A simple integral

$\begin{aligned} I & = \int_0^{2\pi} \sqrt{a(1-\color{#3D99F6}\cos t)} \ dt \\ & = \int_0^{2\pi} \sqrt{a\left(1-\color{#3D99F6}\left(1-2\sin^2 \frac t2\right)\right)} \ dt \\ & = \int_0^{2\pi} \sqrt{2a} \sin \frac t2 \ dt \\ & = -2 \sqrt{2a} \cos \frac t2 \bigg|_0^{2\pi} \\ & = \boxed{4\sqrt{2a}} \end{aligned}$

$\int_{0}^{2\pi} \sqrt{a(1 - cost)} dt = 2 \sqrt{2a} \int_{0}^{2\pi} sin(\dfrac{t}{2}) * \dfrac{1}{2} dt =$

$-2 \sqrt{2a} * cos(\dfrac{t}{2})|_{0}^{2\pi} = \boxed{4 \sqrt{2a}}$