Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics IV / 3 / a / i / Solution

For a piecewise continuously differentiable parametric curve, the arc length is the integral of speed:The Pythagorean theorem gives the length of an infinitesimal displacement as $\sqrt{d{x}^2+d{y}^2}$. Integration counts traversal length, including repeated portions if the parametrization retraces them. Piecewise continuous derivatives, or suitable absolute continuity, justify the formula; mere differentiability as stated in the source is not by itself the usual sufficient regularity hypothesis.