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:

I = $\int_{0}^{lnx} \int_{1}^{2}$ f(x,y)dy dx$So you see that the dy dx of the end has not become dx dy. However we are still to find the new limits of integration. So now this time the outer limits would belong to y and so we need constants for that now. From the above graph we can see that the smallest value of y is 0 where as the largest value of y is ln(2). Thus the limits of y would be from 0 to ln(2). (We can see that from the graph).

Now, coming to the inner limits since we already established that the outer limits belong to y, so obviously the inner limits would belong to x.

Also, since these are inner limits, we need x in terms of some function of y. The function given to us in the question is y = ln(x) (the upper limit of y given in the original question). Solving that for x we have, x = $e^y$. From the graph we also see that the right hand side limit of the x axis for the region R is the line x=2. Thus the upper limit of x would be 2 and the lower limit would be $e^y$.

Therefore now our new integral with reversed order would looks follows:

I = $\int_{0}^{ln2}\int_{e^y}^{2} f(x,y)dy dx$

Thus we see that we have successfully changed the order of integration for the given example problem.