# VisualEditor:TestMath

From MediaWiki.org

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

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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

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- 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:^{[1]}

- (as in the Pythagorean theorem or the Euclidean norm), and
- ,

where atan2 is a common variation on the arctangent function defined as

References