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Concept
Riemann integral
Summary
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, or simulated using Monte Carlo Integration.
Let f be a non-negative real-valued function on the interval [a, b], and let S be the region of the plane under the graph of the function f and above the interval [a, b]. 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 S. 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 S. By taking better and better approximations, we can say that "in the limit" we get exactly the area of S under the curve.
When f(x) can take negative values, the integral equals the signed area between the graph of f and the x-axis: that is, the area above the x-axis minus the area below the x-axis.
A partition of an interval [a, b] is a finite sequence of numbers of the form
Each [xi, xi + 1] 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 P(x, t) of an interval [a, b] is a partition together with a choice of a sample point within each sub-interval: that is, numbers t0, ..., tn − 1 with ti ∈ [xi, xi + 1] for each i. The mesh of a tagged partition is the same as that of an ordinary partition.
Suppose that two partitions P(x, t) and Q(y, s) are both partitions of the interval [a, b]. We say that Q(y, s) is a refinement of P(x, t) if for each integer i, with i ∈ [0, n], there exists an integer r(i) such that xi = yr(i) and such that ti = sj for some j with j ∈ [r(i), r(i + 1)].
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