Stewart – Calculus 8e ET Chapter 6 Form E
____ 10. Find the volume of a pyramid with height and base an equilateral triangle with side a = .
Select the correct answer.
a.
b.
c.
d.
e.
11. 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.
12. 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
Stewart – Calculus 8e ET Chapter 6 Form E
13. 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 .
____ 14. 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.
x2 – y2 = 81, x 0, y = –9, y = 9
a.
972
b.
1458
c.
1944
d.
2916
15. 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
Stewart – Calculus 8e ET Chapter 6 Form E
16. Use the method of cylindrical shells to find the volume of the solid generated by revolving the
region about the indicated axis or line.
12345 x
1
2
3
4
5
y
y-axis
____ 17. A tank is full of water. Find the work required to pump the water out of the outlet.
Select the correct answer.
a.
b.
c.
d.
e. None of these
18. 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?
19. 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.
Stewart – Calculus 8e ET Chapter 6 Form E
____ 20. Evaluate the integral. Select the correct answer.
a.
b.
c.
d.
Stewart – Calculus 8e ET Chapter 6 Form E
Answer Key
Stewart – Calculus 8e ET Chapter 6 Form E
Stewart – Calculus 8e ET Chapter 6 Form F
____ 1. Sketch the region enclosed by the curves Find the area of
the region correct to two decimal places. Select the correct answer.
a.
b.
c.
d.
e.
f.
2. Find the area of the region bounded by the given curves.
3. Find the number b such that the line divides the region bounded by the curves and
into two regions with equal area.
4. Find the area of the shaded region.
y=x+1
y–x=
2
–1
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
Stewart – Calculus 8e ET Chapter 6 Form F
____ 5. 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
6. 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.
____ 7. 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.
– = 1, x 0, y = – 1, y = 1
a.
8
3
b.
4
c.
4
3
d.
2
8. 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.
9. 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.
Stewart – Calculus 8e ET Chapter 6 Form F
____ 10. 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. Select the correct answer.
a.
b.
c.
d.
e.
11. 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 = 9 – x2, y = 0; the line
____ 12. Find the volume of the solid generated by revolving the region bounded by the graphs of the
equations about the indicated line. Select the correct answer.
y = 1 – , y = 0; the line y = 6
a.
112
15
b.
224
5
c.
448
15
d.
224
15
Stewart – Calculus 8e ET Chapter 6 Form F
____ 13. 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. Select the correct answer.
a.
b.
c.
d.
e.
14. Use cylindrical shells to find the volume of the solid.
A sphere of radius .
Stewart – Calculus 8e ET Chapter 6 Form F
____ 15. 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 p 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 .
Select the correct answer.
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 indicated line.
y = 4 – x2, y = 0; the line y = 6
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?
18. A heavy rope, ft long, weighs lb/ft and hangs over the edge of a building 110 ft high. How
much work is done in pulling the rope to the top of the building?
Stewart – Calculus 8e ET Chapter 6 Form F
____ 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? Select the correct answer.
a.
b.
c.
d.
e.
20. A steel girder weighing 800 lb is hoisted from ground level to the roof of a 30-ft building using a
chain that weighs 2 lb/running foot. Find the work done.
Stewart – Calculus 8e ET Chapter 6 Form F
Answer Key
Stewart – Calculus 8e ET Chapter 6 Form G
1.
Find the area of the region bounded by the given curves.
____ 2. Find the number b such that the line divides the region bounded by the curves and
into two regions with equal area. Select the correct answer.
a.
b.
c.
d.
e.
3. Find the area of the shaded region.
y =
y = x –
–6 x2+6x
x
/
_
`
12x
1
2
3
–1
y
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 =