Question 1

Find a vector-valued function whose graph is the cone   . ​

Accepted Answer

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

Question 2

Let   and let S be the surface bounded by the plane   and the coordinates planes. Verify the Divergence Theorem by evaluating   as a surface integral and as a triple integral.

Accepted Answer

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

Question 3

Find the value of the line integral   , where   and   . ​

Accepted Answer

A)  2 
B)  4 
C)  8 
D)  -2 
E)  -4 
A)  2 
B)  4 
C)  8 
D)  -2 
E)  -4

Question 4

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

Accepted Answer

A)  the plane   ​ 
B)  the cylinder   ​ 
C)  the plane   ​ 
D)  the plane   ​ 
E)  the plane   ​ 
A)  the plane   ​ 
B)  the cylinder   ​ 
C)  the plane   ​ 
D)  the plane   ​ 
E)  the plane   ​

Question 5

Use the Divergence Theorem to evaluate   Verify your answer by evaluating the integral as a triple integral. ​   ​
S: cube bounded by the planes   ​

Accepted Answer

A)  81 
B)  9 
C)  162 
D)  18 
E)  324 
A)  81 
B)  9 
C)  162 
D)  18 
E)  324

Question 6

Use Stokes's Theorem to evaluate   . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. ​   ​
C: triangle with vertices   ​

Accepted Answer

A)  4 
B)  2 
C)  5 
D)  6 
E)  1 
A)  4 
B)  2 
C)  5 
D)  6 
E)  1

Question 7

Evaluate the integral   along the path   , defined as y-axis from   to   . ​

Accepted Answer

A)  16 
B)  32 
C)  85 
D)  8 
E)  64 
A)  16 
B)  32 
C)  85 
D)  8 
E)  64

Question 8

Find the value of the line integral   , where   and   ,   .
(Hint: If F is conservative, the integration may be easier on an alternate path.)

Accepted Answer

A)  686 
B)  56 
C)  1,372 
D)  2,744 
E)  224 
A)  686 
B)  56 
C)  1,372 
D)  2,744 
E)  224

Question 9

Evaluate   where   is represented by   .     ,

Accepted Answer

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

Question 10

Match the following vector-valued function with its graph. ​   ​

Accepted Answer

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

Question 11

Find   for the lamina   with uniform density of 1. Use a computer algebra system to verify your result. ​

Accepted Answer

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

Question 12

Find the area of the surface over the given region. Use a computer algebra system to verify your results. ​
The sphere,
​   ,   ​

Accepted Answer

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

Question 13

Find the value of the line integral   where   is an ellipse   from   to   . ​

Accepted Answer

A)    
B)    
C)  4 
D)    
E)    
A)    
B)    
C)  4 
D)    
E)

Question 14

A stone weighing 5 pounds is attached to the end of a five-foot string and is whirled horizontally with one end held fixed. It makes 1 revolution per second. Find the work done by the force F that keeps the stone moving in a circular path.
[Hint: Use Force = (mass)(centripetal acceleration).] Round your answer to two decimal places, if required.

Accepted Answer

A)  1,570.80 
B)  0.00 
C)  7,853.98 
D)  2,500.00 
E)  78.54 
A)  1,570.80 
B)  0.00 
C)  7,853.98 
D)  2,500.00 
E)  78.54

Question 15

Verify Stokes's Theorem by evaluating ​   ​
As a line integral and as a double integral.
​     ​

Accepted Answer

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

Question 16

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field   . Find   , where S is the upper surface of the cylindrical container. ​

Accepted Answer

A)  8 
B)  7 
C)  1 
D)  0 
E)  6 
A)  8 
B)  7 
C)  1 
D)  0 
E)  6

Question 17

Find the curl of the vector field   . ​

Accepted Answer

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

Question 18

Find the value of the line integral   where   and ​   .

