Riemann integral

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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. While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define. For a great many functions and practical applications, the Riemann integral can also be readily evaluated by using the fundamental theorem of calculus or (approximately) by numerical integration.

Some of the technical deficiencies in Riemann integration can be remedied by the Riemann–Stieltjes integral, and most of these disappear with the Lebesgue integral.

Contents

Overview

Let f be a non-negative real-valued function of the interval [a,b], and let S = {(x,y) | 0 < y < f(x)} 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). We are interested in measuring the area of S. Once we have measured it, we will denote the area 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.

Note that where ƒ can be both positive and negative, the integral corresponds to signed area under the graph of ƒ; that is, the area above the x-axis minus the area below the x-axis.

Definition

Partitions of an interval

A partition of an interval [a,b] is a finite sequence a = x_0 < x_1 < x_2 < \cdots < x_n = b. Each [xi,xi + 1] is called a subinterval of the partition. The mesh of a partition is defined to be the length of the longest subinterval [xi,xi + 1], that is, it is max(xi + 1xi) where 0 \le i \le n - 1. It is also called the norm of the partition.

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