Arc Length - Finding Arc Lengths By Integrating

Finding Arc Lengths By Integrating

See also: Differential geometry of curves

Consider 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.