2013 RVHS Promo Q3

The diagram shows the sketch of the curve \(C\), \({{\left( y-1 \right)}^{2}}=x\sqrt{{{x}^{2}}-1}\), with the vertex at \(\left( 1,1 \right)\).

1. Write down the equation of the graph when \(C\) is translated \(1\) unit in the negative \(y\)-direction.[1]
2. The shaded region \(R\), bounded by \(C\) and the vertical line, \(x=a\), is rotated through \(\pi \) radians about the line \(y=1\). By using the substitution \(u=\sqrt{{{x}^{2}}-1}\), or otherwise, find the exact volume obtained in terms of \(a\).[5]