Section 8.2 - Volumes by Slicing
7.3
Solids of Revolution
Find the volume of the solid generated by revolving the regions
bounded by y  x  x 2 and y  0 about the x-axis.
1
1

  r dx    x  x
2
0
0

2 2
dx
Find the volume of the solid generated by revolving the regions
bounded by y  2sin2x, 0  x  2 about the x-axis.
/2


0
/2
r dx     2sin2x  dx
2
0
2
Find the volume of the solid generated by revolving the regions
bounded by y  21 x, x  0, y  2 about the y-axis.
2
2
  r dy     2y  dy
2
0
0
2
Find the volume of the solid generated by revolving the regions
bounded by y  3  x 2 , y  1 about the line y = -1.
2
2

  r dx    3  x
2
2
2
2
   1
2
dx
CALCULATOR REQUIRED
The volume of the solid generated by revolving the first quadrant
region bounded by the curve y  e x / 2 and the lines x = ln 3 and
y = 1 about the x-axis is closest to
a) 2.79
b) 2.82
c) 2.85
ln3

   ex / 2

0
d) 2.88
  1  dx
2
2
e) 2.91
CALCULATOR REQUIRED
The volume of the solid generated by rotating about the x-axis
the region enclosed between the curve y  3x 2 and the line
y  6x is given by
3
A.   6x  3x

dx

dx
2 2
0
2
B.   6x  3x
2 2
0
2
C.   9x 4  36x 2  dx
0
2
D.   36x 2  9x 4  dx
0
2
E.   6x  3x 2  dx
0
CALCULATOR REQUIRED
Let R be the region in the first quadrant above by the graph of
f  x   2Arc tan x and below by the graph of y = x. What is the volume
of the solid generated when R is rotated about the x-axis?
A. 1.21
B. 2.28
C. 2.69
D. 6.66
E. 7.15
CALCULATOR REQUIRED
Let R be the region in the first quadrant that is enclosed by the
graph of f  x   ln  x  1 , the x-axis and the line x = e. What is
the volume of the solid generated when R is rotated about the
line y = -1?
A. 5.037
B. 6.545
C. 10.073
D. 20.146
E. 28.686
e


 ln  x  1  1  1 dx  20.14627352  D
0
2
2
Cross Sections
Let R be the region marked in the first quadrant enclosed by
the y-axis and the graphs of y  4  x 2 and y  1  2sinx
as shown in the figure below
a) Setup but do not evaluate the
integral representing the volume
R
of the solid generated when R
is revolved around the x-axis.
1.102

 
0
 4  x2

  1  2sinx   dx
2
2
b) Setup, but do not evaluate the
integral representing the volume
of the solid whose base is R and
whose cross sections perpendicular
to the x-axis are squares.
1.102
  4  x   1  2sinx 
2
0
2
dx
CALCULATOR REQUIRED
The region S is represented by the area between the graphs of
f  x   0.5x 2  2x  4 and g  x   2  4  4x  x 2 . Write, but do
not evaluate, a definite integral which represents:
a. the volume of a solid with base S if each cross section of
the solid perpendicular to the x-axis is a semi-circle.
  g x  f  x 
dx



20
2

2
4
b. the volume generated by rotating region S around the line
y = 5.
4
  5  f  x     5  g  x   dx
0
2
2
NO CALCULATOR
The base of a solid is the region in the first quadrant bounded by
the curve y  sin x for 0  x  . If each cross section of the
solid perpendicular to the x-axis is a square, the volume of the
solid, in cubic units, is:
A. 0
B. 1
C. 2
D. 3
E. 4


0

2

sinx dx   sin xdx   cos x |0    1   1  2  C
0
NO CALCULATOR
The base of a solid is a right triangle whose perpendicular sides
have lengths 6 and 4. Each plane section of the solid perpendicular
to the side of length 6 is a semicircle whose diameter lies in the
plane of the triangle. The volume, in cubic units, of the solid is:
A. 2
B. 4
C. 8 
D. 16
E. 24
2

2


6
6
x
1  3 
1
2
 
dx


x
dx


2 0 2 
18 0


 3 6

x |0
54
 4  B
CALCULATOR REQUIRED
Let R be the region in the first quadrant under the graph of
y
8
3
x
for 1, 8
a) Find the area of R.
b) The line x = k divides the region R into two regions. If the
part of region R to the left of the line is 5/12 of the area of
the whole region R, what is the value of k?
c) Find the volume of the solid whose base is the region R
and whose cross sections cut by planes perpendicular
to the x-axis are squares.
Let R be the region in the first quadrant under the graph of
8
y  3 for 1, 8
x
a) Find the area of R.
8

1
8
3
x
dx  36
Let R be the region in the first quadrant under the graph of
8
y  3 for 1, 8
x
b) The line x = k divides the region R into two regions. If the
part of region R to the left of the line is 5/12 of the area of
the whole region R, what is the value of k?
k

1
8
5
dx 
 36
3
12
x
12x 2 / 3 |1k  15
A
12k2/ 3  12  15
k  3.375
Let R be the region in the first quadrant under the graph of
8
y  3 for 1, 8
x
c) Find the volume of the solid whose base is the region R
and whose cross sections cut by planes perpendicular
to the x-axis are squares.
2
 8 
1  2 x  dx  192
8
Let R be the region in the first quadrant bounded above by the
2
graph of f(x) = 3 cos x and below by the graph of g  x   e x
a) Setup, but do not evaluate, an integral expression in terms of
a single variable for the volume of the solid generated when
R is revolved about the x-axis.
0.836


0
 
 3cos x 2  e x2


2
 dx

b) Let the base of a solid be the region R. If all cross sections
perpendicular to the x-axis are equilateral triangles, setup,
but do not evaluate, an integral expression of a single
variable for the volume of the solid.
3
4
0.836

0
3cos x  e  dx
x2
2