Arc length

For a real-valued function, the arc-length between two points is:

$$L={\int }_a^b\sqrt{1+(f^{\prime }(x){)}^2}dx$$

For a vector-valued function, the arc-length travelled for $t_1\le t\le t_2$ is:

$$L={\int }_{t_1}^{t_2}\sqrt{(f^{\prime }(t){)}^2+(g^{\prime }(t){)}^2}dt={\int }_{t_1}^{t_2}\Vert {\vec{r}}^{\prime }(t)\Vert dt$$

We also (rather helpfully) define a time dependent arc-length function:

$$s(t)={\int }_{t_1}^t\Vert {\vec{r}}^{\prime }(t)\Vert dt$$

And we say that a curve is parametrised with respect to arc-length if $\Vert {\vec{r}}^{\prime }\Vert =1$. The differential arc-length is:

$$dl=d{l}_x\hat{x}+d{l}_y\hat{y}=ds$$