(i)[6]
Show that the exact value of $\int_0^{\frac{1}{3}\pi} \left(\cos^2 x + \frac{1}{\cos^2 x}\right) dx$ is $\frac{1}{6}\pi + \frac{9}{8}\sqrt{3}$.
(ii)[4]
The diagram shows the curve $y = \cos x + \frac{1}{\cos x}$ for $0 \leq x \leq \frac{1}{3}\pi$. The shaded region is enclosed by the curve and the lines $x = 0$, $x = \frac{1}{3}\pi$ and $y = 0$. Determine the exact volume of the solid formed when the shaded region is fully rotated about the $x$-axis.