Use cylindrical shells to find the volume of the solid generated when the region bounded by
curves = 4 −
ଶ
, = 3 is revolved about line = 1.
Answers
Answer:
Volume of a Solid of Revolution: Cylindrical Shells
Sometimes finding the volume of a solid of revolution using the disk or washer method is difficult or impossible.
For example, consider the solid obtained by rotating the region bounded by the line
y
=
0
and the curve
y
=
x
2
−
x
3
about the
y
−
axis.
Solid obtained by rotating the region bounded by the cubic curve y=x^2-x^3 around the y-axis.
Figure 1.
The cross section of the solid of revolution is a washer. However, in order to use the washer method, we need to convert the function
y
=
x
2
−
x
3
into the form
x
=
f
(
y
)
,
which is not easy.
In such cases, we can use the different method for finding volume called the method of cylindrical shells. This method considers the solid as a series of concentric cylindrical shells wrapping the axis of revolution.
With the disk or washer methods, we integrate along the coordinate axis parallel to the axes of revolution. With the shell method, we integrate along the coordinate axis perpendicular to the axis of revolution.
As before, we consider a region bounded by the graph of the function
y
=
f
(
x
)
,
the
x
−
axis, and the vertical lines
x
=
a
and
x
=
b
,
where
0
≤
a
<
b
.