Arclength is the length of a section of a differentiable curve. Finding the length of an arc is useful in many applications, for the length of a curve can represent distance traveled, work, etc. It is commonly represented as $S$ or the differential $ds$ if one is differentiating or integrating with respect to change in arclength.

If one knows the vector function or parametric equations of a curve, finding the arclength is simple, as it can be given by the sum of the lengths of the tangent vectors to the curve or

$$ \int_a^b |\vec{F}'(t)| \; dt=S $$

Note that $t$ is an independent parameter. In Cartesian coordinates, arclength can be calculated by the formula

$$ S=\int_a^b \sqrt{1+(f'(x))^2} \; dx $$

This formula is derived by viewing arclength as the Riemann sum

$$ \lim_{\Delta x\rightarrow\infty}\sum_{i=1}^n \sqrt{1+f'(x_i)^2}\; \Delta x $$

The term being summed is the length of an approximating secant to the curve over the distance $\Delta x$ . As $\Delta x$ vanishes, the sum approaches the arclength, as desired. Arclength can also be derived for polar coordinates from the general formula for vector functions given above. The result is $$ L = \int_a^b \sqrt{r(\theta)^2 + (r'(\theta))^2}\; d\theta$$