1.24: Past-paper question 7

Figure 2 shows part of the curve $\displaystyle S$ with equation $\displaystyle y = \left( \mathrm{cos} 3\theta + \sqrt{3} \mathrm{sin} 3\theta \right)^{\frac{1}{2}}$

where $\displaystyle m \leq \theta \leq n$

The curve $\displaystyle S$ meets the $\displaystyle x$-axis at the point with coordinates $\displaystyle (m, 0)$ and at the point with coordinates $\displaystyle (n, 0)$

(a) Find the exact value of $\displaystyle m$ and the exact value of $\displaystyle n$

The finite region $\displaystyle R$, shown shaded in Figure 2, is bounded by the curve $\displaystyle S$, and the $\displaystyle x$-axis in the region $\displaystyle m \leq \theta \leq n$

The region $\displaystyle R$ is rotated through $\displaystyle 2\pi$ radians about the $\displaystyle x$-axis.

(b) Use calculus to find the exact volume of the solid generated.