Finite volume, infinite surface

Rotate the curve y = 1/x around the x-axis from x = 1 to infinity. The resulting horn has a finite volume — π cubic units — but an infinite surface area. You could pour π cubic units of paint into it; you could never paint its outside.

View

x_max, cutoff (log scale)

10.0000

1.5101001000

Live values

V (cut)π · (1 − 1/x_max) = 2.82743A (exact)15.1774A ≥ 2π ln(x_max)14.4676V / π0.900000

As x_max → ∞: V → π (finite), A → ∞ (the harmonic integral diverges). The lower bound 2π · ln(x_max) makes the divergence visible.

Painter's paradox

ε (paint thickness)0.0100000

Paint to coat at εA · ε = 0.151774Paint to fill (V)2.82743

Honest paint has nonzero thickness. Coating an infinite surface then already costs infinite volume, and the paradox dissolves.

← Back to the atlas