Constant strip-area density of a surface of revolution

Forthe area density after integrating over $\theta$ is $2\pi \varphi \sqrt{1+{\varphi }^{\prime 2}}$. If every height interval has area $2\pi r$ times its length, continuity forcesWhere $0<\varphi <r$, the sign of ${\varphi }^{\prime }$ is constant and $\sqrt{r^2-{\varphi }^2}$ has derivative $\pm 1$. Hence the profile lies on a circle of radius $r$.