In calculus an iterated integral is the result of applying integrals to a function of more than one variable (for example or ) in a way that each of the integrals considers some of the variables as given constants. For example, the function, if is considered a given parameter can be integrated with respect to, . The result is a function of and therefore its integral can be considered. If this is done, the result is the iterated integral
It is key for the notion of iterated integral that this is different, in principle, to the multiple integral
Although in general these two can be different there is a theorem that, under very mild conditions, gives the equality of the two. This is Fubini's theorem.
The alternative notation for iterated integrals
is also used.
Iterated integrals are computed following the operational order indicated by the parentheses (in the notation that uses them). Starting from the most inner integral outside.
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