1.20: Past-paper question 9

(a) Using a formula given on page 2, show that

(b) Hence show that

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers to be found.

Figure 3 shows part of the curve $\displaystyle C_1$ with equation $\displaystyle y = 2 \mathrm{cos}^2 \theta - 1$ and part of the curve $\displaystyle C_2$ with equation $\displaystyle y = -\mathrm{cos} \theta$

Point $\displaystyle B$ is the intersection of $\displaystyle C_1$ and $\displaystyle C_2$ as shown in Figure 3

Point $\displaystyle A \left( \frac{3\pi}{4}, 0 \right)$ is the intersection of $\displaystyle C_1$ with the $\displaystyle \theta$-axis as shown in Figure 3

Point $\displaystyle E \left( \frac{\pi}{2}, 0 \right)$ is the intersection of $\displaystyle C_2$ with the $\displaystyle \theta$-axis as shown in Figure 3

The finite region $\displaystyle R$, shown shaded in Figure 3, is bounded by the $\displaystyle \theta$-axis, $\displaystyle C_1$ and $\displaystyle C_2$

(c) Use calculus to find, in its simplest form, the exact area of $\displaystyle R$