L37 Volume of Solid of Revolution I
Finding the Volume of the Solid of
Revolution{Disk/Washer
A
solid of revolution
is a solid obtained by
rotating a plane area around some straight line
(the axis).
Two common methods for
nding the volume of a
solid of revolution are the
disk method
and the
shell method
of integration.
To apply these methods, it is easiest to:
1. Draw the graph in question;
2. Identify the area that is to be revolved about the
axis of revolution;
3. Determine the volume of either a disk-shaped slice
or a cylindrical shell of the solid;
4. Integrate all the in nitely many disks or shells .
1

Disk method
The volume
V
of the solid formed by rotating a plane
area about the
x
axis is given by
V
=
Z
b
a
A
(
x
)
dx
=
Z
b
a
f
2
(
x
)
dx
and about the
y
axis by
V
=
Z
b
a
A
(
y
)
dy
=
Z
b
a
g
2
(
y
)
dy
where
A
(
x
) and
A
(
y
) is the cross-sectional area of
the solid.
2