Riemann Sums

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.
  • Left-hand:
  • Midpoint:
  • Right-hand:
A trapezoidal sum differs from the previous 3 in that is the average of the endpoints of each interval evaulated in
  • Trapezoidal: where
In lower and upper sums, where and such that:
  • Lower: is the infimum over each interval
  • Upper: is the supremum over each interval
As , these approximations tend to the actual area between the function and x-axis which is given by: