Determine the volume of the set $$\{ (x_1,x_2) \in \mathbb{R}^2\mid \left| x_1- \sqrt{2}x_2 \right| \leq 1, \left| x_1- \sqrt{3}x_2 \right| \leq 1 \}$$
We can find that $ \sqrt{2}x_2-1 \leq x_1 \leq 1+ \sqrt{2}x_2$ and $ \sqrt{3}x_2-1 \leq x_1 \leq 1+ \sqrt{3}x_2$. Hence,
$\left\{\begin{matrix} x_1 \in [\sqrt{3}x_2-1,\sqrt{2}x_2+1] \; \text{if} \; x_2>0\\ x_1 \in [\sqrt{2}x_2-1,\sqrt{3}x_2+1] \; \text{if} \; x_2<0 \end{matrix}\right. $
So the volume of this set is $\int_{0}^\infty \int_{\sqrt{3}x_2-1}^{\sqrt{2}x_2+1} 1 dx_1 dx_2+\int_{-\infty}^0 \int_{\sqrt{2}x_2-1}^{\sqrt{3}x_2+1} 1 dx_1 dx_2$?