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:
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Height = value of the function
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Radius = distance from axis
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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:
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radius = x (distance from y-axis)
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height = f(x) or difference of functions
So:
Rotation around the x-axis
Use horizontal slices, thickness dy:
Same idea:
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radius = distance from axis
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height = function expressed in terms of y
Derivation (Simple Explanation)
A thin shell has:
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Radius = r
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Height = h
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Thickness = Δx
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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:
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radius = x
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height = $$x^2$$
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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:
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radius: |x – k|
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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:
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radius: |y – k|
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height: g(y)−h(y)
V=$$2\pi\int_{a}^{b}|y-k|\cdot(g(y)-h(y))dy$$