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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.
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...
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 ...
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
This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right. Conversely, the term "quadrature" may also be used for the solution of differential equations: "solving by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals.
A way to obtain the formula The method's formula can be derived as follows: Consider the function f ( x ) {\displaystyle f(x)} which describes our cross-section of the solid, now the integral of the function can be described as a Riemann integral: ∫ a b f ( x ) d x = lim n → ∞ ∑ i = 1 n f ( a + i Δ x ) Δ x {\displaystyle \int \limits ...
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 .