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 $t \in \left(a..b\right)$.

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

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle s\)	\(=\)	\(\displaystyle \int_a^b \sqrt{1 + \left({\frac {\mathrm dy}{\mathrm dx} }\right)^2}\ \mathrm d x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of arc length
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \int_a^b \sqrt{\left(\frac {\frac {\mathrm dx}{\mathrm dt} }{\frac {\mathrm dx}{\mathrm dt} }\right)^2 + \left(\frac {\frac {\mathrm dy}{\mathrm dt} }{\frac {\mathrm dx}{\mathrm dt} }\right)^2}\ \mathrm d x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	because $\left(\dfrac {\frac {\mathrm dx}{\mathrm dt} }{\frac {\mathrm dx}{\mathrm dt} }\right)^2 = 1$, and from corollary to chain rule
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \int_a^b \sqrt{\left({\frac {\mathrm dx}{\mathrm dt} }\right)^2 + \left({\frac {\mathrm dy}{\mathrm dt} }\right)^2} \frac 1 {\frac {\mathrm dx}{\mathrm dt} }\ \mathrm d x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	factor $\dfrac {\mathrm dx}{\mathrm dt}$ out of the radicand. No absolute value is needed as length cannot be negative.
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \int_a^b \sqrt{\left({\frac {\mathrm dx}{\mathrm dt} }\right)^2 + \left({\frac {\mathrm dy}{\mathrm dt} }\right)^2} \frac {\mathrm dt}{\mathrm dx} \ \mathrm d x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Derivative of an Inverse Function
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \int_a^b \sqrt{\left({\frac {\mathrm dx}{\mathrm dt} }\right)^2 + \left({\frac {\mathrm dy}{\mathrm dt} }\right)^2} \ \mathrm d t\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Integration by Substitution

$\blacksquare$

Also see

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

Sources

• Roland E. Larson,  Robert P. Hostetler and Bruce H. Edwards: Calculus: 8th Edition (2005) $\S 10.3$