Thank you! I think you're right that's what they wanted. I was getting confused because I thought the first integral equation was a "special case" of the Euler equation but it turns out they're two different things.
I've got that length of a curve on the surface is:
$$L=\int_{-\infty}^{\infty}\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}dx$$
So the function to extremise is:
$$f(\rho,\rho')=\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}$$
Where...