2018 EJC P1 Q3

The parametric equations of curve \(C\), is given as \(x=at\), \(y=a{{t}^{3}}\), where \(a\) is a positive constant.

1. The point \(P\) on the curve has parameter \(p\) and the tangent to the curve at point \(P\) cuts the \(y\)-axis at \(S\) and the \(x\)-axis at \(T\). The point \(M\) is the midpoint of \(ST\). Find a Cartesian equation of the curve traced by \(M\) as \(p\) varies.[5]
2. Find the exact area bounded by curve \(C\), the line \(x=0\), \(x=3\) and \(x\)-axis, giving your answer in terms of \(a\).[3]