Editing Improper integral (section)

==Multivariable improper integrals==
The improper integral can also be defined for functions of several variables. The definition is slightly different, depending on whether one requires integrating over an unbounded domain, such as <math>\R^2</math>, or is integrating a function with singularities, like <math>f(x,y)=\log\left(x^2+y^2\right)</math>.

===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>

===Improper integrals with singularities===
If ''f'' is a non-negative function which is unbounded in a domain ''A'', then the improper integral of ''f'' is defined by truncating ''f'' at some cutoff ''M'', integrating the resulting function, and then taking the limit as ''M'' tends to infinity.  That is for <math>M>0</math>, set <math>f_M=\min\{f,M\}</math>.  Then define
:<math>\int_A f = \lim_{M\to\infty}\int_A f_M</math>
provided this limit exists.

===Functions with both positive and negative values===
These definitions apply for functions that are non-negative.  A more general function ''f'' can be decomposed as a difference of its positive part <math>f_+=\max\{f,0\}</math> and negative part <math>f_-=\max\{-f,0\}</math>, so
:<math>f=f_+-f_-</math>
with <math>f_+</math> and <math>f_-</math> both non-negative functions.  The function ''f'' has an improper Riemann integral if each of <math>f_+</math> and <math>f_-</math> has one, in which case the value of that improper integral is defined by
:<math>\int_Af = \int_Af_+ - \int_A f_-.</math>
In order to exist in this sense, the improper integral necessarily converges absolutely, since
:<math>\int_A|f| = \int_Af_+ + \int_Af_-.</math><ref>{{harvnb|Cooper|2005|loc=p. 538}}: "We need to make this stronger definition of convergence in terms of |''f''(''x'')| because cancellation in the integrals can occur in so many different ways in higher dimensions."</ref><ref>{{harvnb|Ghorpade|Limaye|2010|loc=p. 448}}: "The relevant notion here is that of unconditional convergence." ... "In fact, for improper integrals of such functions, unconditional convergence turns out to be equivalent to absolute convergence."</ref>