Solve the double integral of $e^{x/y}dydx$

Question

I am asked to solve the double integral
$$\int_0^3 \int_{x}^3 e^{x/y} dy dx$$

I have tried doing it through "u-sub" and integration by parts, however, I cannot seem to solve it. I have also tried polar coordinates and have had no luck.

Answer 1

Use Fubini's Theorem and change integration order:

$$\int_0^3\int_x^3 e^{x/y}dydx=\int_0^3\int_0^y e^{x/y}dxdy=\left.\int_0^3y e^{x/y}dy\right|_0^y=\int_0^3y(e-1)=$$

$$=\left.\frac{e-1}2y^2\right|_0^3=\frac{9(e-1)}2$$