Volumes of revolution

Math

Definition

Volumes of revolution are 3D solids formed by rotating a 2D region around a line (axis of revolution). Calculus provides methods to compute these volumes, most commonly the disk/washer method and the shell method.

How It Works

  1. Sketch the region and identify the axis of revolution.
  2. Choose the method: disk/washer (perpendicular slices) or shell (parallel slices).
  3. Set up the integral with the correct formula and bounds.
  4. Evaluate the integral to find the volume.

Examples

  • Rotating y = x² from x = 0 to x = 2 around the x-axis creates a paraboloid shape
  • Using the washer method to find the volume when the region between y = x and y = x² is rotated around the x-axis
  • Using the shell method to revolve a region around the y-axis
Key Fact

Disk method: V = π∫[f(x)]² dx; Washer: V = π∫([R]² − [r]²) dx; Shell: V = 2π∫x·f(x) dx

Related Terms

Riemann sums Area between curves Antiderivatives Fundamental theorem

Study This Concept

Practice volumes of revolution with free review games in these units:

AP Calculus AB Unit 8: Applications of Integration