Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.
One Dimension - Clearly this structure is infinitely long.
Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$
Three Dimensions - Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \frac{1}{1}) = \pi$
So this structure has infinite length and infinite surface area. However it has finite volume, which just does not make sense.
Even more interesting, the "walls" of this structure are infinitely thin. Since the volume is finite, we could fill this structure with a finite amount of paint. To fill the structure the paint would need to cover the complete surface area of the inside of this structure. Since the "walls" are infinitely thin, why would a finite amount of paint not be able to cover the outside of the "walls" too?
Please help me make sense of this whole thing.