Stumbling through mathematics: Gabriel, blow your horn!

Gabriel's Horn is a mathematical oddity. It has a finite volume with an infinite surface area. Below is a Maple 16 plot of Gabriel's Horn:

Gabriel's Horn is formed by rotating the function f(x) = 1/x about the x-axis. The volume of Gabriel's Horn is a finite number and is calculated as follows:

V = ∫^∞₁ π 1/x² dx = π

The surface area of Gabriel's Horn is tricker to calculate, but it turns out to be infinite:

S = ∫^∞₁ 2π(1/x)√(1 + (1/x²)²) dx > ∫^∞₁ 2π(1/x) dx = ∞

Thus, you have the strange result of an object that can be filled with a finite amount of paint, but requires an infinite amount to paint the outside surface!