Question 1

Find the mass of the solid with density   and the given shape.

Accepted Answer

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

Question 2

Compute the Riemann sum for the given function and region, a partition with n equal-sized rectangles and the given evaluation rule.

Accepted Answer

A)  -8 
B)  8 
C)  0 
D)  -36 
A)  -8 
B)  8 
C)  0 
D)  -36

Question 3

Using an appropriate coordinate system, evaluate the integral   where Q is the region above z = 0 bounded by the cone   and the sphere   .

Accepted Answer

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

Question 4

Find the volume of the solid bounded by the given surfaces.

Accepted Answer

A)    
B)    
C)  40 
D)  52 
A)    
B)    
C)  40 
D)  52

Question 5

Find a constant c such that   is a joint pdf on the region bounded by   and

Accepted Answer

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

Question 6

Evaluate the integral   , where Q is the region with z > 0 bounded by   and   .

Accepted Answer

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

Question 7

Find the area of the region bounded by

Accepted Answer

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

Question 8

Use an appropriate coordinate system to find the volume of the solid lying along the positive z-axis and bounded by the cone   and the sphere   .

Accepted Answer

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

Question 9

Which if the following could represent the triple integral   in cylindrical coordinates where Q is the region below   and above   ?

Accepted Answer

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

Question 10

Find the surface area of the portion of   below   .

Accepted Answer

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

Question 11

Set up and evaluate the integral   where Q is the region bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2.

Accepted Answer

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

Question 12

Evaluate the iterated integral after changing coordinate systems.

Accepted Answer

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

Question 13

Evaluate the iterated integral   by changing coordinate systems.

Accepted Answer

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

Question 14

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 15

Use a double integral to find the area of the region bounded by   and   .

Accepted Answer

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

Question 16

Find the center of mass of the solid with density   and the given shape.

Accepted Answer

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

Question 17

Find the surface area of the portion of the surface   in the first octant.

Accepted Answer

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

Question 18

Compute the volume of the solid bounded by   ,   ,   ,   and   .

Accepted Answer

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

Question 19

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 20

Evaluate the iterated integral by converting to polar coordinates.

Accepted Answer

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

Find the mass of the solid with density and the given shape.

Compute the Riemann sum for the given function and region, a partition with n equal-sized rectangles and the given evaluation rule.

Using an appropriate coordinate system, evaluate the integral where Q is the region above z = 0 bounded by the cone and the sphere .

Find the volume of the solid bounded by the given surfaces.

Find a constant c such that is a joint pdf on the region bounded by and

Evaluate the integral , where Q is the region with z > 0 bounded by and .

Find the area of the region bounded by

Use an appropriate coordinate system to find the volume of the solid lying along the positive z-axis and bounded by the cone and the sphere .

Which if the following could represent the triple integral in cylindrical coordinates where Q is the region below and above ?

Find the surface area of the portion of below .

Set up and evaluate the integral where Q is the region bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2.

Evaluate the iterated integral after changing coordinate systems.

Evaluate the iterated integral by changing coordinate systems.

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

Use a double integral to find the area of the region bounded by and .

Find the center of mass of the solid with density and the given shape.

Find the surface area of the portion of the surface in the first octant.

Compute the volume of the solid bounded by , , , and .

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

Evaluate the iterated integral by converting to polar coordinates.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

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 14: Multiple Integrals

Find the mass of the solid with density and the given shape.

Compute the Riemann sum for the given function and region, a partition with n equal-sized rectangles and the given evaluation rule.

Using an appropriate coordinate system, evaluate the integral where Q is the region above z = 0 bounded by the cone and the sphere .

Find the volume of the solid bounded by the given surfaces.

Find a constant c such that is a joint pdf on the region bounded by and

Evaluate the integral , where Q is the region with z > 0 bounded by and .

Find the area of the region bounded by

Use an appropriate coordinate system to find the volume of the solid lying along the positive z-axis and bounded by the cone and the sphere .

Which if the following could represent the triple integral in cylindrical coordinates where Q is the region below and above ?

Find the surface area of the portion of below .

Set up and evaluate the integral where Q is the region bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2.

Evaluate the iterated integral after changing coordinate systems.

Evaluate the iterated integral by changing coordinate systems.

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

Use a double integral to find the area of the region bounded by and .

Find the center of mass of the solid with density and the given shape.

Find the surface area of the portion of the surface in the first octant.

Compute the volume of the solid bounded by , , , and .

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

Evaluate the iterated integral by converting to polar coordinates.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

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