Arc Length

Topics: Curve - Calculus

---

(theorem)

Let $\sigma$ be a curve that is drawn by the following vector-valued function:

$$r(t)=⟨x_1(t),x_2(t),\dots ,x_n(t)⟩$$

The arc length of $\sigma$ is given by:

$$s={\int }_a^b\sqrt{[x_1^{\prime }(t){]}^2+[x_2^{\prime }(t){]}^2+\cdots +[x_n^{\prime }(t){]}^2}dt$$

We can use arc length to parametrise a function nicely.