433 Chapter 7: Applications of Integration
7.3 Volume: The Shell Method
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use the shell method to set up and evaluate the integral that gives the volume of the
solid generated by revolving the plane region about the y-axis.
a.
b.
c.
d.
e.
____ 2. Use the shell method to set up and evaluate the integral that gives the volume of the solid
generated by revolving the plane region about the y-axis.
a.
b.
c.
7.3 Volume: The Shell Method
434
d.
e.
____ 3. Use the shell method to set up and evaluate an integral that gives the volume of the solid
generated by revolving the plane region about the y-axis.
a.
b.
c.
d.
e.
____ 4. Use the shell method to set up and evaluate the integral that gives the volume of the solid
generated by revolving the plane region bounded by
about the y-axis. Round your answer to three decimal places.
5.987
0.561
2.047
0.972
2.917
____ 5. Use the shell method to set up and evaluate the integral that gives the volume of the solid
generated by revolving the plane region about the x-axis.
a.
b.
c.
d.
e.
435
Chapter 7: Applications of Integration
____
6.
Use the shell method to set up and evaluate the integral that gives the volume of the
solid generated by revolving the plane region about the x-axis.
a.
b.
c.
d.
e.
____ 7. Use the shell method to set up and evaluate the integral that gives the volume of the solid
generated by revolving the plane region about the x-axis.
a.
b.
c.
d.
e.
7.3 Volume: The Shell Method
436
____
8.
Use the shell method to set up and evaluate the integral that gives the volume of the
solid generated by revolving the plane region bounded by
about the
x-axis.
a.
b.
c.
d.
e.
____
9.
Use the shell method to find the volume of the solid generated by revolving the plane
region about the line
.
a.
b.
c.
d.
e.
____ 10.
Use the shell method to find the volume of the solid generated by revolving the plane
region bounded by
about the line
.
a.
b.
c.
d.
e.
437
Chapter 7: Applications of Integration
____
11.
Use the disk or shell method to find the volume of the solid generated by revolving
the region bounded by the graphs of the equations about the line
.
a.
b.
c.
d.
e.
____ 12.
Use the disk or the shell method to find the volume of the solid generated by
revolving the region bounded by the graphs of the equations
about the x-axis.
a.
b.
c.
d.
e.
____ 13.
Use the disk or the shell method to find the volume of the solid generated by
revolving the region bounded by the graphs of the equations
about the line
.
a.
b.
c.
d.
e.
7.3 Volume: The Shell Method
438
____ 14.
Use the disk or shell method to find the volume of the solid generated by revolving
the region bounded by the graphs of the equations about the line
.
a.
b.
c.
d.
e.
____ 15.
Use the disk or the shell method to find the volume of the solid generated by
revolving the region bounded by the graphs of the equations
about the
x-axis. Round your answer to two decimal places.
a.
674.33
b.
17.75
c.
96.05
d.
301.76
e.
453.96
____ 16. Use the disk or shell method to find the volume of the solid generated by revolving the
region in the first quadrant bounded by the graph of the equation about the given line.
(i) the x-axis; (ii) the y-axis
a.
b.
c.
d.
439 Chapter 7: Applications of Integration
e.
____ 17. A solid is generated by revolving the region bounded by and
about the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so
that one-fourth of the volume is removed. Find the diameter of the hole. Round your answer
to three decimal places.
10.733
11.167
4.392
2.536
8.785
____ 18. A solid is generated by revolving the region bounded by and
about the y-axis. A hole, centered along the axis of revolution, is drilled through this
solid so that one-third of the volume is removed. Find the diameter of the hole. Round your
answer to three decimal places.
11.547
5.774
4.867
18.838
9.734
7.3 Volume: The Shell Method
440
7.3 Volume: The Shell Method
Answer Section
1.
2.
11.
13.
14.
17.
441 Chapter 7: Applications of Integration
7.4 Arc Length and Surfaces of Revolution
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the arc length of the graph of the function
over the interval [14,
16].
a.
b.
c.
d.
e.
____
2.
Find the arc length of the graph of the function
over the interval [0,6]
.
a.
b.
c.
d.
e.
____
3.
Find the arc length of the graph of the function
over the interval [1,2].
a.
b.
c.
d.
e.
7.4 Arc Length and Surfaces of Revolution
442
____
4.
Find the arc length of the graph of the function
over the interval
[1,64].
a.
b.
c.
d.
e.
____
5.
Find the arc length of the graph of the function
over the interval
.
a.
b.
c.
d.
e.
____
6.
Find the arc length of the graph of the function
over the interval
.
a.
b.
c.
d.
e.
443
Chapter 7: Applications of Integration
____
7.
Use the integration capabilities of a graphing utility to approximate the arc length of
the graph of
on the interval
. Round your answer to three decimal places.
