Arc length

The arc length is defined as the distance between to points along a section of a curve. A curve in a plane can be approximated by dividing this continuous curve into line segments, connecting the points \(t_i\) on a curve, creating a polygonal path. The total length of the curve can then be approximated by summing the lengths of each linear segment, in the limit of infinitesimal small segments this becomes \[\begin{equation} L(f) \approx \lim_{n \to \infty} \sum_i^N | f(t_i) - f(t_{i-1}) | \thinspace , \end{equation}\] where \(t_i = a + i \Delta t\) for \(i=0, \dots, N\). Dividing and multiplying by \(\Delta t\) results in \[\begin{equation} \lim_{n \to \infty} \sum_i^N \frac{ | f(t_i) - f(t_{i-1}) | }{\Delta t} \Delta t \thinspace . \end{equation}\] The Mean Value Theorem states that for a function continuous on closed interval \([a, b]\) and differentiable on open interval \(]a,b[\), there is a number \(c\) such that for \(a<c<b\) \[\begin{equation} f'(c) = \frac{ f(b) - f(a) }{b-a} \end{equation}\] such that \[\begin{equation} \lim_{n \to \infty} \sum_i^N \frac{ | f(t_i) - f(t_{i-1}) | }{\Delta t} \Delta t = \int_a^b | f'(t) | dt = L(f) \thinspace . \end{equation}\]