Find: $$\int_1^2(\int_{\sqrt{x}}^xe^{\frac{x}{y}} dy)dx + \int_2^4(\int_{\sqrt{x}}^2e^{\frac{x}{y}} dy)dx$$
I have tried to go in this order and reached some $ye^{\sqrt{y}}$ integral and completely got stuck.
So I thought of changing the order of integration.
I drew the areas of each integral and reached:
$$\int_1^{\sqrt{2}}(\int_{y}^{y^2}e^{\frac{x}{y}} dx)dy + \int_{\sqrt{2}}^{2}(\int_{y}^{2}e^{\frac{x}{y}} dx)dy + \int_{\sqrt{2}}^{2}(\int_{2}^{y^2}e^{\frac{x}{y}} dx)dy$$
And I also got integrals $ye^{\frac{2}{y}}$ which were hard for me to calculate.
I'm wondering if there's any trick I haven't seen or I have mistakes while changing the order of integration.
Any help is really appreciated, thanks in advance!
Best Answer
Please see the region shaded in the diagram.
So with change of order, the integral will be
$\displaystyle \int_1^2 \int_y^{y^2} e^{\frac{x}{y}} \ dx \ dy$
This will require you to integrate $\displaystyle y \ e^y$ in the second integral, which can be done by Integration by parts.
Edit: If you combine your second and third integral, and then with the first, you get the same integral I wrote above. You just considered change of order for each sub-region separately and so ended up with three integrals.