2015 VJC Promo Q10

Sketch the graph of \({{\left( x-93 \right)}^{2}}+{{y}^{2}}={{15}^{2}}\), showing the coordinates of the points with the maximum and minimum \(y\)-values. [2]

The shape of a circular tube can be modelled by rotating the region bounded by \({{\left( x-93 \right)}^{2}}+{{y}^{2}}={{15}^{2}}\) completely about the \(y\)-axis.

Show the volume of the tube is given by \(372\pi \displaystyle\displaystyle \displaystyle\int_{-15}^{15}{\sqrt{{{15}^{2}}-{{y}^{2}}}}\ \mathrm{d}y\).[3]

By using the substitution \(y=15\sin \theta \), find the exact volume of the tube.[5]