Question 1

Evaluate the cylindrical coordinate integral.
-

Accepted Answer

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

Question 2

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

Accepted Answer

A)    $\pi$ 
B)    $\pi$ 
C)  15$\pi$ 
D)    $\pi$ 
A)    $\pi$ 
B)    $\pi$ 
C)  15$\pi$ 
D)    $\pi$

Question 3

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

Accepted Answer

A)    
B)    
C)    
D)  25,000 
A)    
B)    
C)    
D)  25,000

Question 4

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

Accepted Answer

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

Question 5

Find the area of the region specified in polar coordinates.
-the region enclosed by the curve r = 8 - 7 cos $\theta$

Accepted Answer

A)    $\pi$ 
B)    $\pi$ 
C)  81$\pi$ 
D)  113$\pi$ 
A)    $\pi$ 
B)    $\pi$ 
C)  81$\pi$ 
D)  113$\pi$

Question 6

Evaluate the double integral over the given region.
-  R = {(x, y): 0 $\le$x$\le$6, 0$\le$y$\le$4}

Accepted Answer

A)  ln 7   
B)  ln 7 ln 5 
C)  5 ln 7 
D)  ln 35 
A)  ln 7   
B)  ln 7 ln 5 
C)  5 ln 7 
D)  ln 35

Question 7

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

Question 8

Integrate the function f over the given region.
-f(x, y) =   over the region bounded by the x-axis, line   and curve

Accepted Answer

A)  8 
B)  1 
C)  7 
D)  9 
A)  8 
B)  1 
C)  7 
D)  9

Question 9

Evaluate the spherical coordinate integral.
-

Accepted Answer

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

Question 10

Find the center of mass of a thin plate of constant density covering the given region.
-The region bounded by the parabola x =   and the line x = 4

Accepted Answer

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

Question 11

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.
-z = 8x + 4y + 7; R = {(x, y): 0 $\le$ x $\le$ 1, 1 $\le$ y $\le$ 3}

Accepted Answer

A)  28 
B)  26 
C)  36 
D)  38 
A)  28 
B)  26 
C)  36 
D)  38

Question 12

Find the center of mass of a thin plate covering the given region with the given density function.
-The region bounded by the parabola y = 36 -   and the x-axis, with density  (x) = 6

Accepted Answer

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

Question 13

Use spherical coordinates to find the volume of the indicated region.
-the region that lies inside the sphere   and outside the cylinder

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

A)  16,807 $\pi$ 
B)  117,649 $\pi$ 
C)  33,614 $\pi$ 
D)  235,298 $\pi$ 
A)  16,807 $\pi$ 
B)  117,649 $\pi$ 
C)  33,614 $\pi$ 
D)  235,298 $\pi$

Question 16

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)  5838 
B)  3892 
C)  7784 
D)  11,676 
A)  5838 
B)  3892 
C)  7784 
D)  11,676

Question 17

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

Accepted Answer

A)  9 $\pi$ 
B)  36 $\pi$ 
C)  3 $\pi$ 
D)  18 $\pi$ 
A)  9 $\pi$ 
B)  36 $\pi$ 
C)  3 $\pi$ 
D)  18 $\pi$

Question 18

Find the average value of the function f over the given region.
-f(x, y) = 5x + 10y over the triangle with vertices   ,   , and

Accepted Answer

A)    
B)  15 
C)  25 
D)  13 
A)    
B)  15 
C)  25 
D)  13

Question 19

Solve the problem.
-Find the average height of the part of the paraboloid   that lies above the   .

Accepted Answer

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

Question 20

Find the center of mass of a thin plate of constant density covering the given region.
-The region in the first and fourth quadrants enclosed by the curves   and   and by the lines   and

Accepted Answer

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

Evaluate the cylindrical coordinate integral. -

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

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

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

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 8 - 7 cos $\theta$

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y): 0 \le x \le 6, 0 \le y \le 4}$ R = {(x, y): 0 $\le$ x $\le$ 6, 0 $\le$ y $\le$ 4}

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

Integrate the function f over the given region. -f(x, y) = over the region bounded by the x-axis, line and curve

Evaluate the spherical coordinate integral. -

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the parabola x = and the line x = 4

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. -z = 8x + 4y + 7; R = {(x, y): 0 $\le$ x $\le$ 1, 1 $\le$ 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 parabola y = 36 - and the x-axis, with density (x) = 6

Use spherical coordinates to find the volume of the indicated region. -the region that lies inside the sphere and outside the cylinder

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

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

Change the order of integration and evaluate the integral. -

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) = 5x + 10y over the triangle with vertices , , and

Solve the problem. -Find the average height of the part of the paraboloid that lies above the .

Find the center of mass of a thin plate of constant density covering the given region. -The region in the first and fourth quadrants enclosed by the curves and and by the lines and

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 cylindrical coordinate integral. -

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

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

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

Find the area of the region specified in polar coordinates. -the region enclosed by the curve r = 8 - 7 cos θ\thetaθ

Evaluate the double integral over the given region. - R = {(x, y): 0 ≤\le≤ x ≤\le≤ 6, 0 ≤\le≤ y ≤\le≤ 4}

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

Integrate the function f over the given region. -f(x, y) = over the region bounded by the x-axis, line and curve

Evaluate the spherical coordinate integral. -

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the parabola x = and the line x = 4

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. -z = 8x + 4y + 7; R = {(x, y): 0 ≤\le≤ x ≤\le≤ 1, 1 ≤\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 parabola y = 36 - and the x-axis, with density (x) = 6

Use spherical coordinates to find the volume of the indicated region. -the region that lies inside the sphere and outside the cylinder

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

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

Change the order of integration and evaluate the integral. -

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) = 5x + 10y over the triangle with vertices , , and

Solve the problem. -Find the average height of the part of the paraboloid that lies above the .

Find the center of mass of a thin plate of constant density covering the given region. -The region in the first and fourth quadrants enclosed by the curves and and by the lines and

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 = 8 - 7 cos $\theta$

Evaluate the double integral over the given region. - $Evaluate the double integral over the given region. - R = {(x, y): 0 \le x \le 6, 0 \le y \le 4}$ R = {(x, y): 0 $\le$ x $\le$ 6, 0 $\le$ y $\le$ 4}

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. -z = 8x + 4y + 7; R = {(x, y): 0 $\le$ x $\le$ 1, 1 $\le$ y $\le$ 3}