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 !?