\iint\limits_{D}x^2e^{-y^2}dxdy=\int_0^1dy\int_0^yx^2e^{-y^2}dx=\int_0^1\frac{1}{3}y^3e^{-y^2}dy=\frac{1}{6}\int_0^1ze^{-z}dz(z=y^2)=\frac{1}{6}(1-\frac{2}{e})=\frac{1}{6}-\frac{1}{3e}
