Arc Length for Parametric Functions Examples

The parametric equations

x(t) = cos 2t, y(t) = sin 2t

trace the unit circle. Use the arc length formula to find the circumference of the unit circle.

The circumference of the unit circle is 2π, so we know after evaluating the integral we should get 2π. This problem didn't tell us values for t, which means we have to figure them out ourselves. These equations trace the unit circle once if we take 0 ≤ t ≤ π.

We have

So the length of the curve is

Since sin²(2t) + cos²(2t) = 1, we can evaluate this integral by hand.

Boom. This is the right answer.

In the example, if we had picked 0 ≤ t ≤ 2π, we wouldn't have gotten the right answer. It's important to pick the correct values for t so that the part of the curve we're interested in gets traced exactly once.