Section 6.2:
Volumes
The volume of a solid of revolution is obtained by evaluating:
V
=
Ax
( 29
dx
a
b
∫
, if the solid is obtained by rotating a plane region about the
x
axis
or a line parallel to the
x
axis, where
a
,
b
[ ]
corresponds to variable
x
.
V
=
A
(
y
)
dy
c
d
∫
, if the solid is obtained by rotating a plane region about the
y
axis
or a line parallel to the
y
axis, where
c
,
d
[ ]
corresponds to variable
y
.
The crosssectional area,
Ax
( 29
or
Ay
( 29
, is obtained as follows:
•
If the crosssection is a disk, then obtain the radius of the disk (as either a function
of
x
or a function of
y
, whichever is appropriate), and use
A
=π
radius
( 29
2
.
•
If the crosssection is a washer, then obtain the inner radius and the outer radius of
the washer (as either a function of
x
or a function of
y
, whichever is appropriate),
and use
A
=π
ou
ter rad
ius
( 29
2

inner rad
ius
( 29
2
[ ]
.
Any radius that is required is always measured from the axis of revolution.
Example.
Find the volume of the solid obtained when the region bounded by the
x
axis,
the
y
axis, and the line
y
=
x
+
3
is rotated about the
x
axis.
Solution.
The region is sketched below.
Since the axis of revolution is the
x
axis, slice perpendicular to the
x
axis and
obtain a piece with width
∆
x
.
Since
∆
x
becomes
dx
in the definition of the definite
integral, we need to evaluate
V
=
Ax
( 29
dx
a
b
∫
, where
a
,
b
[ ]
=
3
,0
[ ]
.
In other words, we
need the function for area expressed in terms of
x
.
Because the plane region lies directly against the axis of revolution, the