Surface area of revolution
Objective. Model rotated surfaces by integration.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The surface area generated by rotating a curve y=f(x) about the x-axis between x=a and x=b is given by S = ∫2πy√(1+(dy/dx)²)dx, combining the circumference 2πy of each circular slice with the arc length element √(1+(dy/dx)²)dx. This formula models the surface as made of many thin circular bands, each approximating a frustum, whose areas are summed via integration. For example, rotating y=√x about the x-axis from x=0 to 4 requires finding dy/dx=1/(2√x) and substituting into the surface area formula.
Worked example
Find the surface area formed when y = x is rotated about the x-axis from x=0 to x=3, using the surface of revolution formula.
- 1dy/dx = 1, so √(1+(dy/dx)²) = √2
- 2Surface area: S = ∫₀^3 2πx√2 dx
- 3Integrate: 2π√2 ∫₀^3 x dx = 2π√2 [x²/2]₀^3
- 4Evaluate: 2π√2 × 9/2 = 9π√2
Answer: 9π√2
Concept mastery
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Practice
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Start practisingCommon mistake. Students use the arc length formula alone, forgetting to multiply by 2πy for the circumference of rotation. Surface area of revolution always includes both the arc length element and the 2πy (or 2πx) factor for the radius being rotated.
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