= Solution
{c}
For a piecewise <continuously differentiable> <parametric curve>, the <arc length> is the integral of <speed>:
$$
\boxed{s=\int_a^b\sqrt{[x'(t)]^2+[y'(t)]^2}\,dt.}
$$
The <Pythagorean theorem> gives the length of an infinitesimal displacement as $\sqrt{dx^2+dy^2}$. 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.
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