Volumes of Revolution

This lesson covers volumes of revolution — finding the volume of a solid formed by rotating a curve around an axis — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to set up and evaluate integrals for rotation about both the x-axis and the y-axis.

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The Concept

When a region of a graph is rotated 360° (a full turn) about the x-axis or y-axis, it sweeps out a three-dimensional solid called a solid of revolution.

Think of it like a potter's wheel: the curve is the profile, and spinning it creates the 3D shape.

Examples:

• Rotating y = r (a horizontal line) about the x-axis creates a cylinder.
• Rotating y = x about the x-axis creates a cone.
• Rotating y = √(r² - x²) about the x-axis creates a sphere of radius r.

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Volume of Revolution About the x-axis

When the curve y = f(x) is rotated 360° about the x-axis between x = a and x = b:

V = π ∫ from a to b of y² dx

Derivation