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15.3 Conservative Vector Fields and Independence of Path 919
15.3 Conservative Vector Fields and Independence of Path
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Calculate the line integral along for and C is any path
starting at the point and ending at .
a.
b.
c.
d.
e.
____ 2. Calculate the line integral along for and C is any path
starting at the point and ending at .
a.
b.
c.
d.
e.
920 Chapter 15: Vector Analysis
____ 3. For the vector field , find the value of A for which the
field is conservative.
a. 5
b. 6
c. 4
d. 25
e. is not conservative for any value of A.
____ 4. Determine whether or not the vector field is conservative.
a. conservative
b. not conservative
____ 5. Find the value of the line integral , where and
.
(Hint: If F is conservative, the integration may be easier on an alternate path.)
a. 3,072
b. 192
c. 6,144
d. 6,144
e. 6,912
____ 6. Find the value of the line integral on the closed path consisting of line
segments from to from to and then from to where
.
(Hint: If F is conservative, the integration may be easier on an alternate path.)
a. 99
b. 12
c. 36
d. 30
e. 0
15.3 Conservative Vector Fields and Independence of Path 921
____ 7. Find the value of the line integral .
(Hint: If is conservative, the integration may be easier on an alternate path.)
a. 243
b. 192
c. 108
d. 0
e. 12
____ 8. Find the value of the line integral .
(Hint: If is conservative, the integration may be easier on an
alternate path.)
a.
922 Chapter 15: Vector Analysis
b.
c.
d.
e.
____ 9. Find the value of the line integral where C is an ellipse
from to .
(Hint: If is conservative, the integration may be easier on an alternate
path.)
a.
b.
c. 8
d.
e.
15.3 Conservative Vector Fields and Independence of Path 923
____ 10. Find the value of the line integral , where and
.
(Hint: If F is conservative, the integration may be easier on an alternate path.)
a. 4
b. 8
c. 4
d. –4
e. –8
____ 11. Find the value of the line integral where and
.
(Hint: If F is conservative, the integration may be easier on an alternate path.)
a. –24
b. 48
c. –48
d. 24
e. 0
____ 12. Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a
computer algebra system to verify your results.
C: a smooth curve from to
a. 30
b. 15
c. 13
d. 56
e. 169
924 Chapter 15: Vector Analysis
____ 13. Evaluate the line integral using the Fundamental
Theorem of Line Integrals, where C is the line segment from to .
a. –2
b. –1
c. 2
d. 1
e. 0
____ 14. Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a
computer algebra system to verify your results.
C: circle clockwise from to
a.
b.
c.
d.
e.
____ 15. Evaluate the line integral using the Fundamental
Theorem of Line Integrals, where C is the smooth curve from to .
a. 274
b. 298
c. 300
d. 30
e. 272
15.3 Conservative Vector Fields and Independence of Path 925
____ 16. Find the work done by the force field in moving an object from P to Q.
a. 27,992
b. 55,985
c. 83,978
d. 41,989
e. 69,982
____ 17. A stone weighing 2 pounds is attached to the end of a four-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.
a. 201.06
b. 0.00
c. 804.25
d. 256.00
e. 12.57
926 Chapter 15: Vector Analysis
15.3 Conservative Vector Fields and Independence of Path
Answer Section
MULTIPLE CHOICE
