2023 SAJC P1 Q5

The diagram below shows the region \(R\) bounded by the circle with equation \({{\left( x-4 \right)}^{2}}+{{y}^{2}}=9\), the line with equation \(y=-\sqrt{5}x+3\sqrt{5}\) and the \(x\)-axis. The line and the circle meets at the point \(A\) with coordinates \(\left( 2,\sqrt{5} \right)\).

Show that the volume \(V\), formed when \(R\) is rotated through \(2\pi \) radians about the \(y\)-axis is given by

\(\pi \displaystyle\displaystyle \displaystyle\int_{0}^{\sqrt{5}}{\left[ \frac{{{\left( y-3\sqrt{5} \right)}^{2}}}{5}-\left( 25-{{y}^{2}}-8\sqrt{9-{{y}^{2}}} \right) \right]}\mathrm{ d}y\).

Hence, evaluate \(V\), leaving your answer correct to \(3\) decimal places.[5]