3 Fig. 3.1 shows the curve with equation \(y = x ^ { 2 } + 3\). The region \(R\), shown shaded, is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\). A uniform solid of revolution S is formed by rotating the region R through \(2 \pi\) about the \(x\)-axis.
The volume of \(S\) is \(\frac { 202 } { 5 } \pi\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{feb9a438-26b0-41d3-b044-6acd6efccde0-3_392_547_511_246}
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\caption{Fig. 3.1}
\end{figure}
\section*{(a) In this question you must show detailed reasoning.}
Show that the \(x\)-coordinate of the centre of mass of S is \(\frac { 395 } { 303 }\).
S is fixed to a cylinder of base radius 3 units and height 2 units to form the uniform solid D . The smaller circular face of S is joined to the top circular face of the cylinder, as shown in Fig. 3.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{feb9a438-26b0-41d3-b044-6acd6efccde0-3_394_556_1491_244}
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\caption{Fig. 3.2}
\end{figure}
(b) Find the distance of the centre of mass of D from its smaller circular face.
D is placed with its smaller circular face in contact with a rough plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is given that D does not slip.
(c) Determine whether D topples.