# arc length

Arclength is the 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 , as it can be given by the sum of the lengths of the tangent vectors to the curve or

 $\int_{a}^{b}|\vec{F}^{\prime}(t)|\;dt=S$

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

 $S=\int_{a}^{b}\sqrt{1+(f^{\prime}(x))^{2}}\;dx$

This formula is derived by viewing arclength as the Riemann sum

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

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

 $L=\int_{a}^{b}\sqrt{r(\theta)^{2}+(r^{\prime}(\theta))^{2}}\;d\theta$
 Title arc length Canonical name ArcLength Date of creation 2013-03-22 12:02:43 Last modified on 2013-03-22 12:02:43 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 14 Author mathcam (2727) Entry type Algorithm Classification msc 26B15 Synonym length of a curve Related topic Rectifiable Related topic IntegralRepresentationOfLengthOfSmoothCurve Related topic StraightLineIsShortestCurveBetweenTwoPoints Related topic PerimeterOfEllipse Related topic Evolute2 Related topic Cycloid