8. The curve \(C\) which passes through \(O\) has polar equation
$$r = 4 a ( 1 + \cos \theta ) , \quad - \pi < \theta \leq \pi .$$
The line \(l\) has polar equation
$$r = 3 a \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } .$$
The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in the diagram.
- Prove that \(P Q = 6 \sqrt { } 3 a\).
The region \(R\), shown shaded in the diagram, is bounded by \(l\) and \(C\).
- Use calculus to find the exact area of \(R\).
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