Editing Improper integral (section)

===Improper integrals over arbitrary domains===
If <math>f:\R^n\to\R</math> is a non-negative function that is Riemann integrable over every compact cube of the form <math>[-a,a]^n</math>, for <math>a>0</math>, then the improper integral of ''f'' over <math>\R^n</math> is defined to be the limit
:<math>\lim_{a\to\infty}\int_{[-a,a]^n}f,</math>
provided it exists.

A function on an arbitrary domain ''A'' in <math>\mathbb R^n</math> is extended to a function <math>\tilde{f}</math> on <math>\R^n</math> by zero outside of ''A'':
:<math>\tilde{f}(x)=\begin{cases}f(x)& x\in A\\
0 & x\not\in A
\end{cases}</math>
The Riemann integral of a function over a bounded domain ''A'' is then defined as the integral of the extended function <math>\tilde{f}</math> over a cube <math>[-a,a]^n</math> containing ''A'':  
:<math>\int_A f = \int_{[-a,a]^n}\tilde{f}.</math>
More generally, if ''A'' is unbounded, then the improper Riemann integral over an arbitrary domain in <math>\mathbb R^n</math> is defined as the limit:
:<math>\int_Af=\lim_{a\to\infty}\int_{A\cap [-a,a]^n}f=\lim_{a\to\infty}\int_{[-a,a]^n}\tilde{f}.</math>