Let f : D → R be a function defined on a subset, D, of the real line, R. Let I = [a, b] be a closed interval contained in D, and let
be a partition of I, where
The Riemann sum of f over I with partition P is defined as
The choice of in the interval is arbitrary.
Example: Specific choices of give us different types of Riemann sums:
- If for all i, then S is called a left Riemann sum.
- If for all i, then S is called a right Riemann sum.
- If for all i, then S is called a middle Riemann sum.
- The average of the left and right Riemann sum is the trapezoidal sum.
- If it is given that
- where is the supremum of f over , then S is defined to be an upper Riemann sum.
- Similarly, if is the infimum of f over , then S is a lower Riemann sum.
The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by: