CalcRef

Applications of Integrals · Arc Length & Surface Area

Surface Area of Revolution

S=2π∫abf(x)1+[f′(x)]2 dxS = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2}\, dx

The surface area generated by revolving y = f(x) about the x-axis.

Conditions. f(x) ≥ 0 on [a, b].

Variables

SymbolNameUnit
aLeft bound—
bRight bound—

Worked examples

Find the surface area when y = x on [0, 1] is revolved about the x-axis.
  1. f'(x) = 1. √(1+1) = √2
  2. S = 2π ∫₀¹ x · √2 dx = 2π√2 [x²/2]₀¹ = π√2

Answer: π√2 ≈ 4.443

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