What is "iterated integral"

In calculus an iterated integral is the result of applying integrals to a function of more than one variable (for example f(x, y) or f(x, y, z)) in a way that each of the integrals considers some of the variables as given constants. For example, the function f(x, y), if y is considered a given parameter, f(x, y) can be integrated with respect to x, ∫f(x, y)dx. The result is a function of y and therefore its integral can be considered. If this is done, the result is the iterated integral

∫(∫f(x, y) dx) dy.
It is key for the notion of iterated integral that this is different, in principle, from the multiple integral

∬f(x, y) dx dy.
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

∫dy∫f(x, y) dx
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.