Question 1

Evaluate   along the path C, defined as counterclockwise along the circle   from   to   . ​

Accepted Answer

A)    
B)    
C)  162 
D)  0 
E)  324 
A)    
B)    
C)  162 
D)  0 
E)  324

Question 2

Use Green's Theorem to evaluate the integral   where   is the boundary of the region lying inside the rectangle bounded by   ,   ,   ,   and outside the square bounded by   , and   . ​

Accepted Answer

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

Question 3

Let   and let S be the graph of   oriented counterclockwise. Use Stokes's Theorem to evaluate   . ​

Accepted Answer

A)  0 
B)    
C)    
D)    
E)    
A)  0 
B)    
C)    
D)    
E)

Question 4

Evaluate   , where   is   . ​

Accepted Answer

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

Question 5

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) ​   ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 6

Find the mass of the surface lamina S of density   . ​   ​

Accepted Answer

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

Question 7

Evaluate the line integral along the given path.

Accepted Answer

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

Question 8

Find a piecewise smooth parametrization of the path C given in the following graph.

Accepted Answer

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

Question 9

Evaluate   , where   and S is given by   ,   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 10

Calculate the line integral along   for   and C is any path starting at the point   and ending at   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 11

Find the divergence of the vector field   given by   .

Accepted Answer

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

Question 12

Use the Divergence Theorem to evaluate   and find the outward flux of   through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Accepted Answer

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

Question 13

Find the area of the lateral surface over the curve   in the xy-plane and under the surface   where Lateral surface   . ​   line from   to   .

Accepted Answer

A)   
B)  0 
C)    
D)    
E)    
A)   
B)  0 
C)    
D)    
E)

Question 14

Determine whether the vector field is conservative.

Accepted Answer

A)  conservative 
B)  not conservative 
A)  conservative 
B)  not conservative

Question 15

Use Green's Theorem to evaluate the integral ​   ​
For the path C: boundary of the region lying between the graphs of   and   .
​

Accepted Answer

A)    ​ 
B)    ​ 
C)    
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    
D)    ​ 
E)    ​

Question 16

Use a computer algebra system and the result "The centroid of the region having area A bounded by the simple closed path   is   ,   '' to find the centroid of the region bounded by the graphs of   and   . ​

Accepted Answer

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

Question 17

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by   and   . ​

Accepted Answer

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

Question 18

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 19

Find the divergence of the vector field at the given point.   ,

Accepted Answer

A)  153 
B)  155 
C)  173 
D)  161 
E)  157 
A)  153 
B)  155 
C)  173 
D)  161 
E)  157

Question 20

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function   and sketch the graph. ​

Accepted Answer

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

Evaluate along the path C, defined as counterclockwise along the circle from to .

Use Green's Theorem to evaluate the integral where is the boundary of the region lying inside the rectangle bounded by , , , and outside the square bounded by , and .

Let and let S be the graph of oriented counterclockwise. Use Stokes's Theorem to evaluate .

Evaluate , where is .

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)

Find the mass of the surface lamina S of density .

Evaluate the line integral along the given path.

Find a piecewise smooth parametrization of the path C given in the following graph.

Evaluate , where and S is given by , .

Calculate the line integral along for and C is any path starting at the point and ending at .

Find the divergence of the vector field given by .

Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Find the area of the lateral surface over the curve in the xy-plane and under the surface where Lateral surface . line from to .

Determine whether the vector field is conservative.

Use Green's Theorem to evaluate the integral For the path C: boundary of the region lying between the graphs of and .

Use a computer algebra system and the result "The centroid of the region having area A bounded by the simple closed path is , '' to find the centroid of the region bounded by the graphs of and .

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by and .

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function .

Find the divergence of the vector field at the given point. ,

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function and sketch the graph.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Filters

Exam 15: Vector Anal

Evaluate along the path C, defined as counterclockwise along the circle from to . ​

Use Green's Theorem to evaluate the integral where is the boundary of the region lying inside the rectangle bounded by , , , and outside the square bounded by , and . ​

Let and let S be the graph of oriented counterclockwise. Use Stokes's Theorem to evaluate . ​

Evaluate , where is . ​

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) ​ ​

Find the mass of the surface lamina S of density . ​ ​

Evaluate the line integral along the given path.

Find a piecewise smooth parametrization of the path C given in the following graph.

Evaluate , where and S is given by , . ​

Calculate the line integral along for and C is any path starting at the point and ending at . ​

Find the divergence of the vector field given by .

Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Find the area of the lateral surface over the curve in the xy-plane and under the surface where Lateral surface . ​ line from to .

Determine whether the vector field is conservative.

Use Green's Theorem to evaluate the integral ​ ​ For the path C: boundary of the region lying between the graphs of and . ​

Use a computer algebra system and the result "The centroid of the region having area A bounded by the simple closed path is , '' to find the centroid of the region bounded by the graphs of and . ​

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by and . ​

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function . ​

Find the divergence of the vector field at the given point. ,

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function and sketch the graph. ​

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Filters

Evaluate along the path C, defined as counterclockwise along the circle from to .

Use Green's Theorem to evaluate the integral where is the boundary of the region lying inside the rectangle bounded by , , , and outside the square bounded by , and .

Let and let S be the graph of oriented counterclockwise. Use Stokes's Theorem to evaluate .

Evaluate , where is .

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)

Find the mass of the surface lamina S of density .

Evaluate , where and S is given by , .

Calculate the line integral along for and C is any path starting at the point and ending at .

Find the area of the lateral surface over the curve in the xy-plane and under the surface where Lateral surface . line from to .

Use Green's Theorem to evaluate the integral For the path C: boundary of the region lying between the graphs of and .

Use a computer algebra system and the result "The centroid of the region having area A bounded by the simple closed path is , '' to find the centroid of the region bounded by the graphs of and .

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by and .

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function .

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function and sketch the graph.