Riemann Sum - Wikipedia

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

• If x i ∗ = x i − 1 {\displaystyle x_{i}^{*}=x_{i-1}}   for all i, the method is the left rule[2][3] and gives a left Riemann sum.
• If x i ∗ = x i {\displaystyle x_{i}^{*}=x_{i}}   for all i, the method is the right rule[2][3] and gives a right Riemann sum.
• If x i ∗ = ( x i + x i − 1 ) / 2 {\displaystyle x_{i}^{*}=(x_{i}+x_{i-1})/2}   for all i, the method is the midpoint rule[2][3] and gives a middle Riemann sum.
• If f ( x i ∗ ) = sup f ( [ x i − 1 , x i ] ) {\displaystyle f(x_{i}^{*})=\sup f([x_{i-1},x_{i}])}   (that is, the supremum of f {\textstyle f}   over [ x i − 1 , x i ] {\displaystyle [x_{i-1},x_{i}]}  ), the method is the upper rule and gives an upper Riemann sum or upper Darboux sum.
• If f ( x i ∗ ) = inf f ( [ x i − 1 , x i ] ) {\displaystyle f(x_{i}^{*})=\inf f([x_{i-1},x_{i}])}   (that is, the infimum of f over [ x i − 1 , x i ] {\displaystyle [x_{i-1},x_{i}]}  ), the method is the lower rule and gives a lower Riemann sum or lower Darboux sum.

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.