Question 1

Find the area of the region specified in polar coordinates.
-the region enclosed by the curve r = 4 sin 2$\theta$

Accepted Answer

A)    $\pi$ 
B)  2$\pi$ 
C)  4$\pi$ 
D)  8$\pi$ 
A)    $\pi$ 
B)  2$\pi$ 
C)  4$\pi$ 
D)  8$\pi$

Question 2

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

Accepted Answer

A)  4 
B)  120 
C)  60 
D)  8 
A)  4 
B)  120 
C)  60 
D)  8

Question 3

Find the volume of the indicated region.
-the region that lies under the paraboloid   and above the triangle enclosed by the lines     , and

Accepted Answer

A)    
B)  8505 
C)  76,545 
D)  945 
A)    
B)  8505 
C)  76,545 
D)  945

Question 4

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, 0$\le$ y $\le$ 1}

Accepted Answer

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

Question 5

Evaluate the integral.
-

Accepted Answer

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

Question 6

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 semicircle y =

Accepted Answer

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

Question 7

Solve the problem.
-Let D be the region that is bounded below by the cone   and above by the sphere   Set up the triple integral for the volume of D in cylindrical coordinates.

Accepted Answer

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

Question 8

Evaluate the improper integral.
-

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Find the volume of the indicated region.
-the region bounded by the paraboloid   , the cylinder   , and the

Accepted Answer

A)    $\pi$ 
B)  5000 $\pi$ 
C)     $\pi$ 
D)  2500 $\pi$ 
A)    $\pi$ 
B)  5000 $\pi$ 
C)     $\pi$ 
D)  2500 $\pi$

Question 11

Evaluate the improper integral.
-

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 two quadrants of the disk   to the origin.

Accepted Answer

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

Question 13

Find the area of the region specified in polar coordinates.
-the region enclosed by the curve r = 9 cos 3$\theta$

Accepted Answer

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

Question 14

Find the center of mass of a thin plate of constant density covering the given region.
-The region bounded by the parabola y = 9 -   and the x-axis

Accepted Answer

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

Question 15

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

Accepted Answer

A)  9 
B)  2 
C)  4 
D)  18 
A)  9 
B)  2 
C)  4 
D)  18

Question 16

Find the center of mass of a thin plate of constant density covering the given region.
-The region bounded by y =   and y = 3

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

Change the order of integration and evaluate the integral.
-

Accepted Answer

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

Question 19

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

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 = {(x, y): 1 $\le$ x $\le$ 5, 1$\le$ y $\le$ 5}

Accepted Answer

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

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 4 sin 2 $\theta$

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 that lies under the paraboloid and above the triangle enclosed by the lines , and

Evaluate the integral. -

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 semicircle y =

Solve the problem. -Let D be the region that is bounded below by the cone and above by the sphere Set up the triple integral for the volume of D in cylindrical coordinates.

Evaluate the improper integral. -

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

Find the volume of the indicated region. -the region bounded by the paraboloid , the cylinder , and the

Evaluate the improper integral. -

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

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 9 cos 3 $\theta$

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the parabola y = 9 - and the x-axis

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

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by y = and y = 3

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

Change the order of integration and evaluate the integral. -

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

Find the average value of the function f over the given region. -f(x, y) = $Find the average value of the function f over the given region. -f(x, y) = ; R = {(x, y): 1 \le x \le 5, 1 \le y \le 5}$ ; R = {(x, y): 1 $\le$ x $\le$ 5, 1 $\le$ y $\le$ 5}

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 area of the region specified in polar coordinates. -the region enclosed by the curve r = 4 sin 2 θ\thetaθ

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 that lies under the paraboloid and above the triangle enclosed by the lines , and

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, 0 ≤\le≤ y ≤\le≤ 1}

Evaluate the integral. -

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 semicircle y =

Solve the problem. -Let D be the region that is bounded below by the cone and above by the sphere Set up the triple integral for the volume of D in cylindrical coordinates.

Evaluate the improper integral. -

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

Find the volume of the indicated region. -the region bounded by the paraboloid , the cylinder , and the

Evaluate the improper integral. -

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

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 9 cos 3 θ\thetaθ

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the parabola y = 9 - and the x-axis

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

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by y = and y = 3

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

Change the order of integration and evaluate the integral. -

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

Find the average value of the function f over the given region. -f(x, y) = ; R = {(x, y): 1 ≤\le≤ x ≤\le≤ 5, 1 ≤\le≤ y ≤\le≤ 5}

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 = 4 sin 2 $\theta$

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

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 9 cos 3 $\theta$

Find the average value of the function f over the given region. -f(x, y) = $Find the average value of the function f over the given region. -f(x, y) = ; R = {(x, y): 1 \le x \le 5, 1 \le y \le 5}$ ; R = {(x, y): 1 $\le$ x $\le$ 5, 1 $\le$ y $\le$ 5}