Question 1

Evaluate the integral.
-

Accepted Answer

A)  23.2 
B)  7 
C)  8 
D)  9 
A)  23.2 
B)  7 
C)  8 
D)  9

Question 2

Evaluate the double integral over the given region.
-  R = {(x, y): 0 $\le$x$\le$1, 0$\le$y$\le$2}

Accepted Answer

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

Question 3

Find the area of the region specified in polar coordinates.
-the region inside both   and

Accepted Answer

A)  16($\pi$ - 1) 
B)  8($\pi$ - 2) 
C)  8($\pi$ - 1) 
D)  16($\pi$ - 2) 
A)  16($\pi$ - 1) 
B)  8($\pi$ - 2) 
C)  8($\pi$ - 1) 
D)  16($\pi$ - 2)

Question 4

Evaluate the integral.
-

Accepted Answer

A)  27 
B)  9 
C)    
D)    
A)  27 
B)  9 
C)    
D)

Question 5

Solve the problem.
-Find the center of mass of the region of density   bounded by the paraboloid   and the xy-plane.

Accepted Answer

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

Question 6

Find the volume of the indicated region.
-the region bounded by the coordinate planes, the parabolic cylinder   , and the plane

Accepted Answer

A)  6000 
B)  3000 
C)    
D)  8000 
A)  6000 
B)  3000 
C)    
D)  8000

Question 7

Evaluate the double integral over the given region.
-  R =

Accepted Answer

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

Question 8

Find the average value of over the given region.  
-  over the rectang ular solid in the first octant bounded by the coordinate planes and the planes       ,,

Accepted Answer

A)  16 
B)  4 
C)  32 
D)  8 
A)  16 
B)  4 
C)  32 
D)  8

Question 9

Change the order of integration and evaluate the integral.
-

Accepted Answer

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

Question 10

Find the area of the region specified in polar coordinates.
-the region enclosed by the curve r = 5 + cos $\theta$

Accepted Answer

A)  13 $\pi$ 
B)  17 $\pi$ 
C)  26 $\pi$ 
D)     $\pi$ 
A)  13 $\pi$ 
B)  17 $\pi$ 
C)  26 $\pi$ 
D)     $\pi$

Question 11

Write an equivalent double integral with the order of integration reversed.
-

Accepted Answer

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

Question 12

Provide an appropriate response.
-What form do planes perpendicular to the y-axis have in spherical coordinates?

Accepted Answer

A)     = a csc $\theta$ csc   
B)    = a csc $\theta$ 
C)     = a sin   sin $\theta$ 
D)    = a csc $\theta$ 
A)     = a csc $\theta$ csc   
B)    = a csc $\theta$ 
C)     = a sin   sin $\theta$ 
D)    = a csc $\theta$

Question 13

Find the Jacobian for the given transformation
-

Accepted Answer

A)  -100 
B)  2400 
C)  -24 
D)  100 
A)  -100 
B)  2400 
C)  -24 
D)  100

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

A)  4   - 68 
B)    -   
C)    - 1 
D)    
A)  4   - 68 
B)    -   
C)    - 1 
D)

Question 16

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

Accepted Answer

A)  -   
B)  - 45 
C)  -   
D)  - 405 
A)  -   
B)  - 45 
C)  -   
D)  - 405

Question 17

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

Accepted Answer

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

Question 18

Evaluate the double integral over the given region.
-  R = {(x, y): 0 $\le$x$\le$1, 0$\le$y$\le$1}

Accepted Answer

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

Question 19

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

Accepted Answer

A)  160   64 -   ln 4097 
B)  320   64 -   ln 4097 
C)  160   64 - 5 ln 4097 
D)  320   64 - 5 ln 4097 
A)  160   64 -   ln 4097 
B)  320   64 -   ln 4097 
C)  160   64 - 5 ln 4097 
D)  320   64 - 5 ln 4097

Question 20

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)  36 
B)  324 
C)  2592 
D)  288 
A)  36 
B)  324 
C)  2592 
D)  288

Evaluate the integral. -

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y): 0 \le x \le 1, 0 \le y \le 2}$ R = {(x, y): 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 2}

Find the area of the region specified in polar coordinates. -the region inside both and

Evaluate the integral. -

Solve the problem. -Find the center of mass of the region of density bounded by the paraboloid and the xy-plane.

Find the volume of the indicated region. -the region bounded by the coordinate planes, the parabolic cylinder , and the plane

Evaluate the double integral over the given region. - R =

Find the average value of over the given region. - over the rectang ular solid in the first octant bounded by the coordinate planes and the planes ,,

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 = 5 + cos $\theta$

Write an equivalent double integral with the order of integration reversed. -

Provide an appropriate response. -What form do planes perpendicular to the y-axis have in spherical coordinates?

Find the Jacobian for the given transformation -

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

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

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

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

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y): 0 \le x \le 1, 0 \le y \le 1}$ R = {(x, y): 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1}

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

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

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

Evaluate the integral. -

Evaluate the double integral over the given region. - R = {(x, y): 0 ≤\le≤ x ≤\le≤ 1, 0 ≤\le≤ y ≤\le≤ 2}

Find the area of the region specified in polar coordinates. -the region inside both and

Evaluate the integral. -

Solve the problem. -Find the center of mass of the region of density bounded by the paraboloid and the xy-plane.

Find the volume of the indicated region. -the region bounded by the coordinate planes, the parabolic cylinder , and the plane

Evaluate the double integral over the given region. - R =

Find the average value of over the given region. - over the rectang ular solid in the first octant bounded by the coordinate planes and the planes ,,

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 = 5 + cos θ\thetaθ

Write an equivalent double integral with the order of integration reversed. -

Provide an appropriate response. -What form do planes perpendicular to the y-axis have in spherical coordinates?

Find the Jacobian for the given transformation -

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

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

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

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

Evaluate the double integral over the given region. - R = {(x, y): 0 ≤\le≤ x ≤\le≤ 1, 0 ≤\le≤ y ≤\le≤ 1}

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

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

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

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y): 0 \le x \le 1, 0 \le y \le 2}$ R = {(x, y): 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 2}

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 5 + cos $\theta$

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y): 0 \le x \le 1, 0 \le y \le 1}$ R = {(x, y): 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1}