Calculus
Advanced

Surface of Revolution

Surface area generated by revolving a curve about the x-axis.

Formula

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

Variables

fFunction
a, bLimits

Example

Rotating y = x on [0,1] about the x-axis gives S = 2π∫x√2 dx = π√2 ≈ 4.44.

Did You Know?

Spin a curve around an axis and this integral gives the area of the surface it sweeps out.

integralsurfacerevolution