(Solved): EXAMPLE 3 Find the area under one arch of the cycloid \[ x=r(\theta-\sin (\theta)) \quad y=r(1-\co ...

EXAMPLE 3 Find the area under one arch of the cycloid \[ x=r(\theta-\sin (\theta)) \quad y=r(1-\cos (\theta)) \] Video Example \( (())) \) \[ A=\int_{0}^{2 \pi r} y d x=\int_{0}^{2 \pi}(\quad) d \theta \] \[ =r^{2} \int_{0}^{2 \pi}(1-\cos (\theta))^{2} d \theta \] \[ \begin{array}{l} =r^{2} \int_{0}^{2 \pi}\left(1-2 \cos (\theta)+\cos ^{2}(\theta)\right) d \theta \\ =r^{2} \int_{0}^{2 \pi}\left[1-2 \cos (\theta)+\frac{1}{2}(1+\cos (2 \theta))\right] d \theta \\ \left.\left.=r^{2}\right]_{0}^{2 \pi}\right)= \\ =r^{2}(\quad \end{array} \]