Section 2.3 Volumes of Revolution: Cylindrical Shells
References.
- Calculus, Early Transcendentals by Stewart, Section 6.3.
Introduction.
With some solids of revolution, computing the volume through slicing perpendicular to the axis of rotation involves solving equations to get a formula for the slice area, which can be difficult or even impossible.
This happens for example when the region is bounded above and below by curves described as functions of \(x\) and rotation is about the \(y\)-axis.
Volume Formula using Shells for Solids of Rotation about a Vertical Axis.
In the previous section, we saw that when a plane region bounded above and below by curves is rotated about a horizontal axis, thin vertical strips of the region get rotated into disks or washers, whose approximate volumes \(\pi r^2 \Delta x\) or \(\pi (R^2 - r^2) \Delta x\) —we could then combine (integrate) to get the total volume.
When instead such a region is rotated about a vertical axis, the same thin vertical strips between the curves get rotated into “shells”. The shells are still annuli like the washers in the previous section, but far taller and far thinner between the inner and outer radii.
We can again approximate the volume of each shell, so our basic strategy still works: “sum” the shell volumes with an integral.
One difference is that the shell position \(x\) is now the radius at which the shell is located from the rotation axis, so that its values are never negative: we only need to rotate strips to one side of the rotation axis (say \(x>0\)) to get the whole solid.
Volume Formula with Shells from \(y\)-axis Rotation.
If we start with a plane region lying
- vertically in \(g(x) \leq y \leq f(x)\text{,}\) and
- horizontally in \(a \leq x \leq b\) with \(a \geq 0\text{,}\)
the region can be divided into thin vertical strips with
- height \(f(x)-g(x)\text{,}\) and
- width \(\Delta x\text{.}\)
When the region is rotated about vertical axis \(y=0\text{,}\) rotating the strip at distance \(x\) from the \(y\)-axis sweeps out a cylindrical shell of height \(f(x)-g(x)\text{,}\) radius \(x\text{,}\) and thickness \(\Delta x\text{.}\) Each shell has volume \(\Delta V\) of approximately the area of its outer surface times its thickness, which is “circumference times height times thickness”, or
\begin{equation*}
\Delta V \approx 2 \pi x \times [f(x)-g(x) ] \times \Delta x
\end{equation*}
for the shell of radius \(x\text{,}\) \(0 \leq a \leq x \leq b\text{.}\)
In the limit \(\Delta x \to 0\text{,}\) this gives “infinitesimally thin shells” of “infinitesimal volume”
\begin{equation*}
dV = 2 \pi x \; [f(x)-g(x)] \; dx \;, 0 \leq a \leq x \leq b,
\end{equation*}
and the total volume is the integral of these over the relevant values of radius \(x\text{:}\)
\begin{equation*}
V = \int_{x=a}^b dV = \int_{x=a}^b 2 \pi x \; [f(x)-g(x)] \; dx.
\end{equation*}
- Sketch the region between the curve \(y=2x^2-x^3\) and the \(x\)-axis in the right half-plane only. (When rotating a region about an axis, we only need a region on one side of the rotation axis!)
- Mark a typical vertical strip of width \(\Delta x\) within the region.
- Indicate rotation of this strip about the \(y\)-axis on this sketch.
- Sketch the typical cylindrical shell produce by rotation of the above strip.
- Sketch the solid obtained by rotating the above region about the \(y\)-axis.
- Find a definite integral expression for the volume of this solid, using cylindrical shells.
- Compute the volume of this solid.
Volume Formula with Shells in \(0 \leq y \leq f(x)\).
In this case of a solid produced by rotation about the \(y\)-axis of the region between a curve \(y=f(x)>0\) and the \(x\)-axis, with the original region having \(x\) values in the range \(a \leq x \leq b\text{,}\) the volume is given by
\begin{equation*}
V = \int_{x=a}^b 2 \pi x \, f(x) \, dx.
\end{equation*}
As in previous sections, sometimes the relevant range of \(x\)-values is not stated explicitly, but comes naturally from the geometry of the situation.
Example 2.3.2.
- Sketch the region between the curves \(y=x\) and \(y=x^2\text{.}\)
- Mark a typical vertical strip in this region, noting the range of possible \(x\)-values where such strips lie occur, \(a \leq x \leq b\text{.}\)
- Sketch the cylindrical shell produced by rotating this vertical strip about the \(y\)-axis.
- Find a definite integral expression for the volume of this solid, and evaluate the volume.
Again, with practice, a single 2D sketch is enough, so the 3D sketching can be avoided.
The vertical axis of rotation need not always be the \(y\)-axis \(x=0\text{;}\) it can be any vertical line \(x=c\text{.}\) Then the radius of each shell is \(|x-a|\text{,}\) but the absolute value should be eliminated from the formulas by working out the sign of \(x-a\text{;}\) — once again a sketch can help.
Example 2.3.3.
- Find the volume of the solid obtained by rotating the region bounded by \(y=x-x^2\) and \(y=0\) about the line \(x=2\text{.}\)
- Make a single 2D sketch of the curves, the region, the axis of rotation and a typical vertical strip, and label the strip with the radius of the shell produced by rotating it about the axis.
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