  The aim of this paper is to associate a measure for certain sets of paths in
the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then,
working on parameterized surfaces with a specific Riemannian metric, we define
and calculate the integral of the length over the set of paths obtained as the
image of the initial paths in $\mathbb{R}^2$ under the given parameterization.
Moreover, we prove that this integral is given by the average of the lengths of
the external paths times the measure of the set of paths if and only if the
surface has Gaussian curvature equal to zero.

Differential Geometry