In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable) in such a way that each of the integrals considers some of the variables as given constants is considered a given parameter and therefore its integral can be considered. If this is done, the result is the iterated integral

The alternative notation for iterated integrals

is also used.

In the notation that uses parentheses, iterated integrals are computed following the operational order, the innermost integrand is computed first.

Versus multiple integrals

It is key for the notion of iterated integrals that this is different, in principle, from the multiple integral

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

Examples

A simple computation

For the iterated integral

the integral

is computed first and then the result is used to compute the integral with respect toy.

This example omits the constants of integration. After the first integration with respect tox, we would rigorously need to introduce a "constant" function ofy. That is, If we were to differentiate this function with respect to x, any terms containing onlyy would vanish, leaving the original integrand. Similarly for the second integral, we would introduce a "constant" function ofx, because we have integrated with respect toy. In this way, indefinite integration does not make very much sense for functions of several variables.

Lack of commutativity

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.

. Define

, at most one term is different from zero. For this function it happens that

References