2008 AJC Promo Q7

The diagram below shows (not to scale) the region \(R\) which is bounded by the curve \(y=\frac{3}{1+4{{x}^{2}}}\), the line \(y=5x-1\) and the \(y\)-axis. Find the exact area of \(R\).[4]

Another region \(S\) is bounded by the curve \(y=\frac{3}{1+4{{x}^{2}}}\), the lines \(y=3x\), \(x=1\) and the \(x\)-axis. Find the volume generated when \(S\) is rotated through \(2\pi \) about the \(y\)-axis, giving your answer to \(3\) significant figures.[5]