Volumes of revolution
Objective. Model and calculate rotated volumes with definite integrals.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Volumes of revolution find the volume generated when a curve y = f(x) is rotated fully about the x-axis between x = a and x = b, using V = π∫[a to b] y² dx. The cross-section at each x is a circle of radius y, so its area is πy², and integrating accumulates these circular slices. For y = x rotated between x = 0 and x = 2, V = π∫₀² x² dx = π[x³/3]₀² = 8π/3. Always square y before integrating, and state the correct limits.
Worked example
Find the volume of revolution formed when y = x² is rotated about the x-axis between x = 0 and x = 1.
- 1Write V = π∫₀¹ y² dx with y² = x⁴
- 2Integrate: π∫₀¹ x⁴ dx = π[x⁵/5]₀¹
- 3Substitute the limits: π(1/5 − 0)
- 4State V = π/5
Answer: V = π/5
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Practice
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Start practisingCommon mistake. A common error is forgetting to square y, integrating π∫y dx instead of π∫y² dx, which gives the wrong units and value entirely. Always square the function before multiplying by π and integrating.
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