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 continuously differentiable for all $t \in \left({a \,.\,.\, b}\right)$.

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

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


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

for $\dfrac {\mathrm d x}{\mathrm d t} \ne 0$

Also see

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