L38 Volume of Solid of Revolution II{
Shell Method
Shell Method
is another way to calculate the
volume of a solid of revolution when the slice is
parallel to the axis of revolution and you integrate
perpendicular to the axis of revolution.
V
shell
= 2
rhdx
= 2 (shell-radius)(shell-height)
dx
1

The necessary equation for calculating such a
volume depends on which axis is serving as the axis
of revolution.
V
ver
=
Z
A
(
x
)
dx
if the rotation is around a vertical axis of revolution.
V
hor
=
Z
A
(
y
)
dy
if the rotation is around a horizontal axis of revolu-
tion;
Note:
A
shell
= 2 (shell-radius)(shell-height)
2

ex.
Determine the volume of the solid obtained by
rotating the region bounded by
y
= (
x
1)(
x
3)
2
and the
x
axis about the
y
-axis.
(
24
5
)
3

ex.
Determine the volume of the solid obtained by
rotating the region bounded by
y
=
x
2
+
x
and
y
= 0 about the line
y
axis.
(
6
)
4

The remaining examples will have axis of rotation
about axis other than the
x
, and
y
axis:
ex.
Determine the volume of the solid obtained by
rotating the region bounded by
x
= (
y
2)
2
and
y
=
x
about the line
y
=
1.
(
63
2
)
5

ex.
Determine the volume of the solid obtained by
rotating the region bounded by
y
=
x
2
+
x
and
y
= 0 about the line
x
=
3.
(
7
6
)
6

NYTI:
1.
Determine the volume of the solid obtained by
rotating the region bounded by
y
=
p
x
and
y
=
x
about the line
x
=
4.
(
22
15
)
2.
Determine the volume of the solid obtained by
rotating the region bounded by
y
=
p
x
and
y
=
x
about the line
x
= 10.
(
16
5
)
3.
Determine the volume of the solid obtained by
rotating the region bounded by
y
=
p
x
and
y
=
x
about the line
y
=
3.
(
7
6
)
7