The right region is swept in the positive direction. The left region is swept in the negative direction. The space between the two regions is swept in

bothdirections, and so the signed areas swept out cancel each other. Thus the total signed area swept out isA-_{R}A, where_{L}Aand_{R}Aare the areas traversed by the right and left endpoints of the segment, respectively._{L}The rigorous proof is a nice application of Green's Theorem.