Let $\Delta$ be a triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$ in ${\bf R}^2$. I want to compute $I=\int\limits_\Delta x^2\mathrm{e}^{y^2}\;\mathrm{d}A.$ This is what I've done so far: Note that the hypotenuse is given by the line $y=1-x$. Keep $x\in[0,1]$ fixed, then $y$ is between $1$ and $1-x$ this gives $I=\int_0^1\int_1^{1-x}x^2\mathrm{e}^{y^2}\;\mathrm{d}y\;\mathrm{d}x$, which can't be computed. But when $y\in[0,1]$ is fixed, $x$ is between $0$ and $1-y$, which gives $I=\int\limits_0^1\int\limits_1^{1-y}x^2\mathrm{e}^{y^2}\;\mathrm{d}x\;\mathrm{d}y=\frac13\int\limits_0^1(1-y)^3\mathrm{e}^{y^2}\;\mathrm{d}y,$ which gives the same trouble.
As you can see, I keep ending up with some sort of Gaussian integral, which is impossible to compute. Does anyone know how to compute this integral?