Arc Length for Parametric Equations

Theorem
Let $x = \map f t$ and $y = \map g t$ be real functions of a parameter $t$.

Let these equations describe a curve $\mathcal C$ that is continuous for all $t \in \closedint a b$ and continuously differentiable for all $t \in \openint a b$.

Suppose that the graph of the curve does not intersect itself for any $t \in \openint a b$.

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


 * $s = \displaystyle \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \rd t$

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

Also see

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