In calculus, the term ‘order of integration’ pertains to the topic of multiple integrals. When solving multiple integrals, (say double or triple integrals), we evaluate for variable ‘x’ (say) first and then for variable ‘y’, depending on how the question is given to us.

However it is possible to change (in case of triple integrals) or reverse (in case of double integrals) the order of integration. Meaning, we can evaluate using variable y first and then using variable x in case of double integrals. The limits of the integral would change accordingly.

## Statement

Statement for the order of integration rule can be stated symbolically as follows:
$\int_{a}^{b}\int_{f_1(y)}^{f_2(y)}f(x,y)dx dy$

=$\int_{c}^{d}\int_{g_1(y)}^{g_2(y)}f(x,y)dy dx$
Note that in the first expression, the limits of the dx are functions of y,where as in the second expression the limits of dx are constants c and d. Similarly in the first expression the limits of dy are constants and in the second expression they are functions of x. Similar symbolic representation of changing order of integration triple integrals is also possible.

The order of integration rule and the steps for changing the order of integration may vary slightly for double and triple integrals. Let us take a look at them one by one.

## Changing Order of Integration Double Integrals

Many a times while solving double integrals, we encounter situations, where in it seems difficult or impossible to iterate in one direction. In such situations if we reverse the order of integration, then the integration becomes very easy.

This can be best understood with the help of an example:

Example 1: Consider the following double integral:
I = $\int_{1}^{2}\int_{0}^{lnx} f(x,y)dy dx$

Here, the outer limits belong to variable x. Thus 1 $\leqslant$ x $\leqslant$ 2. The other  limit is for the variable y. So, 0 = y = ln(x).

It is important to note here, that the limits of x are constants, where as those of y are functions of x. We saw a similar situation in our symbolic representation earlier as well. We will now use all this information to graph the region that we are trying to integrate.

It would look as follows:

The said region is the pink region R. If we now wish to reverse the order, it would look as follows: