Axisymmetric membrane deformation ($R=R(z)$)

An axisymmetric cylindrical membrane deformation is invariant under rotations about the cylinder axis, so its radius depends only on the axial coordinate. Its surface area is $2\pi \int R\sqrt{1+R_z^2}dz$, and its enclosed volume is $\pi \int R^{2}dz$. These scalar shape constraints are the basis of the fixed-volume capillary spectrum of a cylindrical membrane.