Question 1

Evaluate the improper integral.
-

Accepted Answer

A)  40(ln 40 - 1) 
B)   
C)  40(ln 40 - 2) 
D)

Question 2

Find the average value of the function f over the given region.
-f(x, y) = 10x + 3y; R = {(x, y): 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 10}

Accepted Answer

A)  15 
B)  25 
C)   
D)  10

Question 3

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)  72  
B)  32  
C)  36  
D)  48

Question 4

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

Accepted Answer

A)  1 ln 2 (1 - sin 16) 
B)  2 ln 2 (1 - cos 16) 
C)  1 ln 2 (1 - cos 16) 
D)  2 ln 2 (1 - sin 16)

Question 5

Evaluate the cylindrical coordinate integral.
-

Accepted Answer

A)  48 
B)  32 
C)  16 
D)  24

Question 6

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)  9855 
B)  22,995 
C)  3285 
D)  16,425

Question 7

Evaluate the spherical coordinate integral.
-

Accepted Answer

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

Question 8

Evaluate the integral.
-

Accepted Answer

A)  42 
B)   
C)  21 
D)

Question 9

Solve the problem.
-Let D be the smaller cap cut from a solid ball of radius 4 units by a plane 2 units from the center of the sphere. Set up the triple integral for the volume of D in rectangular coordinates.

Accepted Answer

A)   
B)   
C)   
D)

Question 10

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

Accepted Answer

A)  4  
B)  2  
C)  4  
D)  2

Question 11

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

Accepted Answer

A)  735 
B)   
C)   
D)

Question 12

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

Accepted Answer

A)   
B)   
C)   
D)

Question 13

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

Accepted Answer

A)  18 
B)  27 
C)  54 
D)  36

Question 14

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

Accepted Answer

A)  2(1 + sin 7) 
B)  2(1 - cos 7) 
C)  2 sin 7 
D)  2 cos 7

Question 15

Find the center of mass of a thin plate covering the given region with the given density function.
-The triangular region cut from the first quadrant by the line y = -x + 4, with density  (x) = x

Accepted Answer

A)   
B)  
C)  
D)

Question 16

Use the given transformation to evaluate the integral.
- where R is the interior of the ellipsoid

Accepted Answer

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

Question 17

Evaluate the improper integral.
-

Accepted Answer

A)  24 
B)  72 
C)  48 
D)  8

Question 18

Solve the problem.
-Find the average distance from a point  in the annular region  to the origin.

Accepted Answer

A)   
B)   
C)   
D)

Question 19

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

Accepted Answer

A)  33 
B)  1 
C)  87 
D)  195

Question 20

Find the Jacobian for the given transformation
-

Accepted Answer

A)  1470v 
B)  1260u 
C)  1260v 
D)  1470u

Exam 16: Multiple Integration

Evaluate the improper integral. -

Find the average value of the function f over the given region. -f(x, y) = 10x + 3y; R = {(x, y) : 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 10}

Change the order of integration and evaluate the integral. -

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

Evaluate the cylindrical coordinate integral. -

Change the order of integration and evaluate the integral. -

Evaluate the spherical coordinate integral. - $Evaluate the spherical coordinate integral. - A) \pi B) \pi C) \pi D) \pi$

Evaluate the integral. -

Solve the problem. -Let D be the smaller cap cut from a solid ball of radius 4 units by a plane 2 units from the center of the sphere. Set up the triple integral for the volume of D in rectangular coordinates.

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

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

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

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

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

Find the center of mass of a thin plate covering the given region with the given density function. -The triangular region cut from the first quadrant by the line y = -x + 4, with density (x) = x

Evaluate the improper integral. -

Solve the problem. -Find the average distance from a point in the annular region to the origin.

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

Find the Jacobian for the given transformation -

Exam 16: Multiple Integration

Evaluate the improper integral. -

Find the average value of the function f over the given region. -f(x, y) = 10x + 3y; R = {(x, y) : 0 ≤\le≤ x ≤\le≤ 2, 0 ≤\le≤ y ≤\le≤ 10}

Change the order of integration and evaluate the integral. -

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

Evaluate the cylindrical coordinate integral. -

Change the order of integration and evaluate the integral. -

Evaluate the spherical coordinate integral. -

Evaluate the integral. -

Solve the problem. -Let D be the smaller cap cut from a solid ball of radius 4 units by a plane 2 units from the center of the sphere. Set up the triple integral for the volume of D in rectangular coordinates.

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

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

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

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

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

Find the center of mass of a thin plate covering the given region with the given density function. -The triangular region cut from the first quadrant by the line y = -x + 4, with density (x) = x

Use the given transformation to evaluate the integral. - where R is the interior of the ellipsoid

Evaluate the improper integral. -

Solve the problem. -Find the average distance from a point in the annular region to the origin.

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

Find the Jacobian for the given transformation -

Find the average value of the function f over the given region. -f(x, y) = 10x + 3y; R = {(x, y) : 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 10}

Evaluate the spherical coordinate integral. - $Evaluate the spherical coordinate integral. - A) \pi B) \pi C) \pi D) \pi$