Testwiki:VisualEditor testing/TestMath

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

$P=\{[x_0,x_1),[x_1,x_2),\dots ,[x_{n-1},x_n]\},$

Default $E=\mathrm{\Sigma }m{c}^2$

Default $E=\mathrm{\Sigma }m{c}^2$ $$E=\mathrm{\Sigma }m{c}^2$$

be a partition of I, where

$a=x_0<x_1<x_2<\cdots <x_n=b$$a^2\frac{2}{3}3$

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

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

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 $x_i^{*}=\frac{1}{2}(x_i+x_{i-1})$ 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:

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

$\phi =\mathrm{atan2}(y,x)$,

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

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

References