= Solution
For a piecewise continuously differentiable <contour> $\gamma:[a,b]\to U$, define its <contour integral> by
$$
\boxed{\int_\gamma f(z)\,dz=\int_a^b f(\gamma(t))\gamma'(t)\,dt,}
$$
adding the integrals over the finitely many smooth pieces if necessary. Continuity of $f$ along the contour makes each integrand integrable. The derivative supplies both the direction and the complex line element; replacing it by its modulus would give a different integral with respect to arc length.
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