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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 .
A sequence of Riemann sums over a regular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function. The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero.
Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. [1] They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the ...
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.
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 quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral.
The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. [21] A tagged partition of a closed interval [a, b] on the real line is a finite sequence
The Riemann–Stieltjes integral admits integration by parts in the form () = () () ()and the existence of either integral implies the existence of the other. [2]On the other hand, a classical result [3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1 .