There is no derivative. This is an integral. That 'dx' in the integral may represent nothing but a reminder of what variables you are using. It could also usefully mean a differential. "Slope" is defind a Rise/Run or dy/dx, or simply the change in y given the change in x. 'dx' may mean just the change in x.

Your Pythagorean theorem may make more sense if you would draw the hypotenuse.

As far as solving, ever hear of "Elliptical Integrals"? Not a trivial task. Let's get through the definition of the Riemann Integral, first.