N2010 P1 Q6

The diagram shows the curve with equation \(y=x^3-3x+1\) and the line with equation \(y=1\). The curve crosses the \(x\)-axis at \(x=\alpha\), \(x=\beta\) and \(x=\gamma\) and has turning points at \(x=-1\) and \(x=1\).

1. Find the values of \(\beta\) and \(\gamma\), giving your answers correct to 3 decimal places.[2]
2. Find the area of the region bounded by the curve and the \(x\)-axis between \(x=\beta\) and \(x=\gamma\).[2]
3. Use a non-calculator method to find the area of the shaded region between the curve and the line.[4]
4. Find the set of values of \(k\) for which the equation \(x^3-3x+1=k\) has three real distinct roots.[2]