-\frac{1}{\pi} \int_r^\infty \frac{F'(y)}{\sqrt{y^2-r^2}} \, dy = \int_r^\infty \int_y^\infty \frac{-2 y}{\pi \sqrt{(y^2-r^2) (s^2-y^2)}} f'(s) \, ds dy.
By Fubini's theorem, the last integral equals .