Arc Length for Parametric Equations

Theorem
Let $x=f\left({t}\right)$ and $y=g\left({t}\right)$ be real functions of a parameter $t$.

Let these equations describe a curve $\mathcal C$ that is continuous for all $t \in \left[a..b\right]$ and differentiable for all $t \in \left(a..b\right)$.

Suppose that the graph of the curve does not intersect itself for any $t \in \left(a..b\right)$.

The arc length of $\mathcal C$ between $a$ and $b$ is given by:


 * $s= \displaystyle \int_a^b \sqrt{\left({\frac {\mathrm dx}{\mathrm dt}}\right)^2 + \left({\frac {\mathrm dy}{\mathrm dt}}\right)^2}\ \mathrm d t$

for $\dfrac {\mathrm dx}{\mathrm dt} \ne 0$.

Also see

 * Arc Length
 * Continuously Differentiable Curve Has Finite Arc Length
 * Length of Arc of Cycloid