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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 θ-values 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) x3 + (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 = eθ. The values of θ run from 0 to 2π.

 

See also

Surface area of a surface of revolution

 


  this page updated 28-jul-14
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes