What is an iterated integral?
Answers
In calculus an iterated integral is the result of applying integrals to a function of more than one variable 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,
In 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 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}, {\displaystyle \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 integral that this is different, in principle, from the multiple integral
{\displaystyle \iint 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
{\displaystyle \int dy\int dx\,f(x,y)}
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.