I want to compute $\int_B f\,\mathrm{d}(x,y)$ with $B=\{(x,y)\in\mathbb{R}^2: |x|<y<a, \ a>0\}$ and $f=ye^{x/y}$.
Since we have $|x|<y<a$ I thought to split the integral like this:
$\int_0^a\int_x^af\mathrm{d}y\mathrm{d}x + \int_{-a}^0\int_{-x}^af\mathrm{d}y\mathrm{d}x$ for $x\ge 0$ and $x<0$, respectively.
I hope this is correct, so far. I cannot however, find a closed antiderivative of the double integral. I ever end up with the integralexponentialfunction.
The region you integrate on is as below
And you want to integrate with respect to $x$ first, since it is impossible to integrate the other way. So the integral can be written as
$$\int_0^a\int_{-y}^y f \,dx dy.$$
Can you continue from here?