7 Fig. 7 shows the curve with parametric equations

$$x = \cos \theta , y = \sin \theta - \frac { 1 } { 8 } \sin 2 \theta , 0 \leqslant \theta < 2 \pi$$

The curve crosses the $x$-axis at points $\mathrm { A } ( 1,0 )$ and $\mathrm { B } ( - 1,0 )$, and the positive $y$-axis at C . D is the maximum point of the curve, and E is the minimum point.

The solid of revolution formed when this curve is rotated through $360 ^ { \circ }$ about the $x$-axis is used to model the shape of an egg.

\begin{figure}[h]
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\caption{Fig. 7}
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(i) Show that, at the point $\mathrm { A } , \theta = 0$. Write down the value of $\theta$ at the point B , and find the coordinates of C .\\
(ii) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$.

Hence show that, at the point D,

$$2 \cos ^ { 2 } \theta - 4 \cos \theta - 1 = 0 .$$

(iii) Solve this equation, and hence find the $y$-coordinate of D , giving your answer correct to 2 decimal places.

The cartesian equation of the curve (for $0 \leqslant \theta \leqslant \pi$ ) is

$$y = \frac { 1 } { 4 } ( 4 - x ) \sqrt { 1 - x ^ { 2 } } .$$

(iv) Show that the volume of the solid of revolution of this curve about the $x$-axis is given by

$$\frac { 1 } { 16 } \pi \int _ { - 1 } ^ { 1 } \left( 16 - 8 x - 15 x ^ { 2 } + 8 x ^ { 3 } - x ^ { 4 } \right) \mathrm { d } x .$$

Evaluate this integral.

\hfill \mbox{\textit{OCR MEI C4 2007 Q7 [20]}}