Arc length and speed on a parametric curve

Understand the idea

For a parametric plane curve, both coordinates contribute to the distance traveled during a short parameter interval. The magnitude of the coordinate-derivative vector gives the length accumulated per unit parameter. Integrating that nonnegative quantity gives the length traced over the interval.

Different parametrizations can trace the same geometric arc at different rates. A correct length integral accounts for the rate through both derivatives. If the parameter retraces part of a curve, the integral counts that travel again; it need not equal the length of the set of distinct points.

When simplifying a square root of a square, keep absolute values unless the parameter interval fixes the sign. A result should be at least the straight-line distance between the endpoints. If an elementary antiderivative is unavailable, retain a correct setup or use permitted numerical evaluation.

Worked example

Find the length traced by $\displaystyle x=2\cos t$, $\displaystyle y=2\sin t$ for $\displaystyle 0\le t\le\pi/2$.

1. The coordinate derivatives are $\displaystyle -2\sin t$ and $\displaystyle 2\cos t$.
2. The length integrand is $\displaystyle \sqrt{4\sin^2t+4\cos^2t}=2$.
3. Integrating gives $\displaystyle \int_0^{\pi/2}2\,dt=\pi$.