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.

A trapezoidal sum differs from the previous 3 in that

is the average of the endpoints of each interval evaulated in

In lower and upper sums,

where

and

such that:

, these approximations tend to the actual area between the function and x-axis which is given by: