Infinite solids similar to Gabriel's Horn

Question

Do any solids of revolution exist with properties similar to Gabriel's Horn (i.e. a geometric solid with finite volume but infinite surface area)? Please restrict your answers to functions not in the form $$f(x) = \frac{c}{x^p}$$ I can already think of a great number of functions in that form which satisfy properties similar to Gabriel's Horn. Also, I'd prefer examples with monotonic functions or functions that are strictly positive in the interval $[a,\infty)$ if possible. Thanks!

Answer 1

Take two positive numbers $s\leq t$ and let $g$ be any (differentiable) function such that for any $x\in [a,\infty)$, we have $sx\leq g(x)\leq tx$. Then $f(x)=\frac1{g(x)}$ will be strictly positive and its solid of revolution about the $x$-axis will have the "Gabriel's horn" property. If $g$ is monotonic, so is $f$.

Answer 2

$$f(x)=-\ln |x|$$

Take it on range [-1,1].

The volume is $4\pi$, the surface is infinite.