# VisualEditor:TestMath

Let f : DR 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

$P= \left \{[x_0,x_1),[x_1,x_2),\dots,[x_{n-1},x_{n}] \right \},$
Default ${\textstyle E=\Sigma mc^2}$
Default $E=\Sigma mc^2$ ${\displaystyle E = \Sigma mc^2}$

be a partition of I, where

$a=x_0$a^2{\frac {2} {3}}{3}$

The Riemann sum of f over I with partition P is defined as

$S = \sum_{i=1}^{n} f(x_i^*)(x_{i}-x_{i-1}), \quad x_{i-1}\le x_i^* \le x_i.$

The choice of $x_i^*$ in the interval $[x_{i-1},x_i]$ is arbitrary.

Example: Specific choices of $x_i^*$ give us different types of Riemann sums:

• If $x_i^*=x_{i-1}$ for all i, then S is called a left Riemann sum.
• If $x_i^*=x_i$ for all i, then S is called a right Riemann sum.
• If $x_i^*=\tfrac{1}{2}(x_i+x_{i-1})$ 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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$S = \sum_{i=1}^{n} v_i(x_{i}-x_{i-1}),$
where $v_i$ is the supremum of f over $[x_{i-1},x_i]$, then S is defined to be an upper Riemann sum.
• Similarly, if $v_i$ is the infimum of f over $[x_{i-1},x_i]$, 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]

$r = \sqrt{x^2 + y^2} \quad$ (as in the Pythagorean theorem or the Euclidean norm), and
$\varphi = \operatorname{atan2}(y, x) \quad$,

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

$\operatorname{atan2}(y, x) = \begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0\\ \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\ \arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\ \frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\ -\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\ \text{undefined} & \mbox{if } x = 0 \mbox{ and } y = 0 \end{cases}$

References