A Riemann sum is an approximation of the form . It is most often used to approximate the area under some function on the closed interval . Below are six types of sums: left-hand, midpoint, right-hand, trapezoidal, lower, and upper.
In these sums, represents the width of each rectangle (AKA interval), defined by . The parameter that changes depending on the type of sum is . This determines where the function is evaluated and thus calculates the height of each rectangle.
- Trapezoidal: where
- Lower: is the infimum over each interval
- Upper: is the supremum over each interval