Volume of ellipsoid by revolving an ellipse

Short answer

Revolve about x-axis: V = 4πab²/3 cubic units. Revolve about y-axis: V = 4πa²b/3 cubic units, using V = π∫ y² dx or π∫ x² dy respectively.

Find the volume of the ellipsoid formed by the revolution of an ellipse x²/a² + y²/b² = 1.

Ellipse: x²/a² + y²/b² = 1. Revolving it about an axis gives an ellipsoid of revolution (a spheroid).

The solid is symmetrical about the y-axis, so V = π ∫_(−a)^a y² dx = 2π ∫₀^a y² dx, with y² = b²(1 − x²/a²):

V = 2π b² ∫₀^a (1 − x²/a²) dx
  = 2π b² [x − x³/(3a²)]₀^a
  = 2π b² (a − a/3)

V = 4πab²/3.

Similarly V = 2π a² ∫₀^b (1 − y²/b²) dy = 2πa² (b − b/3), so

V = 4πa²b/3.

For a sphere (a = b = r) both give 4πr³/3, as expected.

✓ Answer checked by Claude Opus 5.5 (high effort)

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