Iterated integral
Encyclopedia
In calculus
an iterated integral is the result of applying integrals to a function of more than one variable (for example
or 
) 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
can be integrated with respect to
, 
. The result is a function of 
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, to 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
.
The alternative notation for iterated integrals
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.
the integral
is computed first and then the result is used to compute the integral with respect to y.
It should be noted, however, that this example omits the constants of integration. After the first integration with respect to x, we would rigorously need to introduce a "constant" function of y. That is, If we were to differentiate this function with respect to x, any terms containing only y would vanish, leaving the original integrand. Similarly for the second integral, we would introduce a "constant" function of x, because we have integrated with respect to y. In this way, indefinite integration does not make very much sense for functions of several variables. While the antiderivatives of single variable functions differ at most by a constant, the antiderivatives of multivariable functions differ by at most unknown single-variable terms, which could have a drastic effect on the behavior of the function.
Let a sequence
, such that 
. Let 
be continuous functions not vanishing in the interval 
and zero elsewhere, such that 
for every 
. Define
In the previous sum, at each specific
, at most one term is different from zero.
For this function it happens that
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
an iterated integral is the result of applying integrals to a function of more than one variable (for example
or ) 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 parameterParameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
can be integrated with respect to
, . The result is a function of and therefore its integral can be considered. If this is done, the result is the iterated integralIt is key for the notion of iterated integral that this is different, in principle, to the multiple integral
Multiple integral
The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...
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
Fubini's theorem
In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
.
The alternative notation for iterated integrals
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.
A simple computation
For the iterated integralthe integral
is computed first and then the result is used to compute the integral with respect to y.
It should be noted, however, that this example omits the constants of integration. After the first integration with respect to x, we would rigorously need to introduce a "constant" function of y. That is, If we were to differentiate this function with respect to x, any terms containing only y would vanish, leaving the original integrand. Similarly for the second integral, we would introduce a "constant" function of x, because we have integrated with respect to y. In this way, indefinite integration does not make very much sense for functions of several variables. While the antiderivatives of single variable functions differ at most by a constant, the antiderivatives of multivariable functions differ by at most unknown single-variable terms, which could have a drastic effect on the behavior of the function.
The order is important
The order in which the integrals are computed is important in iterated integrals. Examples in which the different orders lead to different results are usually for complicated functions as the one that follows.Let a sequence
, such that . Let be continuous functions not vanishing in the interval and zero elsewhere, such that for every . DefineIn the previous sum, at each specific
, at most one term is different from zero.For this function it happens that