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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. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868.^[1] 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.

Overview

Let f be a non-negative real-valued function on the interval , and let S be the region of the plane under the graph of the function f and above the interval . See the figure on the top right. This region can be expressed in set-builder notation as

$${\displaystyle S=\left\{(x,y)\,:\,a\leq x\leq b\,,\,0<y<f(x)\right\}.}$$

We are interested in measuring the area of S. Once we have measured it, we will denote the area in the usual way by

$${\displaystyle \int _{a}^{b}f(x)\,dx.}$$

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.

Definition

Partitions of an interval

A partition of an interval is a finite sequence of numbers of the form

$${\displaystyle a=x_{0}<x_{1}<x_{2}<\dots <x_{i}<\dots <x_{n}=b}$$

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,

$${\displaystyle \max \left(x_{i+1}-x_{i}\right),\quad i\in .}$$

A tagged partition P(x, t) of an interval is a partition together with a choice of a sample point within each sub-interval: that is, numbers t₀, ..., t_{n − 1} with t_i ∈ 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 . We say that Q(y, s) is a refinement of P(x, t) if for each integer i, with i ∈ , there exists an integer r(i) such that x_i = y_r(i) and such that t_i = s_j for some j with j ∈ . That is, a tagged partition breaks up some of the sub-intervals and adds sample points where necessary, "refining" the accuracy of the partition.

We can turn the set of all tagged partitions into a directed set by saying that one tagged partition is greater than or equal to another if the former is a refinement of the latter.

Riemann sum

Let f be a real-valued function defined on the interval . The Riemann sum of f with respect to the tagged partition x₀, ..., x_n together with t₀, ..., t_{n − 1} is^[2]

$${\displaystyle \sum _{i=0}^{n-1}f(t_{i})\left(x_{i+1}-x_{i}\right).}$$

Each term in the sum is the product of the value of the function at a given point and the length of an interval. Consequently, each term represents the (signed) area of a rectangle with height f(t_i) and width x_{i + 1} − x_i. The Riemann sum is the (signed) area of all the rectangles.

Closely related concepts are the lower and upper Darboux sums. These are similar to Riemann sums, but the tags are replaced by the infimum and supremum (respectively) of f on each sub-interval:

$${\displaystyle {\begin{aligned}L(f,P)&=\sum _{i=0}^{n-1}\inf _{t\in }f(t)(x_{i+1}-x_{i}),\\U(f,P)&=\sum _{i=0}^{n-1}\sup _{t\in x_{i},x_{i+1}}f(t)(x_{i+1}-x_{i}).\end{aligned}}}$$

If f is continuous, then the lower and upper Darboux sums for an untagged partition are equal to the Riemann sum for that partition, where the tags are chosen to be the minimum or maximum (respectively) of f on each subinterval. (When f is discontinuous on a subinterval, there may not be a tag that achieves the infimum or supremum on that subinterval.) The Darboux integral, which is similar to the Riemann integral but based on Darboux sums, is equivalent to the Riemann integral.

Riemann integraledit

Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be integrable (or more specifically Riemann-integrable). The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.^[3]

One important requirement is that the mesh of the partitions must become smaller and smaller, so that it has the limit zero. If this were not so, then we would not be getting a good approximation to the function on certain subintervals. In fact, this is enough to define an integral. To be specific, we say that the Riemann integral of f exists and equals s if the following condition holds:

For all ε > 0, there exists δ > 0 such that for any tagged partition x₀, ..., x_n and t₀, ..., t_{n − 1} whose mesh is less than δ, we have

$${\displaystyle \left|\left(\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})\right)-s\right|<\varepsilon .}$$

Unfortunately, this definition is very difficult to use. It would help to develop an equivalent definition of the Riemann integral which is easier to work with. We develop this definition now, with a proof of equivalence following. Our new definition says that the Riemann integral of f exists and equals s if the following condition holds: Zdroj:https://en.wikipedia.org?pojem=Riemann_integral
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