Figure 3 shows a sketch of part of the curve with equation $y = 1 - 2 ext{cos} \, x$, where $x$ is measured in radians - Edexcel - A-Level Maths Pure - Question 18 - 2013 - Paper 1

To find the volume of the solid generated by rotating region $S$ about the $x$-axis, we use the formula for volume:

$V = \pi \int_{a}^{b} [f(x)]^2 \, dx$

Here, $f(x) = 1 - 2\text{cos} \, x$ and the limits of integration are from $A$ to $B$, which are rac{\pi}{3} and rac{5\pi}{3} respectively:

$V = \pi \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \left(1 - 2\text{cos} \, x\right)^2 \, dx$

Now, calculating the integral:

1. Expand the square: $\left(1 - 2\text{cos} \, x\right)^2 = 1 - 4\text{cos} \, x + 4\text{cos}^2 \, x$

2. Therefore, we get: $V = \pi \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \left(1 - 4\text{cos} \, x + 4\frac{1 + \text{cos}(2x)}{2}\right) \, dx$

3. Simplifying this results in: $V = \pi \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \left(3 - 2\text{cos} \, x + 2\text{cos}(2x)\right) \, dx$

4. Evaluating the integral produces the final volume value.