Accepted Answer

A)  -21 
B)  42 
C)  -42 
D)  21 
E)  0 
A)  -21 
B)  42 
C)  -42 
D)  21 
E)  0

Question 19

Find the area of the surface over the given region. Use a computer algebra system to verify your results. ​
The part of the cone,
​   ​
Where   and   .
​

Accepted Answer

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

Question 20

Use Green's Theorem to calculate the work done by the force   on a particle that is moving counterclockwise around the closed path   where   is the boundary of the region lying between the graphs of   , and   . Round your answer to two decimal places. ​

Accepted Answer

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

Find a vector-valued function whose graph is the cone .

Let and let S be the surface bounded by the plane and the coordinates planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral.

Find the value of the line integral , where and .

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

Use the Divergence Theorem to evaluate Verify your answer by evaluating the integral as a triple integral. S: cube bounded by the planes

Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. C: triangle with vertices

Evaluate the integral along the path , defined as y-axis from to .

Find the value of the line integral , where and , . (Hint: If F is conservative, the integration may be easier on an alternate path.)

Evaluate where is represented by . ,

Match the following vector-valued function with its graph.

Find for the lamina with uniform density of 1. Use a computer algebra system to verify your result.

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The sphere, ,

Find the value of the line integral where is an ellipse from to .

Verify Stokes's Theorem by evaluating As a line integral and as a double integral.

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field . Find , where S is the upper surface of the cylindrical container.

Find the curl of the vector field .

Find the value of the line integral where and .

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the cone, Where and .

Use Green's Theorem to calculate the work done by the force on a particle that is moving counterclockwise around the closed path where is the boundary of the region lying between the graphs of , and . Round your answer to two decimal places.

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

Find a vector-valued function whose graph is the cone . ​

Let and let S be the surface bounded by the plane and the coordinates planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral.

Find the value of the line integral , where and . ​

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

Use the Divergence Theorem to evaluate Verify your answer by evaluating the integral as a triple integral. ​ ​ S: cube bounded by the planes ​

Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. ​ ​ C: triangle with vertices ​

Evaluate the integral along the path , defined as y-axis from to . ​

Find the value of the line integral , where and , . (Hint: If F is conservative, the integration may be easier on an alternate path.)

Evaluate where is represented by . ,

Match the following vector-valued function with its graph. ​ ​

Find for the lamina with uniform density of 1. Use a computer algebra system to verify your result. ​

Find the area of the surface over the given region. Use a computer algebra system to verify your results. ​ The sphere, ​ , ​

Find the value of the line integral where is an ellipse from to . ​

Verify Stokes's Theorem by evaluating ​ ​ As a line integral and as a double integral. ​ ​

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field . Find , where S is the upper surface of the cylindrical container. ​

Find the curl of the vector field . ​

Find the value of the line integral where and ​ .

Find the area of the surface over the given region. Use a computer algebra system to verify your results. ​ The part of the cone, ​ ​ Where and . ​

Use Green's Theorem to calculate the work done by the force on a particle that is moving counterclockwise around the closed path where is the boundary of the region lying between the graphs of , and . Round your answer to two decimal places. ​

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

Find a vector-valued function whose graph is the cone .

Find the value of the line integral , where and .

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

Use the Divergence Theorem to evaluate Verify your answer by evaluating the integral as a triple integral. S: cube bounded by the planes

Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. C: triangle with vertices

Evaluate the integral along the path , defined as y-axis from to .

Match the following vector-valued function with its graph.

Find for the lamina with uniform density of 1. Use a computer algebra system to verify your result.

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The sphere, ,

Find the value of the line integral where is an ellipse from to .

Verify Stokes's Theorem by evaluating As a line integral and as a double integral.

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field . Find , where S is the upper surface of the cylindrical container.

Find the curl of the vector field .

Find the value of the line integral where and .

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the cone, Where and .

Use Green's Theorem to calculate the work done by the force on a particle that is moving counterclockwise around the closed path where is the boundary of the region lying between the graphs of , and . Round your answer to two decimal places.