I have a problem relating to the extremal function for a surface of revolution.

The Euler-Lagrange equation is

Surface area of revolution =

So

[eqn. (1)] is a function of y and y' only and so the Euler-Lagrange equation is

because x is absent from F.

Mr F says: This is wrong. It should be where C is a constant. This is a first integral of the Euler-Lagrange equation.
But in a similar problem

[eqn. (2)] and the E-L equation is stated as

because x is not absent from F....

why is eqn. (1) independent of x but eqn. (2) isn't !?