Green's theorem doesn't really apply here. GT relates the line integral over some *closed* connected contour that bounds some region (like a circular path that serves as the boundary to a disk). A line segment doesn't form a region since it's completely one-dimensional.
At any rate, we can still compute the line integral just fine. It's just that GT is irrelevant.
We parameterize the line segment by
with
. Then we find the differential:
with.
Here, the line integral is
as required.
At any rate, we can still compute the line integral just fine. It's just that GT is irrelevant.
We parameterize the line segment by
with
with
Here, the line integral is
as required.