jmc

Consider the three right triangles $T_1,T_2,T_3$ formed by the line $AB$, the segment connecting the bottom left corner of the smallest square to the upper right corner of the largest square, and a side of the smallest, medium, and largest squares, respectively. Since all three triangles share an angle, it follows that they must be similar. Notice that the base of $T_3$ is equal to $2+4+6=12$, and its height is equal to $6$. This, the height-to-base ratio of each of $T_1$ and $T_2$ is equal to $6/12=1/2$. Since the base of $T_1$ is $2$ and the base of $T_2$ is $2+4=6$, it follows that their heights are, respectively, $2\cdot (1/2)=1$ and $6\cdot (1/2)=3$. The shaded region is a trapezoid with bases $1$ and $3$ and altitude $4$, and area $\frac{4(1+3)}{2}=\boxed{8}$.