Given $E$ is a region bound by $$z=1 \ \text{and} \ z=\frac{1}{\sqrt{x^2+y^2}}, \ \text{for} \ 1\le z\le \infty,$$ we are asked to show that $E$ has infinite surface area but finite volume. How should I go about this?
I have already worked out that this region forms a kind of Gabriel's horn, which extends upwards from the plane $z=1$ forever.
I have thought to use a surface integral to determine that it has an infinite surface area, but I'm not quite sure how to approach it. I take it that I should set $S$ as the surface with equation $z=\frac{1}{\sqrt{x^2+y^2}}$ but I'm not very confident in working that out.
Finding the volume, I could use a triple integral but I need to work out the boundaries for $x, y$ and $z$, and this is something I'm not very good at.
Is this a good method for this problem? Is there an easier way?