The line integral of the scalar along a path described parametrically by , , is given by
where is arc length, so = , with being the vector form of the parametric representation of the path. However, there are three linear segments for the polygonal line.
A parametric representation of the first line segment is
=
so that
=
and the line integral along this segment is given by
A parametric representation of the second line segment is
=
so that
=
and the line integral along this segment is given by
A parametric representation of the third line segment is
=
so that
=
and the line integral along this segment is given by
Thus, the line integral along the polygonal path is the sum