Riemann sum
Types of Riemann sums

Specific choices of x i ∗ {\displaystyle x_{i}^{*}} give different types of Riemann sums:

All these Riemann summation methods are among the most basic ways to accomplish numerical integration. Loosely speaking, a function is Riemann integrable if all Riemann sums converge as the partition "gets finer and finer".

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.

Any Riemann sum on a given partition (that is, for any choice of x i ∗ {\displaystyle x_{i}^{*}} between x i − 1 {\displaystyle x_{i-1}} and x i {\displaystyle x_{i}}) is contained between the lower and upper Darboux sums. This forms the basis of the Darboux integral, which is ultimately equivalent to the Riemann integral.
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