2016 NJC Promo Q7

The diagram below shows a shaded region which is bounded by the curve \(C\) with equation \(y=\sec x{{\tan }^{3}}x\) and the line \(L\) with equation \(y=\frac{4\sqrt{2}}{\pi }x\).

1. Verify that the point \(\left( \frac{\pi }{4},\sqrt{2} \right)\) lies on both \(C\) and \(L\).[1]
2. Find the exact volume of the solid generated by revolving the shaded region about the \(x\)-axis through 2\(\pi \). Express your answer in the form \(\frac{1}{42}\left( a{{\pi }^{2}}-b\pi \right)\), where \(a\) and \(b\) are integers.[6]