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14.1 Iterated Integrals and the Area in the Plane
848
14.1 Iterated Integrals and Area in the Plane
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Evaluate the following integral.
a.
b.
c.
d.
e.
____ 2. Evaluate the following integral.
a.
b.
c.
849 Chapter 14: Multiple Integration
d.
e.
____ 3. Evaluate the following integral.
a.
b.
c.
d.
e.
____ 4. Evaluate the following iterated integral.
41.5
38.5
34.5
33.5
37.5
14.1 Iterated Integrals and the Area in the Plane
850
____ 5.
Evaluate the following iterated integral.
a.
b.
c.
d.
e.
____ 6. Evaluate the following iterated integral.
5,901
5,931
5,955
5,970
6,017
____ 7. Evaluate the following iterated integral.
a.
b.
851
Chapter 14: Multiple Integration
c.
d.
e.
____
8.
Evaluate the following improper integral.
The integral does not converge.
____ 9. Evaluate the improper iterated integral .
a.
b.
c.
14.1 Iterated Integrals and the Area in the Plane
852
d.
e.
____ 10. Use an iterated integral to find the area of the region shown in the figure below.
a.
b.
c.
d.
e.
____ 11. Use an iterated integral to find the area of the region bounded by
.
a.
b.
853 Chapter 14: Multiple Integration
The integral is improper and does not converge.
____
12.
Use an iterated integral to find the area of the region bounded by
.
a.
b.
c.
d.
e.
____
13.
Use an iterated integral to find the area of the region bounded by the graphs of the
equations
and
.
a.
b.
c.
d.
e.
14.1 Iterated Integrals and the Area in the Plane
854
____ 14.
Sketch the region R of integration and then switch the order of integration for the
following integral.
a.
b.
c.
d.
e.
855 Chapter 14: Multiple Integration
____ 15. The area of a region R is given by the iterated integral . Switch the
order of integration and show that both orders yield the same area. What is this
area? a.
b.
c.
d.
e.
____ 16. The area of a region R is given by the iterated integrals .
Switch the order of integration and show that both orders yield the same area. What is this area?
32
1025
64
771
63
____ 17. The area of a region R is given by the iterated integral . Switch the order
of integration and show that both orders yield the same area. What is this area?
65
16
21
8
4
14.1 Iterated Integrals and the Area in the Plane
856
____ 18.
Evaluate the iterated integral below. Note that it is necessary to switch the order of
integration.
a.
b.
c.
d.
e.
____ 19. Evaluate the iterated integral by switching the order of
integration. Round your answer to three decimal places.
0.538
0.558
30.538
1.538
4.538
857 Chapter 14: Multiple Integration
14.1 Iterated Integrals and Area in the Plane
Answer Section
14.2 Double Integrals and Volume
858
14.2 Double Integrals and Volume
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Evaluate .
3
8
6
9
5
____ 2. Evaluate . Round your answer to two decimal places.
508.64
307.25
2457.99
2224.97
741.66
____ 3. Set up an integral for both orders of integration, and use the more convenient order to
evaluate the integral below over the region R.
a.
b.
c.
d.
e.
859 Chapter 14: Multiple Integration
____ 4. Use a double integral to find the volume of the indicated solid.
32
16
60
15
24
____ 5. Use a double integral to find the volume of the indicated solid.
a.
14.2 Double Integrals and Volume
860
b.
c.
d.
e.
____ 6. Use a double integral to find the volume of the indicated solid.
none of the above
861 Chapter 14: Multiple Integration
____ 7. Use a double integral to find the volume of the indicated solid.
16
9
4
20
6
____ 8. Set up and evaluate a double integral to find the volume of the solid bounded by the graphs
of the equations given below.
a.
b.
c.
d.
e.
14.2 Double Integrals and Volume
862
____ 9.
Set up and evaluate a double integral to find the volume of the solid bounded by the
graphs of the equations
and
.
1,089
35,937
35,949
11,979
2,178
____ 10. Set up and evaluate a double integral to find the volume of the solid bounded by the
graphs of the equations and in the first octant.
5,184
576
3,456
1,728
1,152
____ 11. Set up and evaluate a double integral to find the volume of the solid bounded by the
graphs of the equations given below.
a.
b.
c.
d.
e.
863
Chapter 14: Multiple Integration
____
12.
Evaluate the iterated integral below. Note that it is necessary to switch the order of
integration.
a.
b.
c.
d.
e.
____ 13. Evaluate the integral by switching the order of
integration. Round your answer to two decimal places.
12.39
19.36
9.73
7.73
26.57
____ 14. Find the average value of over the region R where:
R: rectangle with vertices
a.
b.
14.2 Double Integrals and Volume
864
c.
d.
e.
____ 15. Find the average value of over the region R, where R is a triangle with
vertices .
a.
b.
c.
d.
e.
____ 16. Suppose the Cobb-Douglas production function for an automobile manufacturer is where x
is the number of units of labor and y is the number of units of capital.
Estimate the average production level if the number of units of labor x varies between 150 and 200
and the number of units of capital y varies between 375 and 450. Round your answer to two
decimal places.
229,894.96
15,558.45
55,174.79
24,631.60
5,971.30
865
Chapter 14: Multiple Integration
____
17.
Suppose the temperature in degrees Celsius on the surface of a metal plate is
where x and y are measured in centimeters. Estimate the average temperature
if x varies between 0 and 2 centimeters and y varies between 0 and 7 centimeters.
degrees Celsius
degrees Celsius
degrees Celsius
degrees Celsius
degrees Celsius
14.2 Double Integrals and Volume
866
14.2 Double Integrals and Volume
Answer Section
867 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.
