= Polar arc length
{title2=$ds=\sqrt{r^2+(dr/d\theta)^2}\,d\theta$}
For a piecewise <differentiable> <curve> given in <polar coordinates> by $r=r(\theta)$, differentiate $x=r\cos\theta$ and $y=r\sin\theta$. The cross terms cancel in $dx^2+dy^2$, leaving $ds^2=dr^2+r^2d\theta^2$. Therefore the <arc length> on an increasing parameter interval is
$$
\ell=\int_{\theta_0}^{\theta_1}\sqrt{r(\theta)^2+r'(\theta)^2}\,d\theta.
$$
This formula remains valid through an isolated zero of the speed when the parametrization is piecewise smooth. A square root of a square must retain its <absolute value>.
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