lets $f(r,x)$ a function integrable and diferrentiable, and define
$$\begin{align}L(r)&=\int_0^r\sqrt{1+\left[f_x(r,x)\right]^2}dx\\ A(r)&=\int_0^rf(r,x)dx\end{align}$$
find all $f(r,x)$ such that $L(r)=\frac{d}{dr}A(r)$
i get that
$$\begin{align} \int_0^r\sqrt{1+\left[f_x(r,x)\right]^2}dx&=\frac{d}{dr}\int_0^rf(r,x)dx\\ \int_0^r\sqrt{1+\left[f_x(r,x)\right]^2}dx&=f(r,r)+\int_0^rf_r(r,x)dx \end{align}$$
by trial and error i find that $f(r,x)=\sqrt{r^2-x^2}$, but i dont know how to solve or proof that its unique solution, how to solve this?