Problem

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Figure 1

Figure 1 shows a sketch of the circles $C_1$ and $C_2$

The circle $C_1$ has polar equation

$$\begin{align*} r=\sqrt3\sin\theta \qquad 0\leqslant\theta\leqslant\pi \end{align*}$$

The circle $C_2$ has polar equation

$$\begin{align*} r=3\cos\theta \qquad -\frac{\pi}{2}\leqslant\theta\leqslant\frac{\pi}{2} \end{align*}$$

Circles $C_1$ and $C_2$ intersect at the origin $O$ and at the point $P$.

(a) Determine the polar coordinates of $P$.

The finite region $R$ is bounded by $C_1$ and $C_2$ and is shown shaded in Figure 1.

(b) Use algebraic integration to determine the exact area of $R$, giving your answer in the form $a\pi+b\sqrt3$ where $a$ and $b$ are simplified rational numbers.