Torricelli’s trumpet

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

$$A:=\{(x,y)\in {ℝ}^2\mathrm{⋮}x\ge 1,\mathrm{\hspace{0.33em}0}\le y\le \frac{1}{x}\}$$

rotates about the $x$-axis. It has a finite volume, $\pi$ volume, but the area of its surface is infinite; infact even the area of $A$ is infinite, i.e., the improper integral${\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{1}{x}}𝑑x$ is not convergent.

Torricelli’s trumpet is surprising since it can be filled by a finiteamount of paint, but this paint can never suffice for painting itssurface, no matter how a coat of paint isused!