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Stewart_Calc_7ET ch06sec02
MULTIPLE CHOICE
1. Find the volume of the solid obtained by rotating the region bounded by
about the x-axis.
a.
b.
c.
d.
e.
2. 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
3. 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
4. The region bounded by the given curves is rotated about the specified axis. Find the volume
of the resulting solid by any method. Round your answer to 3 decimal places.
about the y-axis and .
a.
1763.213
b.
None of these
c.
1752.016
d.
1760.025
e.
880.012
5. 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.
6. 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
a.
b.
c.
d.
e.
7. Find the volume of the solid that is obtained by revolving the region about the x-axis.
a.
b.
y = x –( 2 )2+ 3
1 2 3 4 5 6 7–1 x
2
4
6
8
10
12
14
16
18
y
199
5
597
20
c.
d.
8. Find the volume of the solid that is obtained by revolving the region about the line y = .
a.
b.
c.
d.
89
9. 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
a.
b.
199
20
199
10
5
2
y =5/2
y = x y = x3
1x
1
2
3
y
89
42
89
14
89
84
256
512
3
c.
d.
10. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the x-axis.
y = cos x + 1, x = 0, y = 0, x =
a.
+
b.
+
c.
+
d.
+
11. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated line.
y = 4 – , y = 0; the line y = 5
a.
b.
c.
d.
128
256
3
1
2
3
4
3
4
1088
5
1088
15
544
15
2176
15
12. 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.
13. Find the volume of the solid obtained by rotating the region bounded by
about the line
a.
b.
c.
d.
e.
14. Find the volume of a pyramid with height and base an equilateral triangle with side a = .
a.
b.
c.
d.
e.
NUMERIC RESPONSE
1. Find the volume of the frustum of a pyramid with square base of side square top of
side and height
2. The base of S is a circular region with boundary curve Cross-sections
perpendicular to the x axis are isosceles right triangles with hypotenuse in the base.
Find the volume of S.
3. Find the volume common to two spheres, each with radius r = if the center of each
sphere lies on the surface of the other sphere.
4. Use a computer algebra system to find the exact volume of the solid obtained by rotating the
region bounded by the given curves about the specified line.
5. The base of S is the parabolic region Cross-sections perpendicular to
the y axis are squares.
Find the volume of S.
6. Find the volume of a cap of a sphere with radius r = and height h = 3 .
7. True or False?
The volume of a solid torus (the donut-shaped solid shown in the figure) with r = and R =
is
8. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the
region bounded by the given curves about the specified axis.
9. Find the volume of the solid obtained by rotating about the x-axis the region under the curve
from x = to x = .
10. Find the volume of the solid obtained by rotating the region bounded by and
about the line
11. Find the volume of the solid obtained by rotating the region bounded by and
about the y-axis.
12. Find the volume of the solid obtained by rotating the region bounded by the given curves
about the specified axis.
SHORT ANSWER
1. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated axis.
y = , y = 0, x = 3, x = 5; the x-axis
2. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated axis.
, , , ; the x-axis
3. Use a graphing utility to (a) plot the graphs of the given functions and (b) find the
approximate x-coordinates of the points of intersection of the graphs. Then find an
approximation of the volume of the solid obtained by revolving the region bounded by the
graphs of the functions about the x-axis. Round answers to two decimal places.
y = , y = 3 –
4. Sketch a plane region, and indicate the axis about which it is revolved so that the resulting
solid of revolution has the volume given by the integral.
5. 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 = 0, x = 1, x = 12; the y-axis
y
