Volume of Revolution (Disk and Shell Methods)
Formulas for calculating volumes of solids of revolution using the disk method, washer method, and cylindrical shell method.
Disk Method (revolving around x-axis)
V = π ∫[a to b] [f(x)]² dx
Used when the region is revolved around the x-axis and there is no hole in the solid.
Washer Method (revolving around x-axis with hole)
V = π ∫[a to b] { [R(x)]² - [r(x)]² } dx
Where R(x) is the outer radius and r(x) is the inner radius.
Cylindrical Shell Method (revolving around y-axis)
V = 2π ∫[a to b] x · f(x) dx
Used when it is easier to integrate with respect to x while revolving around the y-axis.
Variables
| Symbol | Meaning |
|---|---|
| V | Volume of the solid of revolution |
| f(x) | The function being revolved |
| R(x) | Outer radius (distance from axis to outer curve) |
| r(x) | Inner radius (distance from axis to inner curve) |
| a, b | Bounds of integration |
When to Use Which Method
- Disk/Washer: When slicing perpendicular to the axis of revolution
- Shell: When slicing parallel to the axis of revolution
- If revolving around x-axis and integrating dx → use disk/washer
- If revolving around y-axis and integrating dx → use shell
Example 1 — Disk Method
Find the volume when y = √x (from x=0 to x=4) is revolved around the x-axis
V = π ∫[0 to 4] (√x)² dx = π ∫[0 to 4] x dx
= π [x²/2] from 0 to 4 = π(16/2) = 8π ≈ 25.13
Example 2 — Shell Method
Find the volume when y = x² (from x=0 to x=2) is revolved around the y-axis
V = 2π ∫[0 to 2] x · x² dx = 2π ∫[0 to 2] x³ dx
= 2π [x⁴/4] from 0 to 2 = 2π(4) = 8π ≈ 25.13
Key Notes
- Disk method: V = π ∫[f(x)]² dx: When rotating around the x-axis, each thin vertical slice sweeps out a disk of radius f(x) and thickness dx. Integrating πr² = π[f(x)]² over the interval gives total volume. Use when the solid has no hole.
- Washer method: V = π ∫([f(x)]² − [g(x)]²) dx: When rotating the region between two curves (outer radius f, inner radius g), each slice sweeps a washer (disk with hole). Integrate the difference of squared radii.
- Shell method: V = 2π ∫x·f(x) dx: Integrates cylindrical shells rather than disks. Each thin vertical strip at position x sweeps a shell of radius x, height f(x), and thickness dx — giving volume 2πx·f(x)·dx. Shell method is often simpler when rotating around a vertical axis.
- Choosing the method: Disk/washer: natural when the rotation axis is parallel to the integration variable's axis. Shell: simpler when setting up disk/washer integrals would require solving for x in terms of y or splitting the integral.
- Applications: Volume of revolution calculates the volume of rotationally symmetric objects — turned machine parts (cylinders, cones, goblets), wine barrel capacity, rocket nose cone volume, and the volume of any shape that can be described as a revolved curve.