Question 1

Evaluate the triple integral   .   ,

Accepted Answer

A)    
B)  10 
C)    
D)    
A)    
B)  10 
C)    
D)

Question 2

Evaluate the iterated integral by first changing the order of integration.

Accepted Answer

A)    
B)    
C)    
D)  The integral can be evaluated only numerically. 
A)    
B)    
C)    
D)  The integral can be evaluated only numerically.

Question 3

Use an appropriate coordinate system to compute the volume of the solid below   , above   and inside   .

Accepted Answer

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

Question 4

A function   is a pdf on the three-dimensional region Q if   for all   in Q and   . Find k such that   is a pdf on the region bounded by z = 0, z = 3, x = 0, x = 1, y - z = 0 , and y - z = 3.

Accepted Answer

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

Question 5

Find a transformation from a rectangular region S in the uv-plane to the region R. Show all your work.
R is bounded by y = x2, y = x2 + 3, y = 6 - x2, and y = 3 - x2

Accepted Answer

Answer details may vary.
The r

Question 6

Find the mass and the center of mass of the lamina bounded by   and   with density   .

Accepted Answer

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

Question 7

Convert the equation   into spherical coordinates.

Accepted Answer

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

Question 8

Find the Jacobian of the given transformation.

Accepted Answer

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

Question 9

Evaluate the double integral.

Accepted Answer

A)  352 
B)  416 
C)  -352 
D)  -416 
A)  352 
B)  416 
C)  -352 
D)  -416

Question 10

Suppose that   is the population density of a species of a certain small animal. Estimate the population in the triangular region     and   .

Accepted Answer

A)  4578 
B)  1916 
C)  4704 
D)  1740 
A)  4578 
B)  1916 
C)  4704 
D)  1740

Question 11

Compute the Riemann sum for the given function, the irregular partition shown and midpoint evaluation.

Accepted Answer

A)  48 
B)  33 
C)  -57 
D)  57 
A)  48 
B)  33 
C)  -57 
D)  57

Question 12

Evaluate the iterated integral after changing coordinate systems.

Accepted Answer

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

Evaluate the triple integral . ,

Evaluate the iterated integral by first changing the order of integration.

Use an appropriate coordinate system to compute the volume of the solid below , above and inside .

A function is a pdf on the three-dimensional region Q if for all in Q and . Find k such that is a pdf on the region bounded by z = 0, z = 3, x = 0, x = 1, y - z = 0 , and y - z = 3.

Find a transformation from a rectangular region S in the uv-plane to the region R. Show all your work. R is bounded by y = x², y = x² + 3, y = 6 - x², and y = 3 - x²

Find the mass and the center of mass of the lamina bounded by and with density .

Convert the equation into spherical coordinates.

Find the Jacobian of the given transformation.

Evaluate the double integral.

Suppose that is the population density of a species of a certain small animal. Estimate the population in the triangular region and .

Compute the Riemann sum for the given function, the irregular partition shown and midpoint evaluation.

Evaluate the iterated integral after changing coordinate systems.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Vector Calculus

Second Order Differential Equations

Filters

Exam 15: Multiple Integrals

Evaluate the triple integral . ,

Evaluate the iterated integral by first changing the order of integration.

Use an appropriate coordinate system to compute the volume of the solid below , above and inside .

A function is a pdf on the three-dimensional region Q if for all in Q and . Find k such that is a pdf on the region bounded by z = 0, z = 3, x = 0, x = 1, y - z = 0 , and y - z = 3.

Find a transformation from a rectangular region S in the uv-plane to the region R. Show all your work. R is bounded by y = x2, y = x2 + 3, y = 6 - x2, and y = 3 - x2

Find the mass and the center of mass of the lamina bounded by and with density .

Convert the equation into spherical coordinates.

Find the Jacobian of the given transformation.

Evaluate the double integral.

Suppose that is the population density of a species of a certain small animal. Estimate the population in the triangular region and .

Compute the Riemann sum for the given function, the irregular partition shown and midpoint evaluation.

Evaluate the iterated integral after changing coordinate systems.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Vector Calculus

Second Order Differential Equations

Filters

Find a transformation from a rectangular region S in the uv-plane to the region R. Show all your work. R is bounded by y = x², y = x² + 3, y = 6 - x², and y = 3 - x²