Paradox of the trumpet shape

Suppose you want to paint the surface with a coat of paint with a constant thickness. The volume of paint you will need is (surface area)*(thickness), which is infinite.

However, when you fill the inside with paint, the thickness of the coating is not constant, in fact it decays to 0 as x goes to infinity. This is why a finite volume of paint can fill the inside of the trumpet.

See the Wikipedia article on Gabriel's Horn for more information.

Your issue is trying to compare a 2 dimensional object (surface area) with a 3 dimensional object (volume). Any volume of liquid can be spread thin enough to cover as much surface area as you want (mathematically speaking, there are probably physical limits to this).