Arc
Length of a Curve
The length of a curve or line.
The length of an arc can be found
by one of the formulas below
for any differentiable curve defined by rectangular, polar,
or parametric equations.
For the length of a circular arc, see arc
of a circle.
Formula:



where a and b represent x, y, t, or θvalues as appropriate, and ds can be found as follows.
1. In rectangular form, use whichever of the following is easier:
2. In parametric form, use
3. In polar form, use

Example 1: Rectangular 
Find the length of an arc of the curve y = (1/6) x^{3} +
(1/2) x^{–1} from


x = 1 to x = 2.

Example 2: Parametric 
Find the length of the arc in one period of the cycloid x = t – sin t, y = 1 – cos t. The values of t run from 0 to 2π.

Example 3: Polar 
Find the length of the first rotation of the logarithmic spiral r = e^{θ}. The values of θ run from 0 to 2π.

See
also
Surface area of a surface
of revolution
