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

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

This formula is derived by viewing arclength as the Riemann sum

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