8.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{9bf05d7e-7bb9-40f6-b626-69a8a6eda5a5-10_545_979_285_552}
\end{figure}
The curve shown in Figure 2 has parametric equations
$$x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leqslant t \leqslant 2 \pi$$
- Show that the curve crosses the \(x\)-axis where \(t = \frac { \pi } { 3 }\) and \(t = \frac { 5 \pi } { 3 }\).
The finite region \(R\) is enclosed by the curve and the \(x\)-axis, as shown shaded in Figure 2.
- Show that the area of \(R\) is given by the integral
$$\int _ { \frac { \pi } { 3 } } ^ { \frac { 5 \pi } { 3 } } ( 1 - 2 \cos t ) ^ { 2 } \mathrm {~d} t$$
- Use this integral to find the exact value of the shaded area.