Name: __________________
Class:
Date: _____________
1 Find the area of the region that lies under the given curve:
8 Find the volume of the solid obtained by rotating the region
5
4x + 2 , 0 y =
x Select the correct answer. The choices are rounded to the nearest
thousandth.
2
3
a.
1.981
c.
2.088
e.
2.078
b.
1.978
d.
1.853
f.
1.958
5
bounded by y = x and x = y about the line x = 1.
1
a.
939 77
904 462
b.
c.
904 42
9 Find the volume of a right circular cone with height h = 15 and
base radius r = 2.
2
a.
2
Sketch the region enclosed by y = 7x and y = x + 10 .
Decide whether to integrate with respect to x or y. Draw a typical
approximating rectangle and label its height and width. Then find
the area of the region.
a.
19.213259
c.
86.066297
e.
5.737753
b.
2.868877
d.
34.426519
f.
17.213259
2
20 60 b.
c.
20
10 Find the volume of a cap of a sphere with radius r = 6 and height
h = 0.9.
2
Sketch the region enclosed by x = 5 y and x = y 10 . Decide whether to integrate with respect to x or y. Draw a
typical approximating rectangle and label its height and width.
Then find the area of the region.
a.
54.772256
c.
273.861279
e.
55.772256
b.
18.257419
d.
164.316767
f.
10.954451
a.
5.13 b.
4.86 c.
4.617 4 Find the positive value of c such that the area of the region
2
bounded by the parabolas y = x
is 4608.
2
c and y = c
2
2
x
Select the correct answer.
5 Find the volume of the solid obtained by rotating about the x axis the region under the curve y =
1 from x = 1 to x = 6.
x
a.
Select the correct answer.
a.
7
5 6
b.
7 6
c.
6 Find the volume of the solid obtained by rotating the region in the
2
first quadrant bounded by y = x and y = 7 about the y axis.
a.
7 2
49
2
b.
49 2
c.
7 Find the volume of the solid obtained by rotating the region
bounded by y =
a.
PAGE 1
8 18
4
x and y = x about the line y = 1 .
b.
6 18
c.
8
18
d.
11 Find the volume of a pyramid with height h = 9 and rectangular
base with dimensions 9 and 18.
6
18
54
b.
243
c.
486
Name: __________________
Class:
Date: _____________
12 Cavalieri's Principle states that if a family of parallel planes
gives equal cross sectional areas for two solids S and S then
1
2
16 Use the method of cylindrical shells to find the volume
generated by rotating the region bounded by the given curves
about the specified axis.
the volumes of S and S are equal.
1
2
2
y = x , y = 0 , x = 1 , x = 2 ; about x = 1
True or False:
If r = 3 and h = 9, then the volume of the oblique cylinder shown
in the figure is 81 .
a.
V = 17 6
c.
V = 34 b.
V = 22 d.
V = 15 17 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.
y = sin x , y = 0 , x = 5 , x =
8 ; about the y axis.
8
a.
V =
2 x sin(x) dx
5
True
False
8
b.
13 Find the volume common to two spheres, each with radius r = 8
if the center of each sphere lies on the surface of the other
sphere.
14 Use the method of cylindrical shells to find the volume
generated by rotating the region bounded by the given curves
about the y axis.
V =
2
x sin(x) dx
2
x sin(x) dx
5
5
c.
V =
0
8
d.
V =
2 sin(x) dx
5
y = 1 , y = 0,x = 1,x = 9
x
Select the correct answer.
a. V = 17 c.
b.
V = 7
d.
A sphere of radius r.
V = 16 V = 8
15 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
x = 5 + y , x = 0,y = 1,y = 3
a.
V = 85 c.
V = 160 b.
V = 78 d.
V = 80 PAGE 2
18 Use cylindrical shells to find the volume of the solid.
V = 5 r
3
3
a.
V = 1 r
3
3
b.
V = 2 r
3
3
c.
V = 4 r
3
3
d.
Name: __________________
Class:
19 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 t through the center of a
sphere of radius K and express the answer in terms of h.
Date: _____________
23 Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line.
y = x axis.
1 , x = 6, x = 9, y = 0; about the x
Enter your answer as an expression using the symbol decimal rounded to the nearest hundredths.
or as a
Sketch the region bounded by the given curves.
a.
3
V = 1 h
4
c.
2
V = 1 h
6
b.
3
V = 1 h
6
d.
2
V = 1 h
3
a.
20 Find the area of the region bounded by the curves y = sin x ,
x
y = e , x = 0 , and x = 2
.
21 Evaluate the integral.
1
3x
3
3x dx
1
22 (a) Find the number a such that the line x = a bisects the area
under the curve y =
a =
1 for 1 2
x
x 3.
b.
________
(b) Find the number b such that the line y = b bisects the
area under the curve y =
b =
1 ,1 2
x
x 3.
________
..to be continued
PAGE 3
Name: __________________
Class:
Date: _____________
continuation
Sketch the solid and a typical disk or washer obtained by
revolving the region about the specified line.
c.
a.
d.
b.
..to be continued
e.
PAGE 4
Name: __________________
Class:
continuation
c.
d.
e.
PAGE 5
Date: _____________
Name: __________________
Class:
Date: _____________
continuation
24 Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line.
2
y = 4x , 0 x 2, y = 16; about the y axis.
Enter your answer as an expression using the symbol decimal rounded to the nearest hundredths.
or as a
c.
Sketch the region bounded by the given curves.
a.
d.
b.
e.
..to be continued
PAGE 6
Name: __________________
Class:
Date: _____________
continuation
Sketch the solid and a typical disk or washer obtained by
revolving the region about the specified line.
c.
a.
d.
b.
..to be continued
e.
PAGE 7
Name: __________________
PAGE 8
Class:
Date: _____________