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Stewart_Calc_7ET ch06sec03
MULTIPLE CHOICE
1. Use the method of cylindrical shells to find the volume of solid obtained by rotating the
region bounded by the given curves about the x-axis.
a.
b.
c.
d.
e.
2. 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
a.
c.
128
64
3
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
b.
128
d.
3. 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.
c.
b.
d.
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
64
3
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
4. 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 = 25 – , y = 0; the line x = – 5
a.
c.
b.
d.
5000
3
2 4 6 8–2–4–6–8 x
10
20
30
y
2500
3
2 4 6 8–2–4–6–8 x
10
20
30
y
1250
625
5. Sketch a plane region and indicate the axis about which it is revolved so that the resulting
solid of revolution (found using the shell method) is given by the integral.
2
a.
x-axis
c.
x-axis
b.
d.
2 4 6 8–2–4–6–8 x
10
20
30
y
2 4 6 8–2–4–6–8 x
10
20
30
y
1x
1
y
1x
1
y
y-axis
y-axis
6. Use cylindrical shells to find the volume of the solid.
A sphere of radius .
a.
b.
c.
d.
e.
7. 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 h 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 .
1x
1
y
1x
1
y
a.
b.
c.
d.
e.
8. 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.
Select the correct answer. The choices are rounded to the nearest hundredth.
a.
b.
c.
d.
e.
NUMERIC RESPONSE
1. Use the method of cylindrical shells to find the volume of solid obtained by rotating the
region bounded by the given curves about the x-axis.
2. 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.
3. Use the method of cylindrical shells to find the volume generated by rotating the region
bounded by the given curves about the specified axis.
SHORT ANSWER
1. 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 = 3 , y = 0, x = 1; the y-axis
2. 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 = 0, x = 2, x = 5; the y-axis
3. Use a graphing utility to (a) plot the graphs of the given functions, (b) find the approximate
x-coordinates of the points of intersection of the graphs, and (c) find an approximation of the
volume of the solid obtained by revolving the region bounded by the graphs of the functions
about the y-axis. Round answers to two decimal places.
y = x, y = – , x 0
y
4. 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.
y = 2 , y = 4x – 2, y = 8; the y-axis
5. 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 = , y = x – 1; the line x = 2
6. 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.
y = , y = 2x – 1, y = 4; the y-axis
