Iterated integral
In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example formula_1 or formula_2) in a way that each of the integrals considers some of the variables as given constants. For example, the function formula_1, if formula_4 is considered a given parameter, can be integrated with respect to formula_5, formula_6. The result is a function of formula_4 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 integrals that this is different , in principle, from the multiple integral 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 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 formula_11, the innermost integrand is computed first. Examples. A simple computation. For the iterated integral the integral is computed first and then the result is used to compute the integral with respect to "y". 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. The order is important. The order in which the integrals are computed is important in  iterated integrals, particularly when the integrand is not continuous on the domain of integration. Examples in which the different orders lead to different results are usually for complicated functions as the one that follows. Let a sequence formula_15, such that formula_16. Let formula_17 be a continuous functions not vanishing in the interval formula_18 and zero elsewhere, such that formula_19 for every formula_20. Define In the previous sum, at each specific formula_22, at most one term is different from zero. For this function it happens that