Solution
= Solution
Translation invariance in $y$ lets the surface be written $z=z(x)$. Per unit $y$ length, its area and entropy functionals are
$$
\boxed{\frac AL=\int_{-R}^{R}
\frac{\sqrt{1+z'(x)^2}}{z(x)^2}\,dx,
\qquad
\frac SL=\frac1{4G}\frac AL}.
$$
The boundary conditions are $z(\pm R)=0$, understood with an ultraviolet cutoff, and reflection symmetry gives $z'(0)=0$ at $z(0)=z_{\max}$.