In 1850, a beautiful turn in mathematical analysis is the work by Cauchy and after that soon the work of Bernhard Reimann.The idea of integration in starting one was completely divorced from the derivative and instead use the notion "area under the curve"as a starting point for constructing a rigorous definition of the integral.\\The Reimann integral is today a very important notion in real analysis and also in introductory to calculus. This is used...