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While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.
One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.
Riemann mapping theorem (complex analysis) Riemann series theorem (mathematical series) Riemann's existence theorem (algebraic geometry) Riemann's theorem on removable singularities (complex analysis) Riemann–Roch theorem (Riemann surfaces, algebraic curves) Riemann–Roch theorem for smooth manifolds (differential topology)
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.
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals , meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. [ 1 ]
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