Editing Iterated integral (section)

{{Short description|Type of integral of functions of multiple variables}}
In [[multivariable calculus]], an '''iterated integral''' is the result of applying [[integral]]s to a [[Function (mathematics)|function]] of [[Function of several real variables|more than one variable]] (for example <math>f(x,y)</math> or <math>f(x,y,z)</math>) in such a way that each of the integrals considers some of the variables as given [[Constant (mathematics)|constant]]s. For example, the function <math>f(x,y)</math>, if <math>y</math> is considered a given [[parameter]], can be integrated with respect to <math>x</math>, <math display="inline">\int f(x,y)\,dx</math>. The result is a function of <math>y</math> and therefore its integral can be considered. If this is done, the result is the iterated integral 

:<math>\int\left(\int f(x,y)\,dx\right)\,dy.</math>
It is key for the notion of iterated integrals that this is different, in principle, from the [[multiple integral]]
:<math>\iint f(x,y)\,dx\,dy.</math>

In general, although these two can be different, [[Fubini's theorem]] states that under specific conditions, they are equivalent. 

The alternative notation for iterated integrals
:<math>\int dy \int dx \, f(x,y)</math>
is also used.

In the notation that uses parentheses, iterated integrals are computed following the [[Order of operations|operational order]] indicated by the parentheses starting from the most inner integral outside. In the alternative notation, writing <math display="inline">\int dy \, \int dx \, f(x, y)</math>, the innermost integrand is computed first.