Cylindrical Shell Method

What is the Cylindrical Shell Method?

The cylindrical shell method finds the volume of a solid created when a 2D region is revolved around an axis by summing up the volumes of thin cylindrical shells.

Each shell is like a thin soup can label:

• Height = value of the function

• Radius = distance from axis

• Thickness = dx or dy (depending on integration direction)

Why use Shells instead of Disks/Washers?

The shell method is especially convenient when:

• You rotate around the y-axis and the region is described as y=f(x).
• You rotate around the x-axis and the region is described as x=g(y).
• Solving for inverse functions would be annoying.
• Washers create two separate integrals, but shells need only one.

Formula for Cylindrical Shells

Rotation around the y-axis (most common)

Use vertical slices, thickness dx:

V=$$2\pi\int_{a}^{b}(radian)(height)dx$$

Where:

• radius = x (distance from y-axis)

• height = f(x) or difference of functions

So:

V=$$2\pi\int_{a}^{b}x[f(x)]dx$$​

Rotation around the x-axis

Use horizontal slices, thickness dy:

V=$$2\pi\int_{a}^{b}y[g(y)]dy$$​

Same idea:

• radius = distance from axis 

• height = function expressed in terms of y

Derivation (Simple Explanation)

A thin shell has:

• Radius = r

• Height = h

• Thickness = Δx

• Circumference = 2πr

Volume of thin shell:

Volume=$$2\pi\cdot{h}\cdot\Delta{x}$$

Sum → integral:

V=$$\int2\pi{r}hdx$$

Example 1 — Rotate Around the y-axis

Find the volume generated when the region under

$$y=x^2$$

from x=0 to x=2 is rotated around the y-axis.

Shell Method Setup:

• radius = x

• height = $$x^2$$

• thickness = dx

V=$$2\pi\int_{0}^{2}x(x^2)dx=2\pi\int_{0}^{2}x^3dx$$

=$$2\pi\left[\frac{x^4}{4}\right]_{0}^{2}=2\pi\cdot\frac{16}{4}=8\pi$$

Example 2 — Area Between Two Curves

Region between

$$y=4x , y=x^2$$

rotated around the y-axis.

Height of shell:

height = $$4x-x^2$$

Radius:

r = $$x$$

Intersection:

$$4x=x^2 \longrightarrow x^2-4x=0$$ then $$x(x-4)=0 \longrightarrow x=0 , x=4$$

Integral:

V=$$2\pi\int_{0}^{4}x(4x-x^2)dx$$

When Shell Method Is Required

The AP exam often gives situations where shell method is much better:

Rotating region between two vertical lines around the y-axis

→ washers would require solving for inverse functions.

Rotating region around a line like x=3

→ shell radius becomes |3 – x|, still easy.

Rotating horizontally described region around x-axis

→ shells avoid rewriting in terms of y.

Rotation Around a Vertical Line x=k

If rotating around x=k:

• radius: |x – k|

• height: f(x)−g(x)

Formula:

V=$$2\pi\int_{a}^{b}|x-k|\cdot(f(x)-g(x))dx$$

If the region lies on one side, drop the absolute value.

Rotation Around a Horizontal Line y=k

Use horizontal shells:

• radius: |y – k|

• height: g(y)−h(y)

V=$$2\pi\int_{a}^{b}|y-k|\cdot(g(y)-h(y))dy$$