# Difference between revisions of "Arclength"

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'''Arclength''' is the [[mathematical]] length of a [[curve]], i.e. the distance that would be registered if one traveled along the curve as if it were a road. | '''Arclength''' is the [[mathematical]] length of a [[curve]], i.e. the distance that would be registered if one traveled along the curve as if it were a road. | ||

− | The length can be approximated by selecting a finite number of points along the curve and connecting the points with straight lines. The sum of the line segments can give a lower estimate for the length of the arc. The more points used, the closer the approximation to the curve. Technically, the arclength is the | + | The length can be approximated by selecting a finite number of points along the curve and connecting the points with straight lines. The sum of the line segments can give a lower estimate for the length of the arc. The more points used, the closer the approximation to the curve. Technically, the arclength is the supremum of all such sums, represented as an [[integral]]. For a curve that can be parametrised as <math>\vec{r}(t)</math>, the arclength from point <math>A</math> to <math>B</math> (with <math>t</math> being equal to <math>\alpha</math> and <math>\beta</math> at those points) can be found as |

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+ | <math>S = \int^{B}_{A} \, \text{d}s = \int^{\beta}_{\alpha} \vec{r}'(t) \cdot \vec{r}'(t) \, \text{d}t</math> | ||

[[Category:Calculus]] | [[Category:Calculus]] |

## Latest revision as of 14:31, 14 December 2016

This article/section deals with mathematical concepts appropriate for late high school or early college. |

**Arclength** is the mathematical length of a curve, i.e. the distance that would be registered if one traveled along the curve as if it were a road.

The length can be approximated by selecting a finite number of points along the curve and connecting the points with straight lines. The sum of the line segments can give a lower estimate for the length of the arc. The more points used, the closer the approximation to the curve. Technically, the arclength is the supremum of all such sums, represented as an integral. For a curve that can be parametrised as , the arclength from point to (with being equal to and at those points) can be found as