10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78543314-72b7-4366-98a1-dbb6b852632f-32_385_679_280_694}
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\caption{Figure 2}
\end{figure}
A solid playing piece for a board game is modelled by rotating the curve \(C\), shown in Figure 2, through \(2 \pi\) radians about the \(x\)-axis.
The curve \(C\) has equation
$$y = \sqrt { 1 + \frac { x ^ { 2 } } { 9 } } \quad - 4 \leqslant x \leqslant 4$$
with units as centimetres.
- Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the playing piece is given by
$$S = p \pi \int _ { - 4 } ^ { 4 } \sqrt { 81 + 10 x ^ { 2 } } \mathrm {~d} x + q \pi$$
where \(p\) and \(q\) are constants to be determined.
Using the substitution \(x = \frac { 9 } { \sqrt { 10 } } \sinh u\), or another algebraic integration method, and showing all your working,
- determine the total surface area of the playing piece, giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\)