A Riemann sum is an approximation of the form [math]\sum_{k=1}^{n}{\left[f\left(x_{k} \right )\cdot\Delta{x} \right ]}[/math]. It is most often used to approximate the area under some function [math]f\left(x\right)[/math] on the closed interval [math]\left[a,b\right][/math]. Below are six types of sums: left-hand, midpoint, right-hand, trapezoidal, lower, and upper.