When the left end of the segment (segment of length L) moves between coordinate x₀ and x₀ + 1 (x₀ is integer), the covered area changes as a linear function. The sum/difference of slopes is enough to say how it changes.

You should sweep over the given points, updating something when the sweep is over some point or some point is L to the right from the sweep line. This way you will know the area for any placement you want.