Torricelli’s trumpet is a fictional infinitely long solid of revolution formed when the closed domain

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!