In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. Overview Let be a non-negative real-valued function on the interval , and let be the region of the plane under the graph of the function and above the interval . See the figure on the top right. This region can be expressed in set-builder notation as We are interested in measuring the area of . Once we have measured it, we will denote the area in the usual way by The basic idea of the Riemann integral is to use very simple approximations for the area of . By taking better and better approximations, we can say that "in the limit" we get exactly the area of under the curve. When can take negative values, the integral equals the signed area between the graph of and the -axis: that is, the area above the -axis minus the area below the -axis. Definition Partitions of an interval A partition of an interval is a finite sequence of numbers of the form Each is called a sub-interval of the partition. The mesh or norm of a partition is defined to be the length of the longest sub-interval, that is, A tagged partition of an interval is a partition together with a finite sequence of numbers subject to the conditions that for each , . In other words, it is a partition together with a distinguished point of every sub-interval. The mesh of a tagged partition is the same as that of an ordinary partition. Suppose that two partitions and are both partitions of the interval . We say that is a refinement of if for each integer , with , there exists an integer such that and such that for some with . Said more simply, a refinement of a tagged partition breaks up some of the sub-intervals and adds tags to the partition where necessary, thus it "refines" the accuracy of the partition. We can turn the set of all tagged partitions into a directed set by