imc

Suppose that line $\ell$ is horizontal, and each circle lies either north or south to $\ell .$ We construct the circles one by one: 1. Without the loss of generality, we draw the circle with radius $7$ north to $\ell .$ 2. To maximize the area of region $S,$ we draw the circle with radius $5$ south to $\ell .$ 3. Now, we need to subtract the circle with radius $3$ at least. The optimal situation is that the circle with radius $3$ encompasses the circle with radius $1,$ in which we do not need to subtract more. That is, the two smallest circles are on the same side of $\ell ,$ but can be on either side. The diagram below shows one possible configuration of the four circles: Together, the answer is $\pi \cdot 7^2+\pi \cdot 5^2-\pi \cdot 3^2=\boxed{65\pi }.$