Problem

Figure 1

Figure 1 shows a sketch of the curve with polar equation

$$\begin{align*} r ={}& 10\cos \theta + \tan \theta \qquad 0 \leqslant \theta < \frac{\pi}{2} \end{align*}$$

The point $P$ lies on the curve where $\theta = \frac{\pi}{3}$.

The region $R$ , shown shaded in Figure 1, is bounded by the initial line, the curve and the line $OP$ , where $O$ is the pole.

Use algebraic integration to show that the exact area of $R$ is

$$\begin{align*} \frac{1}{12}(a\pi + b\sqrt{3} + c) \end{align*}$$

where $a$ , $b$ and $c$ are integers to be determined.