Parametric Equations Q1

The figure above shows a curve known as a re-entrant cycloid, with parametric equations

\(x=\theta -4\sin \theta \), \(y=1-2\cos \theta \), \(0\le \theta \le 2\pi \).

The curve crosses the \(x\)-axis at the points \(P\) and \(Q\).

1. Find the value of \(\theta \) at the points \(P\) and \(Q\).
2. Show that the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in the figure above, is given by the integral
   \(\displaystyle\displaystyle \displaystyle\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{1-6\cos \theta +8{{\cos }^{2}}\theta \,\mathrm{d}\theta }\),
   where \({{\theta }_{1}}\) and \({{\theta }_{2}}\) must be stated.
3. Find an exact value for the above integral.