for the given transformation
-

Accepted Answer

A)  4536v 
B)  4536u 
C)  3528u 
D)  3528v 
A)  4536v 
B)  4536u 
C)  3528u 
D)  3528v

Question 6

Find the volume of the indicated region.
-the region in the first octant bounded by the coordinate planes and the planes   ,

Accepted Answer

A)  15000 
B)  150 
C)  1500 
D)    
A)  15000 
B)  150 
C)  1500 
D)

Question 7

Evaluate the integral.
-

Accepted Answer

A)  3 
B)  -   
C)    
D)  -1 
A)  3 
B)  -   
C)    
D)  -1

Question 8

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)   $\pi$ 
B)  69,984$\pi$ 
C)    $\pi$ 
D)    $\pi$ 
A)   $\pi$ 
B)  69,984$\pi$ 
C)    $\pi$ 
D)    $\pi$

Question 9

Choose the one alternative that best completes the statement or answers the question. Evaluate the integral
-

Accepted Answer

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

Question 10

Solve the problem.
-The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let   Assuming that the total annual snowfall (in inches), S(x,y), at   is given by the function S(x,y) = 60   with (x,y)   D, find the average snowfall on D.

Accepted Answer

A) 52.06 inches 
B)  52.44 inches 
C) 51.78 inches 
D)  51.14 inches 
A) 52.06 inches 
B)  52.44 inches 
C) 51.78 inches 
D)  51.14 inches

Question 11

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.
-z =   ; R = {(x, y): 0 $\le$ x$\le$ 1, 1 $\le$ y $\le$ e}

Accepted Answer

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

Question 12

Solve the problem.
-Find the average distance from a point   in the first quadrant of the disk   to the origin.

Accepted Answer

A)  3 
B)    
C)    
D)  6 
A)  3 
B)    
C)    
D)  6

Question 13

Evaluate the integral by changing the order of integration in an appropriate way.
-

Accepted Answer

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

Question 14

Evaluate the cylindrical coordinate integral.
-

Accepted Answer

A)  4$\pi$ + ln 2 
B)  2$\pi$ + ln 2 
C)  4 - ln 2 
D)  2 - ln 2 
A)  4$\pi$ + ln 2 
B)  2$\pi$ + ln 2 
C)  4 - ln 2 
D)  2 - ln 2

Question 15

Use the given transformation to evaluate the integral.
-  where R is the parallelepiped bounded by the planes

Accepted Answer

A)  102 
B)  68 
C)  272 
D)  612 
A)  102 
B)  68 
C)  272 
D)  612

Question 16

Evaluate the improper integral.
-

Accepted Answer

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

Question 17

Reverse the order of integration and then evaluate the integral.
-

Accepted Answer

A)    
B)  40 
C)    
D)  30 
A)    
B)  40 
C)    
D)  30

Question 18

Use cylindrical coordinates to find the volume of the indicated region.
-the region enclosed by the paraboloids   and

Accepted Answer

A)  3888$\pi$ 
B)  2592$\pi$ 
C)  1296$\pi$ 
D)  5184$\pi$ 
A)  3888$\pi$ 
B)  2592$\pi$ 
C)  1296$\pi$ 
D)  5184$\pi$

Question 19

Find the area of the region specified in polar coordinates.
-one petal of the rose curve r = 6 cos 3$\theta$

Accepted Answer

A)  9$\pi$ 
B)  3$\pi$ 
C)  18$\pi$ 
D)  6$\pi$ 
A)  9$\pi$ 
B)  3$\pi$ 
C)  18$\pi$ 
D)  6$\pi$

Question 20

Use the given transformation to evaluate the integral.
-  where R is the parallelogram bounded by the lines

Accepted Answer

A)  34 
B)  17 
C)  13 
D)  26 
A)  34 
B)  17 
C)  13 
D)  26

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

Change the order of integration and evaluate the integral. -

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 6 cos $\theta$

Find the average value of the function f over the given region. -f(x, y) = 10x + 7y over the region bounded by the coordinate axes and the lines and .

Find the Jacobian for the given transformation -

Find the volume of the indicated region. -the region in the first octant bounded by the coordinate planes and the planes ,

Evaluate the integral. -

Change the order of integration and evaluate the integral. -

Choose the one alternative that best completes the statement or answers the question. Evaluate the integral -

Solve the problem. -The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let Assuming that the total annual snowfall (in inches), S(x,y), at is given by the function S(x,y) = 60 with (x,y) D, find the average snowfall on D.

Solve the problem. -Find the average distance from a point in the first quadrant of the disk to the origin.

Evaluate the integral by changing the order of integration in an appropriate way. -

Evaluate the cylindrical coordinate integral. -

Use the given transformation to evaluate the integral. - where R is the parallelepiped bounded by the planes

Evaluate the improper integral. -

Reverse the order of integration and then evaluate the integral. -

Use cylindrical coordinates to find the volume of the indicated region. -the region enclosed by the paraboloids and

Find the area of the region specified in polar coordinates. -one petal of the rose curve r = 6 cos 3 $\theta$

Use the given transformation to evaluate the integral. - where R is the parallelogram bounded by the lines

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

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

Change the order of integration and evaluate the integral. -

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 6 cos θ\thetaθ

Find the average value of the function f over the given region. -f(x, y) = 10x + 7y over the region bounded by the coordinate axes and the lines and .

Find the Jacobian for the given transformation -

Find the volume of the indicated region. -the region in the first octant bounded by the coordinate planes and the planes ,

Evaluate the integral. -

Change the order of integration and evaluate the integral. -

Choose the one alternative that best completes the statement or answers the question. Evaluate the integral -

Solve the problem. -The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let Assuming that the total annual snowfall (in inches), S(x,y), at is given by the function S(x,y) = 60 with (x,y) D, find the average snowfall on D.

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. -z = ; R = {(x, y): 0 ≤\le≤ x ≤\le≤ 1, 1 ≤\le≤ y ≤\le≤ e}

Solve the problem. -Find the average distance from a point in the first quadrant of the disk to the origin.

Evaluate the integral by changing the order of integration in an appropriate way. -

Evaluate the cylindrical coordinate integral. -

Use the given transformation to evaluate the integral. - where R is the parallelepiped bounded by the planes

Evaluate the improper integral. -

Reverse the order of integration and then evaluate the integral. -

Use cylindrical coordinates to find the volume of the indicated region. -the region enclosed by the paraboloids and

Find the area of the region specified in polar coordinates. -one petal of the rose curve r = 6 cos 3 θ\thetaθ

Use the given transformation to evaluate the integral. - where R is the parallelogram bounded by the lines

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 area of the region specified in polar coordinates. -the region enclosed by the curve r = 6 cos $\theta$

Find the area of the region specified in polar coordinates. -one petal of the rose curve r = 6 cos 3 $\theta$