Volume of Revolution Calculator

How to Use This Calculator

1. Enter f(x) and bounds a, b.
2. Choose axis: x-axis uses the disk method; y-axis uses the shell method.
3. Click Calculate for the volume.
4. Use the Washer tab to revolve the region between two curves.
5. The Professional tab shows rotation about y=k and Pappus's theorem.

Formula

Example

Frequently Asked Questions

• What is the disk method for volume of revolution? ▼
  Rotating y=f(x) about the x-axis: V = π ∫ₐᵇ [f(x)]² dx. Each cross-section is a disk of radius f(x), area πr² = π[f(x)]².
• What is the shell method? ▼
  Rotating about the y-axis: V = 2π ∫ₐᵇ x·|f(x)| dx. Each cylindrical shell has radius x, height f(x), and thickness dx; its volume is 2πx·f(x)·dx.
• What is the washer method? ▼
  Between two curves f(x) outer and g(x) inner rotated about x-axis: V = π ∫ₐᵇ [f(x)²−g(x)²] dx. The cross-section is an annulus (washer).
• What is Pappus's centroid theorem? ▼
  Pappus's theorem: the volume of revolution equals 2π times the distance of the centroid from the axis times the area of the region. V = 2π·ȳ·A (for rotation about x-axis).
• How do I rotate about y=k instead of y=0? ▼
  Replace f(x) with f(x)−k in the disk formula: V = π ∫ [f(x)−k]² dx. Our Professional tab handles this shift automatically.

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