Parametric arc length

$L=\int_a^b\sqrt{[x'(t)]^2+[y'(t)]^2}\,dt$ — the total distance traveled along a parametric curve.

The arc length of a parametric curve on $t\in[a,b]$ is $L=\int_a^b\sqrt{[x'(t)]^2+[y'(t)]^2}\,dt$ — the integral of SPEED (the magnitude of the velocity vector $\langle x'(t),y'(t)\rangle$ ) over time, generalizing the $y=f(x)$ arc-length formula (where $x'(t)=1$ reduces it back to the familiar $\sqrt{1+[f'(x)]^2}$ form).

Examples

For $x=\cos t$ , $y=\sin t$ on $[0,2\pi]$ : $\sqrt{\sin^2t+\cos^2t}=1$ , so $L=\int_0^{2\pi}1\,dt=2\pi$ , the unit circle's circumference.