Stewart – Calculus 8e ET Chapter 6 Form C
____ 12. Use the method of disks or washers, or 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.
, , , the x-axis
a.
135
c.
45
b.
90
d.
180
Stewart – Calculus 8e ET Chapter 6 Form C
____ 13. Use cylindrical shells to find the volume of the solid.
A sphere of radius .
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 6 Form C
____ 14. 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 .
a.
b.
c.
d.
e.
____ 15. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the x-axis.
y = 2sin x + 1, x = 0, y = 0, x =
a.
b.
c.
3
d.
3
Stewart – Calculus 8e ET Chapter 6 Form C
____ 16. 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
a.
104
15
b.
208
15
c.
104
5
d.
52
15
____ 17. A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length
of 15 cm. How much work is done in stretching the spring from cm to cm?
a.
b.
c.
d.
e.
____ 18. A tank is full of water. Find the work required to pump the water out of the outlet.
a.
b.
c.
d.
e. None of these
Stewart – Calculus 8e ET Chapter 6 Form C
____ 19. 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?
a.
b.
c.
d.
e.
____ 20. Newton’s Law of Gravitation states that two bodies with masses attract each other with a
force
where r is the distance between the bodies and G is the gravitation constant. If one of bodies is fixed,
find the work needed to move the other from to .
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 6 Form C
Answer Key
Stewart – Calculus 8e ET Chapter 6 Form D
Select the correct answer for each question.
____ 1. Find the area of the region bounded by the given curves.
a. 2
b.
2
3
c. 4
d.
e.
____ 2. Graph the region between the curves and use your calculator to compute the area correct to five decimal
places.
a. 18.3008
b. 3.66016
c. 91.504
d. 3.141257
e. 12.08127
Stewart – Calculus 8e ET Chapter 6 Form D
____ 3. Find the area of the shaded region.
y=x+4
y–x=
2
–4
12345678–1–2–3–4–5–6–7–8 x
1
2
3
4
5
6
7
–1
–2
–3
–4
–5
–6
–7
–8
–9
y
a.
100
3
b.
25
3
c.
50
9
d.
50
3
Stewart – Calculus 8e ET Chapter 6 Form D
____ 4. Find the area of the shaded region.
y =
y = x –
–7 x2+7x
x
/
_
`
12x
1
2
3
–1
y
a.
2
3
b.
4
3
c.
1
7
d. 8
Stewart – Calculus 8e ET Chapter 6 Form D
____ 5. Sketch the region bounded by the graphs of the given equations and find the area of their region.
, ,
a.
1–1 x
1
–1
y
c.
1–1 x
1
–1
y
b.
1–1 x
1
–1
y
d.
1–1 x
1
–1
y
Stewart – Calculus 8e ET Chapter 6 Form D
____ 6. Sketch the region bounded by the graphs of the given equations and find the area of their region.
, ,
a.
10 x
10
–10
y
19
c.
10 x
10
–10
y
19
3
b.
10 x
10
–10
y
38
3
d.
10 x
10
–10
y
38
3
Stewart – Calculus 8e ET Chapter 6 Form D
____ 7. Sketch the region bounded by the graphs of the given equations and find the area of that region.
y = sec2 x + 4, y = 6, x = – , x =
a.
1–1 x
2
4
6
–2
y
–
c.
1–1 x
2
4
6
–2
y
– 2
b.
1–1 x
2
4
6
–2
y
6
d.
1–1 x
2
4
6
–2
y
– 1
Stewart – Calculus 8e ET Chapter 6 Form D
____ 8. Find the volume of the solid obtained by rotating the region bounded by about the
x-axis.
a.
b.
c.
d.
e.
____ 9. Find the volume of the solid obtained by rotating the region bounded by the given curves about the
specified line.
a.
b.
c.
d.
e. None
Stewart – Calculus 8e ET Chapter 6 Form D
____ 10. Find the volume of the solid obtained by rotating the region bounded by the given curves about the
specified axis.
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.
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 6 Form D
____ 12. Find the volume of the solid that is obtained by revolving the region about the x-axis.
y = x –(2)
2
+3
12345678x
2
4
6
8
10
12
14
16
18
20
y
a.
129
5
b.
258
5
c.
387
10
d.
129
10
____ 13. 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.
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 6 Form D
____ 14. 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.
a. 49
b. 16
c. 6
d. 7
e. None of these
____ 15. Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region
under the y-axis the region under the curve.
The choices are rounded to the nearest hundredth.
a.
b.
c.
d.
e.
____ 16. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the x-axis.
y = sin x + 1, x = 0, y = 0, x =
a.
b.
3
2
c.
3
2
d.
Stewart – Calculus 8e ET Chapter 6 Form D
____ 17. A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length
of 15 cm. How much work is done in stretching the spring from cm to cm?
a.
b.
c.
d.
e.
____ 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.
a.
b.
c.
d.
e.
____ 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 )
a.
b.
c.
d.
e.
____ 20. A particle moves a distance of 160 ft along a straight line. As it moves, it is acted upon by a
constant force of magnitude 10 lb in a direction opposite to that of the motion. What is the work
done by the force?
a. –1,600 ft-lb
b.
1
1600
ft-lb
c.
1
16
ft-lb
d.
16
ft-lb
Stewart – Calculus 8e ET Chapter 6 Form D
Answer Key
Stewart – Calculus 8e ET Chapter 6 Form E
____ 1. 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
2. Find the area of the shaded region.
y=x+4
y–x=
2
–4
12345678–1–2–3–4–5–6–7–8 x
1
2
3
4
5
6
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y
3. Sketch the region bounded by the graphs of the given equations and find the area of their region.
, ,
4. 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 E
____ 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 the given curves about the
specified line.
7. The region bounded by the given curves is rotated about the specified axis. Find the volume of the
resulting solid by any method.
about the y-axis
8. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated line.
y = 1 – , y = 0; the line y = 5
9. Find the volume of the solid obtained by rotating the region bounded by about
the line