Polar arc length ($ds=\sqrt{r^2+(dr/d\theta {)}^2}d\theta$)

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 $d{x}^2+d{y}^2$, leaving $d{s}^2=d{r}^2+r^{2}d{\theta }^2$. Therefore the arc length on an increasing parameter interval isThis 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.