The Euler-Lagrange equation is \(\displaystyle {d \over dx}({\partial F \over \partial y'})- {\partial F \over \partial y}=0.\)

Surface area of revolution =\(\displaystyle \int {2\pi y \sqrt{1+(y')^2}}dx\)

So \(\displaystyle F(x,y,y') = {2\pi y \sqrt{1+(y')^2}}dx\) [eqn. (1)] is a function of y and y' only and so the Euler-Lagrange equation is \(\displaystyle F-y'{\partial f \over \partial y'}=0\) because x is absent from F.

But in a similar problem \(\displaystyle F(x,y,y')= (y')^2+yy'+y^2\) [eqn. (2)] and the E-L equation is stated as \(\displaystyle {d \over dx}({\partial F \over \partial y'})- {\partial F \over \partial y}=0\) because x is not absent from F....

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