Lebesgue integral over $\mathbb R^2$ of the function $f(x,y)=2(x-y)e^{-(x-y)^2}\chi_{\{x>0\}}$

Question

Let $f:\mathbb R^2\to \mathbb R$ be given by $$f(x,y)=\begin{cases}2(x-y)e^{-(x-y)^2}& \text{ if }x>0 \\0&\text{ otherwise}\end{cases}$$ Given that $\int^\infty_{-\infty} e^{-z^2} dz=\sqrt \pi$ compute $$\int_{\mathbb R^2}|f(x,y)|dxdy$$ Using Tonelli's theorem I was able to reduce this to $$\int_{\mathbb R}\int_{\mathbb R_+^*}|2(x-y)e^{-(x-y)^2}|dxdy$$ But I have no idea how to proceed. I thought about using a change of variables but I couldn't find how to do that with a Lebesgue-intergral...