Torricelli's trumpet is a fictional infinitely long solid of revolution formed when the closed domain $$ A := \{(x,\,y)\in\mathbb{R}^2\,\vdots\;\; x \ge 1,\; 0 \le y \le \frac{1}{x}\} $$ rotates about the $x$ axis. It has a finite volume, $\pi$ volume units, but the area of its surface is infinite; in fact even the area of $A$ is infinite, i.e., the improper integral $\displaystyle\int_1^\infty\frac{1}{x}\,dx$ is not convergent.

Torricelli's trumpet is surprising since it can be filled by a finite amount of paint, but this paint can never suffice for painting its surface, no matter how thin a coat of paint is used!