How to Build the Cylindrical Shell Method Formula

A shell is circumference times height times thickness

Rotating a thin strip parallel to the axis of rotation produces a thin cylindrical shell. Its approximate volume is

$$(\text{circumference})(\text{height})(\text{thickness}) =2\pi(\text{radius})(\text{height})(\text{thickness}).$$

Adding the shells with an integral gives the volume.

For a region between $y=f(x)$ and $y=g(x)$ rotated about the $y$-axis, vertical strips produce

$$V=2\pi\int_a^b x\,[f(x)-g(x)]\,dx,$$

provided the region lies to the right of the axis. Here $x$ is the shell radius and $f(x)-g(x)$ is the shell height.

Shift the radius when the axis moves

If the same vertical strip is rotated about $x=c$, its radius is the horizontal distance to that line:

$$r(x)=|x-c|.$$

The shell height still comes from top minus bottom. When the entire region lies on one side of the axis, this radius can be used directly in the shell integral.

If the region crosses the axis, extra care is required: points on opposite sides of the axis can generate shells with the same radius, so integrating both sides naively may double-count volume. In that case, describe the shells so that each radius is counted only once, or choose another convenient setup.

For rotation about a horizontal axis, horizontal strips and integration with respect to $y$ often produce shells. The same three factors remain: $2\pi$, radius, and height.

Example with a different region

Rotate the region under $y=2x$ over $0\le x\le1$ about the $y$-axis. A shell has radius $x$ and height $2x$, so

$$\begin{aligned} V &=2\pi\int_0^1x(2x)\,dx\\ &=4\pi\left[\frac{x^3}{3}\right]_0^1\\ &=\frac{4\pi}{3}. \end{aligned}$$

The units are cubic because the shell construction accumulates three-dimensional volume.