Stewart – Calculus 8e ET Chapter 6 Form G
____ 5. Sketch the region enclosed by . Find the area of the region.
Select the correct answer.
a.
32
3
b.
c.
d.
32
3
e.
6. Find the volume of the solid obtained by rotating the region bounded by about the
x-axis.
7. Find the volume of the solid obtained by rotating the region bounded by the given curves about the
specified line.
____ 8. Find the volume of the solid obtained by rotating the region bounded by the given curves about the
specified axis. Select the correct answer.
a.
b.
c.
d.
e. None of these
Stewart – Calculus 8e ET Chapter 6 Form G
9. Find the volume of the solid that is obtained by revolving the region about the line y =
5
2
.
y =5/2
y = x
y = x3
1x
1
2
3
y
10. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations and inequalities about the y-axis.
– = 16, x 0, y = – 4, y = 4
11. Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves. Then
use this information to estimate the volume of the solid obtained by rotating about the y axis the
region enclosed by these curves.
Rounded to the nearest hundredth.
12. Find the volume of a pyramid with height and base an equilateral triangle with side a = .
Stewart – Calculus 8e ET Chapter 6 Form G
____ 13. The base of a solid is a circular disk with radius . Find the volume of the solid if parallel
cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the
base. Select the correct answer.
a. 6
b. 7
c. 16
d. 49
e. None of these
14. Use the method of cylindrical shells to find the volume of solid obtained by rotating the region
bounded by the given curves about the specified axis.
15. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated line. Sketch the region and a representative rectangle.
y = 16 – , y = 0; the line x = – 4
16. Suppose you make napkin rings by drilling holes with different diameters through two wooden balls
(which also have different diameters). You discover that both napkin rings have the same height t as
shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling
a hole with radius d through the center of a sphere of radius D and express the answer in terms of .
Stewart – Calculus 8e ET Chapter 6 Form G
____ 17. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the x-axis. Select the correct answer.
y = 2sin x + 1, x = 0, y = 0, x =
a.
b.
c.
3
d.
3
18. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated line.
y = 1 – x2, y = 0; the line y = 3
____ 19. A bucket weighs lb and a rope of negligible weight are used to draw water from a well that is
ft deep. The bucket starts with 50 lb of water and is pulled up at a rate of 10 ft/s, but water leaks out
of a hole in the bucket at a rate of 0.5 lb/s. Find the work done in pulling the bucket to the top of the
well. Select the correct answer.
a.
b.
c.
d.
e.
20. A particle moves a distance of 190 ft along a straight line. As it moves, it is acted upon by a
constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work
done by the force?
Stewart – Calculus 8e ET Chapter 6 Form G
Answer Key
Stewart – Calculus 8e ET Chapter 6 Form G
Stewart – Calculus 8e ET Chapter 6 Form H
1. Sketch the region enclosed by the curves Find the area of
the region correct to two decimal places.
____ 2. Graph the region between the curves and use your calculator to compute the area correct to five decimal
places. Select the correct answer.
a. 12.08127
b. 91.504
c. 3.141257
d. 3.66016
e. 18.3008
3. Find the area of the region bounded by the given curves.
4. Find the area of the shaded region.
y =
y = x –
–6 x2+6x
x
/
_
`
12x
1
2
3
–1
y
5. Sketch the region bounded by the graphs of the given equations and find the area of their region.
, ,
6. Sketch the region bounded by the graphs of the given equations and find the area of that region.
y = sec2 x + 6, y = 8, x = – , x =
Stewart – Calculus 8e ET Chapter 6 Form H
____ 7. Use integration to find the area of the triangle with the given vertices. Select the correct answer.
(0, 0), (1, 7), (4, 2)
a.
13
2
b. 26
c. 28
d.
13
8. Sketch the region enclosed by . Find the area of the region.
____ 9. Find the volume of the solid obtained by rotating the region bounded by the given curves about the
specified line. Select the correct answer.
a.
b.
c.
d.
e. None
Stewart – Calculus 8e ET Chapter 6 Form H
____ 10. Find the volume of the solid obtained by rotating the region bounded by the given curves about the
specified axis. Select the correct answer.
a.
b.
c.
d.
e. None of these
11. The height of a monument is m. A horizontal cross-section at a distance x meters from the top is
an equilateral triangle with side meters. Find the volume of the monument.
____ 12. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations and inequalities about the y-axis. Select the correct answer.
– = 16, x 0, y = – 4, y = 4
a.
256
b.
512
3
c.
128
d.
256
3
____ 13. The base of a solid is a circular disk with radius . Find the volume of the solid if parallel
cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the
base. Select the correct answer.
a. 6
b. 7
c. 16
d. 49
e. None of these
Stewart – Calculus 8e ET Chapter 6 Form H
14. Use the method of cylindrical shells to find the volume of the solid generated by revolving the
region bounded by the graphs of the equations about the indicated axis. Sketch the region and a
representative rectangle.
y = , y = –x + 4; the y-axis
15. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated line. Sketch the region and a representative rectangle.
y = 16 – , y = 0; the line x = – 4
____ 16. Use cylindrical shells to find the volume of the solid.
A sphere of radius .
Select the correct answer.
a.
b.
c.
d.
e.
____ 17. A spring has a natural length of cm. If a force of N is required to keep it stretched to a length
of cm, how much work is required to stretch it from cm to cm? Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 6 Form H
18. If J of work are needed to stretch a spring from cm to cm and another J are needed to
stretch it from cm to cm, what is the natural length of the spring? Round the answer to
nearest integer.
____ 19. An aquarium m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half
of the water out of the aquarium. (Use the facts that the density of water is )
Select the correct answer.
a.
b.
c.
d.
e.
20. A particle moves a distance of 190 ft along a straight line. As it moves, it is acted upon by a
constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work
done by the force?
Stewart – Calculus 8e ET Chapter 6 Form H
Answer Key
Stewart – Calculus 8e ET Chapter 6 Form H