 Solution

Parametrize the surface by $r(x,\theta )=(x,y(x)\cos \theta ,y(x)\sin \theta )$, with $0\le \theta <2\pi$. Its tangent vectors are orthogonal, of lengths $\sqrt{1+y^{\prime 2}}$ and $y$. Thus the area element is $y\sqrt{1+y^{\prime 2}}dxd\theta$. Integrating over the angle gives