Compute $$\int_{y=0}^{1} \int_{x=y}^{1} \frac{x^2}{y^2} e^{\frac{-x^2}{y}}dxdy.$$
Here's my idea.
Switch the order of integration by Fubini's Theorem. Then compute $$\int_{y=0}^{1} \frac{x^2}{y^2} e^{\frac{-x^2}{y}}dy$$ with $u = \frac{-1}{y}$, $du = \frac{1}{y^2}dy$ to have $$\int_{-\infty}^{-1} x^2e^{ux^2}du = e^{-x^2}$$ but now I get stuck with $$\int_{x=y}^{1}e^{-x^2}dx$$ because this can't be computed.
Does anyone have suggestions?