6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_442_727_237_603}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,
- show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
- show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_478_472_1407_762}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
The mass per unit volume of \(S\) is twice the mass per unit volume of the cone. - Find the distance from \(V\) to the centre of mass of \(T\).
The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
- Find the size of the angle between \(V A\) and the vertical.