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 ) {displaystyle f(x,y)} or f ( x , y , z ) {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 f ( x , y ) {displaystyle f(x,y)} , if y {displaystyle y} is considered a given parameter, can be integrated with respect to x {displaystyle x} , ∫ f ( x , y ) d x {displaystyle int f(x,y)dx} . The result is a function of y {displaystyle y} and therefore its integral can be considered. If this is done, the result is the 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 ) {displaystyle f(x,y)} or f ( x , y , z ) {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 f ( x , y ) {displaystyle f(x,y)} , if y {displaystyle y} is considered a given parameter, can be integrated with respect to x {displaystyle x} , ∫ f ( x , y ) d x {displaystyle int f(x,y)dx} . The result is a function of y {displaystyle y} and therefore its integral can be considered. If this is done, the result is the iterated integral 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.