Riemann integral

In real analysis, the Riemann integral is a rigorous definition of the integral of a function on an interval. It defines the integral by approximating the region under the graph of a function by finite sums of areas of vertical rectangles. For suitable functions, including every continuous function on a closed bounded interval, these Riemann sums approach a single limiting value as the partitions of the interval become finer. That limiting value defines the integral, and Riemann sums that are suitably close to the limit can be used as numerical approximations. Bernhard Riemann introduced the integral in work presented to the faculty at the University of Göttingen in 1854 and published in 1868. It is the integral most commonly introduced in elementary calculus, although in advanced analysis it is often replaced by more general notions such as the Lebesgue integral.