Finding Arc Lengths By Integrating
See also: Differential geometry of curvesConsider a real function f(x) such that f(x) and (its derivative with respect to x) are continuous on . The length s of the part of the graph of f between x = a and x = b can be found as follows:
Consider an infinitesimal part of the curve ds (or consider this as a limit in which the change in s approaches ds). According to Pythagoras' theorem, from which:
If a curve is defined parametrically by x = X(t) and y = Y(t), then its arc length between t = a and t = b is
This is more clearly a consequence of the distance formula where instead of a and, we take the limit. A useful mnemonic is
If a function is defined as a function of x by then it is simply a special case of a parametric equation where and, and the arc length is given by:
If a function is defined in polar coordinates by then the arc length is given by
In most cases, including even simple curves, there are no closed-form solutions of arc length and numerical integration is necessary.
Curves with closed-form solution for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and (mathematically, a curve) straight line. The lack of closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals.
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