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In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann . One very common application is in numerical integration , i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule .
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.
Discrete integral calculus is the study of the definitions, properties, and applications of the Riemann sums. The process of finding the value of a sum is called integration . In technical language, integral calculus studies a certain linear operator .
A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function. We are describing the area of a rectangle, with the width times the height, and we are adding the areas together.
The h-integral is denoted by (). If a and b differ by an integer multiple of h then the definite integral () is given by a Riemann sum of f(x) on the interval [a, b], partitioned into sub-intervals of equal width h. The motivation of h-integral comes from the Riemann sum of f(x).
The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it, where σ and t are real numbers. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) When Re(s) = σ > 1, the function can be written as a converging summation or as an integral:
Taking n → ∞, the sum becomes the Riemann sum for the above integral. A mechanical device that computes area integrals is the planimeter , which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem , converting the ...
The Itô integral can be defined in a manner similar to the Riemann–Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily exist pathwise. Suppose that B is a Wiener process (Brownian motion) and that H is a right-continuous ( càdlàg ), adapted and locally bounded process.