Question 1

Solve the problem.
-Write an iterated triple integral in the order   for the volume of the region enclosed by the paraboloids   and   .

Accepted Answer

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

Question 2

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
-The curve   and the lines   and

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

A)  7776 
B) 864 
C)  15,552 
D)  1728 
A)  7776 
B) 864 
C)  15,552 
D)  1728

Question 5

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

Accepted Answer

A)  300 
B)  1800 
C)  3000 
D)  18000 
A)  300 
B)  1800 
C)  3000 
D)  18000

Question 6

Evaluate the improper integral.
-

Accepted Answer

A)  24 
B)  18 
C)  36 
D)  30 
A)  24 
B)  18 
C)  36 
D)  30

Question 7

Integrate the function f over the given region.
-f(x, y) = xy over the triangular region with vertices (0, 0), ( 5, 0), and (0, 8)

Accepted Answer

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

Question 8

Evaluate the cylindrical coordinate integral.
-

Accepted Answer

A)  16$\pi$ 
B)  48$\pi$ 
C)  8$\pi$ 
D)  32$\pi$ 
A)  16$\pi$ 
B)  48$\pi$ 
C)  8$\pi$ 
D)  32$\pi$

Question 9

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.
-z = 6   y; R = {(x, y): 0 $\le$ x $\le$ 4, 0 $\le$ y $\le$ 3}

Accepted Answer

A)  576 
B)  1256 
C)  2256 
D)  676 
A)  576 
B)  1256 
C)  2256 
D)  676

Question 10

Find the center of mass of a thin plate covering the given region with the given density function.
-The region bounded by the curves y = ±   and the lines x = 1 and x = 9, with density  (x) =

Accepted Answer

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

Question 11

Evaluate the spherical coordinate integral.
-

Accepted Answer

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

Question 12

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

Accepted Answer

A)  1 - cos 5 
B)  - cos 5 
C)  cos 5 
D)  1 + cos 5 
A)  1 - cos 5 
B)  - cos 5 
C)  cos 5 
D)  1 + cos 5

Question 13

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)  27 
B)  72 
C)  18 
D)  36 
A)  27 
B)  72 
C)  18 
D)  36

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

A)  64 $\pi$ 
B)  128 $\pi$ 
C)     $\pi$ 
D)     $\pi$ 
A)  64 $\pi$ 
B)  128 $\pi$ 
C)     $\pi$ 
D)     $\pi$

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

A)  ln 2   
B)  ln 2 
C)  ln 2   
D)  56 ln 2 
A)  ln 2   
B)  ln 2 
C)  ln 2   
D)  56 ln 2

Question 19

Evaluate the improper integral.
-

Accepted Answer

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

Question 20

Find the average value of the function f over the given region.
-f(x, y) =   ; R =

Accepted Answer

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

Solve the problem. -Write an iterated triple integral in the order for the volume of the region enclosed by the paraboloids and .

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. -The curve and the lines and

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

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

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

Evaluate the improper integral. -

Integrate the function f over the given region. -f(x, y) = xy over the triangular region with vertices (0, 0), ( 5, 0), and (0, 8)

Evaluate the cylindrical coordinate integral. -

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded by the curves y = ± and the lines x = 1 and x = 9, with density (x) =

Evaluate the spherical coordinate integral. -

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 ,,

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

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

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

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

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

Evaluate the improper integral. -

Find the average value of the function f over the given region. -f(x, y) = ; R =

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

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Exam 16: Multiple Integration

Solve the problem. -Write an iterated triple integral in the order for the volume of the region enclosed by the paraboloids and .

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. -The curve and the lines and

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

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

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

Evaluate the improper integral. -

Integrate the function f over the given region. -f(x, y) = xy over the triangular region with vertices (0, 0), ( 5, 0), and (0, 8)

Evaluate the cylindrical coordinate integral. -

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. -z = 6 y; R = {(x, y): 0 ≤\le≤ x ≤\le≤ 4, 0 ≤\le≤ y ≤\le≤ 3}

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded by the curves y = ± and the lines x = 1 and x = 9, with density (x) =

Evaluate the spherical coordinate integral. -

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 ,,

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

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

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

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

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

Evaluate the improper integral. -

Find the average value of the function f over the given region. -f(x, y) = ; R =

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