Surface Area of Revolution

Key idea

Area of a surface formed by rotating a curve.

In depth

S = 2π ∫ f(x) √(1 + (f′(x))²) dx. Each small piece of arc length is rotated into a thin band of circumference 2πf(x).

Worked example

y = x on [0, 1] revolved about the x-axis gives a cone with S = 2π∫ x√2 dx = π√2.

Formula

Real-world uses

• Calculating the amount of material needed for a dome, funnel or lampshade

Common mistake

Using f(x) instead of the radius from the axis of rotation.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Calculus

• Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation – Take logs before differentiating products and powers.
• Related Rates – Link rates of change of connected quantities.
• Linear Approximation – Approximate f near a using its tangent line.
• Differentials – Use dy = f′(x)dx for small changes.