Take the curve y = 1/x for x ≥ 1. Spin it around the x-axis. The shape that comes out is a horn that gets infinitely long, but infinitely thin. Compute its volume:
$$ V = \int_1^\infty \pi \left(\frac{1}{x}\right)^2 \,dx = \pi $$
A volume of π. Finite. You can fill it with paint. Now compute its surface area:
$$ A = \int_1^\infty 2\pi \cdot \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} \,dx \;\to\; \infty $$
The integrand for surface area falls like 1/x, which diverges. So this object has finite volume but infinite surface area. You can fill it with paint, but you cannot paint the inside of it.
The "paradox" dissolves once you remember: paint isn't a 2D thing. Real paint has a layer thickness, and the horn gets thinner than any layer eventually. Mathematically though, the contradiction is beautiful — it's the first time most students realize that "infinite" comes in different sizes.