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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{feb9a438-26b0-41d3-b044-6acd6efccde0-3_392_547_511_246}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\end{center}
\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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{feb9a438-26b0-41d3-b044-6acd6efccde0-3_394_556_1491_244}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{center}
\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.

\hfill \mbox{\textit{OCR MEI Further Mechanics B AS 2022 Q3 [10]}}