Iterated integral

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

${\displaystyle \int \left(\int f(x,y)\,dx\right)\,dy.}$

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

${\displaystyle \iint f(x,y)\,dx\,dy.}$

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

${\displaystyle \int dy\int dx\,f(x,y)}$

is also used.

In the notation that uses parentheses, iterated integrals are computed following the operational order indicated by the parentheses starting from the most inner integral outside. In the alternative notation, writing ${\textstyle \int dy\,\int dx\,f(x,y)}$, the innermost integrand is computed first.