Yes, "circulation" must occur on a closed path. This is actually a single path with three different parts. Draw the three graphs given and you should see which closed path this is referring to: the flow along from (0, 0) to (1, 1), along y= 1 from (1, 1) to (0, 1) and along x= 0 from (0, 1) to (0, 0) again.
You could also use Green's theorem to do this. In fact, it would be good idea to do it both ways as a check!