Curve length

Consider the curve given by the graph of a function f on an interval [a,b].

Its length is equal to

Example: Find the length of the curve given by f (x) = cosh(x) for x from the interval [0,1].
Note: This curve is called catenary.

Solution: Picture:

The length is

If the integral is too difficult, sometimes it helps to switch the axes (see Area).

Length of a parametric curve

Consider a parametric curve x = x(t), y = y(t) for t from [α,β].

The length of this curve is equal to

Consider a curve in polar coordinates given by ϱ = ϱ(φ) for φ from [α,β].

The length of this curve is