Volume: Shell Method
Sometimes slicing a shape like bread (Disks) creates messy integrals. Instead, we can peel it like an onion (Cylindrical Shells).
What is the Shell Method?
The Shell Method (also known as the Cylindrical Shell Method) is an alternative to the Disk/Washer method. Instead of slicing the solid into flat disks perpendicular to the axis of rotation, you slice it into thin, vertical "shells" (like the layers of an onion) that are parallel to the axis of rotation.
This method is often much easier when the function is difficult to solve for the other variable, or when the geometry of the Disk Method would require splitting the problem into multiple integrals.
The Shell Method
How it Works: The Geometry
Imagine a thin vertical rectangle under a curve. When you spin it around the -axis, it doesn't make a disk; it makes a hollow, thin-walled cylinder (a "shell").
To find the volume of a shell, we "unroll" it into a flat rectangular sheet:
The Shell Method Formula
If you are rotating a region bounded by around the -axis from to :
The Onion Peeler (Shell Method)
Step-by-Step Process
To solve a volume problem using the shell method, follow these steps:
Worked Example
Volume using Shells
Find the volume when the region bounded by and is rotated about the -axis.
Level Up Examples
Example A: Why Use Shells?
Rotate from to about the -axis.
Disk method needs . Inverting this cubic is impossible algebraically! Shell method uses directly.
Example B: Rotation about x = 2
Rotate on about the line .
Example C: Region Between Curves
Rotate the region between and from to about the -axis.
Practice Quiz
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