6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ddee434-f7e1-4f56-91fc-f487112dbf6b-11_709_1269_292_349}
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\caption{Figure 1}
\end{figure}
Figure 1 shows the curve \(C\) with parametric equations
$$x = 2 \cos \theta - \cos 2 \theta , y = 2 \sin \theta - \sin 2 \theta , \quad 0 \leqslant \theta \leqslant \pi$$
- Show that
$$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = 8 ( 1 - \cos \theta )$$
The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Find the area of the surface generated, giving your answer in the form \(k \pi\), where \(k\) is a rational number.