Question 1

Use spherical coordinates to find the volume of the indicated region.
-the region bounded above by the sphere   and below by the cone

Accepted Answer

A)  2$\pi$(2 -   ) 
B)    $\pi$(2 -   ) 
C)    $\pi$ (2 -   ) 
D)  2$\pi$(2 -   ) 
A)  2$\pi$(2 -   ) 
B)    $\pi$(2 -   ) 
C)    $\pi$ (2 -   ) 
D)  2$\pi$(2 -   )

Question 2

Evaluate the spherical coordinate integral.
-

Accepted Answer

A)   $\pi$ 
B)    $\pi$ 
C)  5$\pi$ 
D)   $\pi$ 
A)   $\pi$ 
B)    $\pi$ 
C)  5$\pi$ 
D)   $\pi$

Question 3

Find the center of mass of a thin plate of constant density covering the given region.
-The region between the x-axis and the curve y = 6   x,   $\le$ x $\le$

Accepted Answer

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

Question 4

Evaluate the integral.
-

Accepted Answer

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

Question 5

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

Accepted Answer

A)  cos 3 - 1 
B)  2(1 - cos 3) 
C)    (1 - cos 3) 
D)    (1 - cos 3) 
A)  cos 3 - 1 
B)  2(1 - cos 3) 
C)    (1 - cos 3) 
D)    (1 - cos 3)

Question 6

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

Accepted Answer

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

Question 7

Solve the problem.
-Find the average height of the paraboloid   above the annular region   in the   .

Accepted Answer

A)    
B)    
C)  78 
D)    
A)    
B)    
C)  78 
D)

Question 8

Evaluate the improper integral.
-

Accepted Answer

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

Question 9

Solve the problem.
-Find the average height of the paraboloid   above the disk   in the   .

Accepted Answer

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

Question 10

Find the volume of the indicated region.
-the region that lies under the plane   and over the triangle bounded by the lines   ,   , and

Accepted Answer

A)  50 
B)  64 
C)  56 
D)  70 
A)  50 
B)  64 
C)  56 
D)  70

Question 11

Solve the problem.
-Find the center of mass of a tetrahedron of density   bounded by the coordinate planes and the plane   .

Accepted Answer

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

Question 12

Evaluate the spherical coordinate integral.
-

Accepted Answer

A)    $\pi$ (15 $\pi$ - 8) 
B)    $\pi$ (15 $\pi$ - 8) 
C)    $\pi$(15 $\pi$ - 16) 
D)   $\pi$ (15 $\pi$ - 16) 
A)    $\pi$ (15 $\pi$ - 8) 
B)    $\pi$ (15 $\pi$ - 8) 
C)    $\pi$(15 $\pi$ - 16) 
D)   $\pi$ (15 $\pi$ - 16)

Question 13

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

Accepted Answer

A)   $\pi$(36 ln 37 + 36) 
B)   $\pi$( 36 ln 37 - 36) 
C)   $\pi$( 37 ln 37 + 36) 
D)  $\pi$( 37 ln 37 - 36) 
A)   $\pi$(36 ln 37 + 36) 
B)   $\pi$( 36 ln 37 - 36) 
C)   $\pi$( 37 ln 37 + 36) 
D)  $\pi$( 37 ln 37 - 36)

Question 14

Find the center of mass of the thin rod with the given density function.
-Find the center of mass of a thin rod that lies along the first-quadrant portion of the circle   if the density of the rod is

Accepted Answer

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

Question 15

Evaluate the integral.
-

Accepted Answer

A)  6075 
B)    
C)  12,150 
D)  18,225 
A)  6075 
B)    
C)  12,150 
D)  18,225

Question 16

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

Accepted Answer

A)  - 72 
B)  - 18 
C)  - 102 
D)  - 12 
A)  - 72 
B)  - 18 
C)  - 102 
D)  - 12

Question 17

Find the volume of the indicated region.
-the region common to the interiors of the cylinders   and

Accepted Answer

A)    
B)    
C)  32 
D)    
A)    
B)    
C)  32 
D)

Question 18

Change the order of integration and evaluate the integral.
-

Accepted Answer

A)  28$\pi$ 
B)  42$\pi$ 
C)  21$\pi$ 
D)  14$\pi$ 
A)  28$\pi$ 
B)  42$\pi$ 
C)  21$\pi$ 
D)  14$\pi$

Question 19

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

Question 20

Find the area of the region specified in polar coordinates.
-the smaller loop of the curve r = 7 + 14 sin $\theta$

Accepted Answer

A)    (2 $\pi$ - 3   ) 
B)    (2 $\pi$ + 3   ) 
C)    (2 $\pi$ - 3   ) 
D)    (2 $\pi$ + 3   ) 
A)    (2 $\pi$ - 3   ) 
B)    (2 $\pi$ + 3   ) 
C)    (2 $\pi$ - 3   ) 
D)    (2 $\pi$ + 3   )

Use spherical coordinates to find the volume of the indicated region. -the region bounded above by the sphere and below by the cone

Evaluate the spherical coordinate integral. -

Evaluate the integral. -

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

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

Solve the problem. -Find the average height of the paraboloid above the annular region in the .

Evaluate the improper integral. -

Solve the problem. -Find the average height of the paraboloid above the disk in the .

Find the volume of the indicated region. -the region that lies under the plane and over the triangle bounded by the lines , , and

Solve the problem. -Find the center of mass of a tetrahedron of density bounded by the coordinate planes and the plane .

Evaluate the spherical coordinate integral. -

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

Find the center of mass of the thin rod with the given density function. -Find the center of mass of a thin rod that lies along the first-quadrant portion of the circle if the density of the rod is

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 common to the interiors of the cylinders and

Change the order of integration and evaluate the integral. -

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

Find the area of the region specified in polar coordinates. -the smaller loop of the curve r = 7 + 14 sin $\theta$

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

Use spherical coordinates to find the volume of the indicated region. -the region bounded above by the sphere and below by the cone

Evaluate the spherical coordinate integral. -

Find the center of mass of a thin plate of constant density covering the given region. -The region between the x-axis and the curve y = 6 x, ≤\le≤ x ≤\le≤

Evaluate the integral. -

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

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

Solve the problem. -Find the average height of the paraboloid above the annular region in the .

Evaluate the improper integral. -

Solve the problem. -Find the average height of the paraboloid above the disk in the .

Find the volume of the indicated region. -the region that lies under the plane and over the triangle bounded by the lines , , and

Solve the problem. -Find the center of mass of a tetrahedron of density bounded by the coordinate planes and the plane .

Evaluate the spherical coordinate integral. -

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

Find the center of mass of the thin rod with the given density function. -Find the center of mass of a thin rod that lies along the first-quadrant portion of the circle if the density of the rod is

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 common to the interiors of the cylinders and

Change the order of integration and evaluate the integral. -

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

Find the area of the region specified in polar coordinates. -the smaller loop of the curve r = 7 + 14 sin θ\thetaθ

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 smaller loop of the curve r = 7 + 14 sin $\theta$