What paths have length?

Abstract

In the classical theory, the length of the curve $y = f(x)(a \leqslant x \leqslant b)$ is determined by computing the integral $\int\limits_{a}^{b} {\sqrt {{1 + f{{\prime }^{2}}(x)dx}} }$. Geometrically, this means that in determining the length of an arc we really compute the area of a plane domain. The length of the circular arc $y = \sqrt {{1 - {{x}^{2}}}} (0 \leqslant x \leqslant b)$ is the area of the plane domain $(0 \leqslant x \leqslant b,0 \leqslant y \leqslant 1\sqrt {{1 - {{x}^{2}}}} )$. If the arc happens to be a quarter of a circle, the domain is not even bounded.