Let $f: I \to (0,\infty)$ be a smooth and positive function. Let $$\Sigma_f = \{(x,f(x)\cos\theta,f(x)\sin\theta) : x \in I, \theta \in \Bbb{R}\}$$ be a solid of revolution generated by rotating the graph of $f$ around the $x$-axis. Determine the distance between $(x,f(x),0)$ and $(y,f(y),0)$.
I don't know how to approach this problem. My definition of distance is taken over all curves $C^1$ by parts (the infimum of these lengths). Maybe I must minimize the arch length between these points. But how to ensure that the arch will be contained in $\Sigma_f$?
Note that both those points are on a single copy of the curve that you're rotating around. Any one of those copies is itself a geodesic, as you can check by looking at the acceleration vector if you travel along the curve with constant speed.