In calculus an iterated integral is the result of applying integrals to a function of more than one variable (for example
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It is key for the notion of iterated integral that this is different, in principle, from 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.
A simple computation
For the iterated integral
the integral
is computed first and then the result is used to compute the integral with respect to y.
It should be noted, however, that this example omits the constants of integration. After the first integration with respect to x, we would rigorously need to introduce a "constant" function of y. That is, If we were to differentiate this function with respect to x, any terms containing only y would vanish, leaving the original integrand. Similarly for the second integral, we would introduce a "constant" function of x, because we have integrated with respect to y. In this way, indefinite integration does not make very much sense for functions of several variables.
The order is important
The order in which the integrals are computed is important in iterated integrals, particularly when the integrand is not continuous on the domain of integration. Examples in which the different orders lead to different results are usually for complicated functions as the one that follows.
Let a sequence
In the previous sum, at each specific