2013 AJC Promo Q8 (b)

The diagram shows a shaded region \(R\) bounded by the curve \({{\left( y-2 \right)}^{2}}=x+1\) and the line \(y+2x=6\).

Find the volume generated when \(R\) is rotated through \(2\pi \) radians about the \(x\)-axis, leaving your answer correct to \(3\) significant figures.[4]