Work out surface area of revolution instantly with clear inputs, formula shown and shareable results.
Rotating a curve about the x-axis sweeps a surface of area 2π∫|f(x)|·√(1 + (f')²) dx: each element is a frustum band of radius f(x) and slant width ds. Pappus's theorem recovers the mean radius as area divided by 2π times arc length.
Surface of revolution
S = 2π ∫ₐᵇ |f(x)| √(1 + (f'(x))²) dx
S = 2π∫x√2 dx = 9π√2 ≈ 39.9822, the lateral area of a cone with radius 3 and slant height 3√2.
Because the band's width follows the slanted curve, not the horizontal projection.