Determine the Dot's Distance

We have $\frac{dx}{dt} = 1 - \cos t$ and $\frac{dy}{dt} = \sin t$ . Let $L$ be the distance, using the arc length formula, we have

$\begin{aligned} L &= \int_0^{2\pi} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \\ &= \int_0^{2\pi}\sqrt{(1-\cos t)^2 + \sin^2t} \, dt \\ &= \int_0^{2\pi} \sqrt{2(1-\cos t)} \, dt \\ &= \int_0^{2\pi} \sqrt{4 \sin^2\left(\frac{t}{2}\right)} \, dt \\ &= \int_0^{2\pi} 2\sin\left(\frac{t}{2}\right) \, dt \\ &= 2\left[-2\cos\left(\frac{t}{2}\right)\right]_0^{2\pi} \\ &= 8 \\ \end{aligned}$