Volume of Revolution (Disk/Washer Method) Calculator

Formula first

Overview

The Disk/Washer Method is a technique in integral calculus used to find the volume of a solid of revolution. It involves integrating the area of infinitesimally thin disks or washers perpendicular to the axis of revolution. For revolution around the x-axis, the formula sums the areas of washers with outer radius R(x) and inner radius r(x) from a to b. This method is particularly effective when the region is bounded by functions that are easily expressed in terms of the variable of integration (x for x-axis revolution, y for y-axis revolution).

Symbols

Variables

R(x) = Outer Radius Function, r(x) = Inner Radius Function, a = Lower Limit of Integration, b = Upper Limit of Integration, V = Volume

Apply it well

When To Use

When to use: Use this method when revolving a 2D region around an axis and the cross-sections perpendicular to the axis of revolution are disks or washers. It's ideal when the functions defining the region are easily expressed in terms of the integration variable (e.g., y=f(x) for x-axis revolution).

Why it matters: This method is fundamental in engineering and physics for calculating volumes of complex shapes, such as machine parts, fluid containers, or architectural elements. It provides a powerful tool for quantifying space occupied by objects with rotational symmetry, essential for design, capacity planning, and material science.

Avoid these traps

Common Mistakes

• Forgetting to square the radii R(x) and r(x).
• Incorrectly identifying R(x) (outer radius) and r(x) (inner radius).
• Using the wrong limits of integration (a, b) for the specified region.
• Applying the formula for the wrong axis of revolution (e.g., using this x-axis formula for y-axis revolution).

One free problem

Practice Problem

Find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=x$ around the x-axis.

Solve for: $V$

Hint: Identify the outer and inner functions, and their intersection points to determine the limits of integration.

The full worked solution stays in the interactive walkthrough.

Keep going

Related Formulas

References

Sources

1. Thomas' Calculus
2. Wikipedia: Solid of revolution
3. Calculus by James Stewart, 8th Edition
4. Stewart's Calculus
5. Stewart, J. (2018). Calculus: Early Transcendentals (8th ed.). Cengage Learning. Chapter 6.2: Volumes by Disks and Washers.