Showing that the double integral of $e^{-xy}$ exists in $\{(x,y):0<x<y<x+x^2\}$

Question

I want to show that $f(x,y)=e^{-xy}$ is Lebesgue integrable in the region $E=\{(x,y):0<x<y<x+x^2\}$ using Fubini's theorem. I thought I could rewrite the function as $$f(x,y)=e^{-xy}\boldsymbol{1}_{x<y<x+x^2}=e^{-xy}\boldsymbol{1}_{x<y}\boldsymbol{1}_{x+x^2}$$ and work in the region $(0,\infty)\times(0,\infty)$ instead. However, I'm stuck on showing that this function is integrable with respect to $x$ for almost all $y$ in the region. Any tips would be much appreciated.

Answer 1

Working on $(0,\infty) \times (0,\infty)$ is indeed a good idea. By Fubini-Tonelli (f is non-negative) you can then integrate in whatever way you want. So you can just integrate over $y$ first. You get $$\int_x^{x+x^2} f \,dy = e^{-x^2} \frac{1-e^{-x^3}}{x}$$ Now the first factor is nicely integrable over $x$ and the second is bounded.