jmc

Plotting $y=|x-3|$ and $y=4-|x-1|,$ we find that the two graphs intersect at $(0,3)$ and $(4,1).$



The region then is a rectangle with side lengths $\sqrt{2}$ and $3\sqrt{2},$ so its area is $(\sqrt{2})(3\sqrt{2})=\boxed{6}.$