7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b36bdc3-a68d-4982-bf23-f780773df5cc-14_259_327_214_868}
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\caption{Figure 2}
\end{figure}
Figure 2 shows the image of a gold pendant which has height 2 cm . The pendant is modelled by a solid of revolution of a curve \(C\) about the \(y\)-axis. The curve \(C\) has parametric equations
$$x = \cos \theta + \frac { 1 } { 2 } \sin 2 \theta , \quad y = - ( 1 + \sin \theta ) \quad 0 \leqslant \theta \leqslant 2 \pi$$
- Show that a Cartesian equation of the curve \(C\) is
$$x ^ { 2 } = - \left( y ^ { 4 } + 2 y ^ { 3 } \right)$$
- Hence, using the model, find, in \(\mathrm { cm } ^ { 3 }\), the volume of the pendant.