Arc Length

Given some piecewise-smooth curve $C$ parametrized by $\vec{r}(t)$ for $\alpha \le t\le \beta$ , the length of the curve called the arc length is given by:

$$s={\int }_{\alpha }^{\beta }\mid \mid {\vec{r}}^{\prime }(t)\mid \mid dt$$

In more detail: Arc Length with Vector Functions