Math
Volumes of revolution.
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 · 4 phases
Step by step.
- Sketch the region and identify the axis of revolution.
- Choose the method: disk/washer (perpendicular slices) or shell (parallel slices).
- Set up the integral with the correct formula and bounds.
- Evaluate the integral to find the volume.
Examples
Real-world.
- 1 Rotating y = x² from x = 0 to x = 2 around the x-axis creates a paraboloid shape
- 2 Using the washer method to find the volume when the region between y = x and y = x² is rotated around the x-axis
- 3 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