Iterated Integrals

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

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

It is key to note that this is different, in principle, from the multiple integral $\iint f(x,y)\,dx\,dy$ . A theorem called Fubini's theorem, however, states that they may be equal under very mild conditions. The alternative notation for iterated integrals $\int dy \int f(x,y)\,dx$ is also used. Iterated integrals are computed following the operational order indicated by the parentheses (in the notation that uses them), starting from the innermost integral and working out.

Use of an iterated integral

An iterated integral can be used to find the volume of the object in the figure.

Example

For the iterated integral $\int\left(\int (x+y) \, dx\right) \, dy$, the integral $\int (x+y) \, dx = \frac{x^2}{2} + yx$ is computed first. The result is then used to compute the integral with respect to $y$:

$\displaystyle{\int \left(\frac{x^2}{2} + yx \right) \, dy = \frac{yx^2}{2} + \frac{xy^2}{2}}$

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 integral. 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 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 unknown single-variable terms, which could have a drastic effect on the behavior of the function.