(a) The curve \(C_{1}\) has equation \(y=-\ln (\cos x)\). Show that the length of the arc of \(C_{1}\) from the point where \(x=0\) to the point where \(x=\frac{1}{3} \pi\) is \(\ln (2+\sqrt{3})\).
(b) The curve \(C_{2}\) has equation \(y=2 \sqrt{ }(x+3)\). The arc of \(C_{2}\) joining the point where \(x=0\) to the point where \(x=1\) is rotated through one complete revolution about the \(x\)-axis. Show that the area of the surface generated is
\(\frac{8}{3} \pi(5 \sqrt{ } 5-8) .\)