An Introductory to Reimann Integral

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 in this way that the function $\varphi$ on $[\alpha, \beta]$, we divide this interval into small subintervals. By using each subinterval $[u_{l-1}, u_{l}], $ we choose any value $a_{l}\in [u_{l-1}, u_{l}]$ and then we find the value of $\varphi(a_{l})$. Geometrical behavior of this is that a row of thin rectangles formed the area between $\varphi$ and the horizontal axis.