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When I took college calculus (more decades ago than I care to admit), we were introduced to a function that would create a surface of revolution having a finite volume but an infinite surface area. In chasing this idea down recently, I have come across the (apparently) classic example called Gabriel's (or Torricelli's) trumpet, resulting from the function $f(x)= \frac{1}{x}$. But my recollection is that the function I was introduced to in the calculus course had the form of a decreasing exponential, very generally, something like $e^{-f(x)}$.

So my question is: Are there functions other than $1\over x$ (and in particular decreasing exponential functions) that create surfaces of revolution that have the property of containing a finite volume, but having an infinite surface area? I've poked around on MSE and Wikipedia, but have found nothing relevant, as queries all tend to lead to discussions of Gabriel's trumpet.

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There are many answers, not just $\frac1x=\exp(-\ln x)$. If $f:[a,\infty)\to[0,\infty)$ is any function such that $\int_a^\infty f(x) dx=\infty$ and $\int_a^1 (f(x))^2 dx<\infty$, its surface of revolution will enclose the finite volume $\pi \int_a^1 (f(x))^2 dx$ but have infinite surface area $2\pi \int_a^1 f(x)\sqrt{1+(f'(x))^2}dx$, as $f(x)\sqrt{1+(f'(x))^2}\ge f(x)$.

If you choose for example $f(x)=x^{-\beta}$, $1>\beta>\frac12$ then $\int_1^\infty f(x)dx=\infty$ for $0<\beta<1$ but $\int_1^\infty f(x)^2dx=\frac1{1-2\beta}<\infty$.

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