6 In this question you must show detailed reasoning.
As shown in Fig. 6.1, the region R is bounded by the lines \(x = 1 , x = 2 , y = 0\) and the curve \(y = 2 x ^ { 2 }\) for \(1 \leq x \leq 2\). A uniform solid of revolution, S , is formed when R is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_725_449_539_751}
\captionsetup{labelformat=empty}
\caption{Fig. 6.1}
\end{figure}
- Show that the volume of S is \(\frac { 124 \pi } { 5 }\).
- Show that the distance of the centre of mass of S from the centre of its smaller circular plane surface is \(\frac { 43 } { 62 }\).
Fig. 6.2 shows S placed so that its smaller circular plane surface is in contact with a slope inclined at \(\alpha ^ { \circ }\) to the horizontal. S does not slip but is on the point of tipping.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_458_565_2014_694}
\captionsetup{labelformat=empty}
\caption{Fig. 6.2}
\end{figure} - Find the value of \(\alpha\), giving your answer in degrees correct to 3 significant figures.