Volumes by the Shell Method

By the end of this lesson, you will be able to compute volumes of revolution using cylindrical shells, choose between the shell and disk/washer methods, and apply the shell method for revolution about either axis.

The Shell Method

Imagine cutting the solid into thin cylindrical shells (like layers of a tree trunk). Each shell has:

• Radius: $x$ (distance from the axis)
• Height: $f(x)$
• Thickness: $dx$
• Volume: $2\pi x \cdot f(x),dx$

For revolution about the $y$-axis:

$$V = \int_a^b 2\pi x f(x),dx \quad (f(x) \geq 0)$$

For revolution about the $x$-axis:

$$V = \int_c^d 2\pi y g(y),dy$$

When to Use Shell vs. Disk/Washer

Shell: easier when:

• You'd need to solve for the inverse function for the disk method
• The region is naturally vertical slices but rotation is about the $y$-axis
• A vertical strip sweeps out a shell

Disk/Washer: easier when:

• The cross-section perpendicular to the axis is a simple circle or annulus
• Horizontal slices for $y$-axis rotation

In general, use whichever avoids inverting the function.

Example 1 — Basic Shell About $y$-Axis

Rotate the region between $y = x^2$ and $y = 0$ on $[0, 2]$ about the $y$-axis.

Using shells (vertical slices, rotating about $y$-axis):

$V = \int_0^2 2\pi x \cdot x^2,dx = 2\pi\int_0^2 x^3,dx = 2\pi\left[\frac{x^4}{4}\right]_0^2 = 2\pi(4) = 8\pi$

Compare with disk method (about $y$-axis): would need $x = \sqrt{y}$ and integrate from $0$ to $4$ in $y$ — more work.

Example 2 — Shell Method for a Region

Rotate the region between $y = x$ and $y = x^2$ on $[0,1]$ about the $y$-axis.

Shell at radius $x$, height = $x - x^2$ (top minus bottom):

$V = 2\pi\int_0^1 x(x-x^2),dx = 2\pi\int_0^1(x^2 - x^3),dx = 2\pi\left[\frac{x^3}{3} - \frac{x^4}{4}\right]_0^1 = 2\pi\left(\frac{1}{3}-\frac{1}{4}\right) = \frac{\pi}{6}$

Example 3 — Shell About $x$-Axis

Rotate $x = y^2$ on $[0,4]$ (in $y$) about the $x$-axis using shells.

Shells at radius $y$, from $y = 0$ to $y = 2$:

Height at $y$ = $4 - y^2$ (horizontal extent from $y^2$ to $4$).

$V = 2\pi\int_0^2 y(4-y^2),dy = 2\pi\int_0^2(4y - y^3),dy = 2\pi\left[2y^2 - \frac{y^4}{4}\right]_0^2 = 2\pi(8-4) = 8\pi$

Example 4 — Shell About a Vertical Axis $x = k$