3.620
5.020
10.540
3.820
5.270
____ 8.
Electrical wires suspended between two towers form a catenary modeled by the
equation
where x and y are measured in meters. The towers are 50
meters apart. Find the length of the suspended cable. Round your answer to three decimal places.
94.421m
50.000 m
64.077 m
40.048 m
47.211 m
____ 9. A barn is 75 feet long and 50 feet wide. A cross section of the roof is the inverted
catenary . Find the number of square feet of roofing on the barn. Round
your answer to the nearest integer.
7.4 Arc Length and Surfaces of Revolution
444
4180 square feet
4200 square feet
2100 square feet
1875 square feet
3750 square feet
____ 10. Find the arc length from clockwise to (2, 2 3 ) along the circle . Round
your answer to four decimal places.
4.1888
0.2618
0.1309
1.8546
2.0944
____ 11. Find the area of the surface generated by revolving the curve about the x-axis.
.
a.
b.
445 Chapter 7: Applications of Integration
c.
d.
e.
____ 12. Find the area of the surface generated by revolving the curve about the x-axis.
.
a.
b.
c.
d.
e.
7.4 Arc Length and Surfaces of Revolution
446
____ 13.
Set up and evaluate the definite integral for the area of the surface formed by
revolving the graph of about the x-axis.
a.
b.
c.
d.
e.
____ 14. Find the area of the surface generated by revolving the curve about the y-axis.
a.
b.
c.
d.
e.
447
Chapter 7: Applications of Integration
____
15.
Set up and evaluate the definite integral for the area of the surface formed by
revolving the graph of
about the y-axis. Round your answer to three decimal places.
58.659
117.319
235.685
5.323
2.661
____ 16. Find the area of the surface generated by revolving the curve about the y-axis.
.
a.
b.
c.
d.
e.
7.4 Arc Length and Surfaces of Revolution
448
____ 17. An ornamental light bulb is designed by revolving the graph of
about the x-axis where x and y are measured in feet. Approximate the amount of glass needed to make
the bulb. (Assume that the glass is 0.016 inches thick.) Round your answer to four decimal places.
a.
b.
c.
d.
e.
____ 18. A bridge has a main span of about 1400 meters. Each of its towers has a height of about
146 meters. Approximate the length of a parabolic cable along the main span if the length C of
the cable is given by . Round your answer to one decimal place.
449 Chapter 7: Applications of Integration
719.8 meters
759.4 meters
2120.7 meters
1439.6 meters
2879.2 meters
7.4 Arc Length and Surfaces of Revolution
450
7.4 Arc Length and Surfaces of Revolution
Answer Section
451 Chapter 7: Applications of Integration
7.5 Work
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Determine the work done by lifting a 140 pound bag of sugar 12 feet.
1,680 ft-lb
16,800 ft-lb
2,600 ft-lb
168 ft-lb
260 ft-lb
____ 2. A force of 10 pounds compresses a 21-inch spring 8 inches. How much work is done in
compressing the spring from a length of 15 inches to a length of 13 inches?
22.5 ft-lb
19.5 ft-lb
17.5 ft-lb
12.5 ft-lb
27.5 ft-lb
____ 3. A force of 265 Newtons stretches a spring 20 centimeters. How much work is done in
stretching the spring from 15 centimeters to 25 centimeters?
28.5 joules
31.5 joules
26.5 joules
21.5 joules
36.5 joules
____ 4. A force of 20 pounds stretches a spring 11 inches in an exercise machine. Find the work
done in stretching the spring 2 feet from its natural position.
ft-lb
ft-lb
ft-lb
ft-lb
ft-lb
7.5 Work
452
____ 5.
An overhead garage door has two springs, one on each side of the door. A force of 12
pounds is required to stretch each spring 1 foot. Because of the pulley system, the springs stretch
only one-half the distance the door travels. The door moves a total of 10 feet and springs are at their
natural length when the door is open. Find the work done by pair of strings.
150 ft-lb
300 ft-lb
2,400 ft-lb
1,200 ft-lb
600 ft-lb
____ 6. Neglecting air resistance and the weight of the propellant, determine the work done in
propelling a six-ton satellite to a height of 200 miles above earth. Assume that Earth has a radius of
4000 miles.
ft-lb
ft-lb
ft-lb
ft-lb
ft-lb
____ 7. Neglecting air resistance and the weight of the propellant, determine the work done in
propelling a 12-ton satellite to a height of 100 miles above Earth. Assume that the Earth has a
radius of 4000 miles.
1,756.10 mi-ton
1,170.73 mi-ton
702.44 mi-ton
936.59 mi-ton
2,575.61 mi-ton
____ 8. A lunar module weighs 16 tons on the surface of Earth. How much work is done in
propelling the module from the surface of the moon to a height of 50 miles? Consider the radius of
the moon to be 1100 miles and its force of gravity to be one-sixth that of Earth.
ft-lb
ft-lb
ft-lb
ft-lb
ft-lb