smc

The region $S$ is a circle radius $T$ and center $(T,0)$. Since $T=t-[t]$, $0\le T<1$. That means the area of the circle is less than $\pi$, and since the region can also be just a dot (achieved when $t$ is integer), the answer is $\boxed{0\le \text{Area }S\le \pi \text{ for all }t}$.