2019 ASRJC Promo Q8

The curve \(C\) is defined by the equations

\(x=2\mathrm{cosec}t\), \(y=5\cos t{{\sin }^{3}}t\), for \(0 < t\le \frac{\pi }{2}\).

The region \(R\) is bounded by \(C\), the \(x\)-axis and \(x=4\) in the first quadrant. Find the exact area of the region \(R\).[6]

The diagram below shows the curves \({{C}_{1}}\) and \({{C}_{2}}\) with equations \(y=2{{\left( x-1 \right)}^{2}}+3\) and \(y=3{{x}^{2}}\) respectively. The region in the first quadrant enclosed by the curves and the \(y\)-axis is denoted by \(S\). Find the volume of the solid generated when the region \(S\) is rotated through \(2\pi \) radians about the \(y\)-axis, giving your answer correct to \(4\) decimal places.[3]