# 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.
is the average of the endpoints of each interval evaulated in
where and such that:
, these approximations tend to the actual area between the function and x-axis which is given by:

- Left-hand:
- Midpoint:
- Right-hand:

- Trapezoidal:
where

- Lower:
is the infimum over each interval - Upper:
is the supremum over each interval

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.
is the average of the endpoints of each interval evaulated in
where and such that:
, these approximations tend to the actual area between the function and x-axis which is given by:

- Left-hand:
- Midpoint:
- Right-hand:

- Trapezoidal:
where

- Lower:
is the infimum over each interval - Upper:
is the supremum over each interval

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