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838 Chapter 14: Multiple Integration
14.3 Change of Variables: Polar Coordinates
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use polar coordinates to describe the region as shown in the figure below:
a.
b.
c.
d.
e.
____ 2. Evaluate the double integral below.
a.
b.
c.
d.
e.
14.3 Change of Variables: Polar Coordinates 839
____ 3. Evaluate the double integral below.
a.
b.
c.
d.
e.
____ 4. Evaluate the double integral below.
a.
b.
c.
d.
e.
840 Chapter 14: Multiple Integration
____ 5. Identify the region of integration for the following integral.
a. d.
b. e.
c.
14.3 Change of Variables: Polar Coordinates 841
____ 6. Evaluate the following iterated integral by converting to polar coordinates.
a.
b.
c.
d.
e.
____ 7. Evaluate the iterated integral by converting to polar
coordinates.
a.
b.
c.
d.
e.
842 Chapter 14: Multiple Integration
____ 8. Evaluate the following iterated integral by converting to polar coordinates.
a.
b.
c.
d.
e.
____ 9. Evaluate the iterated integral by converting to
polar coordinates. Round your answer to four decimal places.
a. 10.5742
b. 13.5742
c. 17.5742
d. 28.5742
e. 14.5742
____ 10. Combine the sum of the two iterated integrals into a single integral by converting to
polar coordinates. Evaluate the resulting iterated integral.
a.
b.
c.
14.3 Change of Variables: Polar Coordinates 843
d.
4
3
e.
4
3
____ 11. Given use polar coordinates to set up
and evaluate the double integral .
a.
b.
c.
d.
e.
____ 12. Use a double integral in polar coordinates to find the volume of the solid in the first
octant bounded by the graphs of the equations given below.
a.
b.
c.
d.
e.
844 Chapter 14: Multiple Integration
____ 13. Use a double integral in polar coordinates to find the volume of the solid inside the
hemisphere
but outside the cylinder
.
a.
b.
c.
d.
e.
____ 14. Find a such that the volume inside the hemisphere and outside
the cylinder is one-half the volume of the hemisphere. Round your answer to four
decimal places.
a.
b.
c.
d.
e.
____ 15. Determine the diameter of a hole that is drilled vertically through the center of the
solid bounded by the graphs of the equations if one-tenth
of the volume of the solid is removed. Round your answer to four decimal places.
a. 1.2245
b. 31.4490
c. 7.2245
d. 5.4490
e. 15.2245
14.3 Change of Variables: Polar Coordinates 845
____ 16. Use a double integral to find the area of the shaded region as shown in the figure
below.
a.
b.
c.
d.
e.
____ 17. Use a double integral to find the area enclosed by the graph of .
846 Chapter 14: Multiple Integration
a.
b.
c.
d.
e.
____ 18. Use a double integral to find the area enclosed by the graph of .
a.
b.
c.
d.
e.
14.3 Change of Variables: Polar Coordinates 847
____ 19. Use a double integral to find the area of the region inside the circle and
outside the cardioid . Round your answer to two decimal places.
a. 46.68
b. 58.34
c. 20.34
d. 55.34
e. 22.34
____ 20. Suppose the population density of a city is approximated by the model
where x and y are measured in miles. Integrate the
density function over the indicated circular region to approximate the population of the city. Round
your answer to the nearest integer.
a. 417,127
b. 417,029
c. 833,901
d. 833,903
e. 416,951
848 Chapter 14: Multiple Integration
14.3 Change of Variables: Polar Coordinates
Answer Section
MULTIPLE CHOICE
14.3 Change of Variables: Polar Coordinates 849
