How do you express a surface of revolution as the graph of a function?

Question

Consider a real function $f$ with domain non-negative real numbers. Let $y = f(x)$, and consider the surface traced out by rotating the graph of $f$ about the $y$-axis. This surface is the graph of some function $y = F(x,z)$. How do we express $F$ in terms of $f$?

Answer 1

The point $(x,z)$ in the plane will have distance $r = \sqrt{x^2 + z^2}$ from the origin. The surface is radially symmetric, and the value of $F$ depends only on the value of $f$ at this distance from the origin. That is, $$F(x,z) = f\left(\sqrt{x^2+z^2}\right)\,.$$