5. (a) Given that \(y = \arctan 3 x\), and assuming the derivative of \(\tan x\), prove that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 1 + 9 x ^ { 2 } }$$
(b) Show that
$$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 3 } } 6 x \arctan 3 x \mathrm {~d} x = \frac { 1 } { 9 } ( 4 \pi - 3 \sqrt { } 3 )$$
(6)
[0pt]
[P5 June 2002 Qn 6]
\section*{6.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{6706ed7f-4575-4898-b757-aee8475b2a30-04_815_431_351_913}
\end{figure}
The curve \(C\) shown in Fig. 1 has equation \(y ^ { 2 } = 4 x , 0 \leq x \leq 1\).
The part of the curve in the first quadrant is rotated through \(2 \pi\) radians about the \(x\)-axis.
(a) Show that the surface area of the solid generated is given by
$$4 \pi \int _ { 0 } ^ { 1 } \sqrt { ( 1 + x ) } d x$$
(b) Find the exact value of this surface area.
(c) Show also that the length of the curve \(C\), between the points \(( 1 , - 2 )\) and \(( 1,2 )\), is given by
$$2 \int _ { 0 } ^ { 1 } \sqrt { \left( \frac { x + 1 } { x } \right) } \mathrm { d } x$$
(d) Use the substitution \(x = \sinh ^ { 2 } \theta\) to show that the exact value of this length is
$$2 [ \sqrt { } 2 + \ln ( 1 + \sqrt { } 2 ) ]$$
[P5 June 2002 Qn 8]