Autonomous vehicles become more and more popular in our daily life. Mobile computing schemes to be installed on these vehicles have drawn a lot of recent research interest. In this paper, we address the important path-planning problem for autonomous vehicles. We introduce and formulate the novel paintbrush coverage problem. We present a theoretical study on the minimum trajectory length of a paintbrush to cover an arbitrary convex region, which is derived as a function of the area of the region and the size of the cover. Three commonly-used patrolling/scouting methods, namely boustrophedon, spiral, and sector, are manifested in details as the potential solutions to the paintbrush coverage problem. The theoretical minimum trajectory lengths any algorithm can achieve are also demonstrated as the benchmarks for different shapes of regions.