length of a curve

length of a curve

Geometrical concept addressed by integral calculus. Methods for calculating exact lengths of line segments and arcs of circles have been known since ancient times. Analytic geometry allowed them to be stated as formulas involving coordinates (see coordinate systems) of points and measurements of angles. Calculus provided a way to find the length of a curve by breaking it into smaller and smaller line segments or arcs of circles. The exact value of a curve's length is found by combining such a process with the idea of a limit. The entire procedure is summarized by a formula involving the integral of the function describing the curve.

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length of a curve [′leŋkth əv ə ′kərv]

(mathematics)

A curve represented byx=x(t),y=y(t) fort₁≤t≤t₂, withx(t₁) =x₁,x(t₂) =x₂,y(t₁) =y₁,y(t₂) =y₂, has length from (x₁,y₁) to (x₂,y₂) given by the integral fromt₁tot₂of the function √((dx/dt)²+ (dy/dt)²).