N2008 P2 Q2

The diagram shows the curve \(C\) with equation \(y^2=x\sqrt{1-x}\). The region enclosed by \(C\) is denoted by \(R\).

1. Write down an integral that gives the area of \(R\), and evaluate this integral numerically.[3]
2. The part of \(R\) above the \(x\)-axis is rotated through \(2\pi\) radians about the \(x\)-axis. By using the substitution \(u=1-x\), or otherwise, find the exact value of the volume obtained.[3]
3. Find the exact \(x\)-coordinate of the maximum point of \(C\).[3]