Choosing a line that halves a parabolic region

For a positive constant \(k\), the line \(y=kx\) and the curve \(y=x^2\) enclose a region \(R_k\).

1. Show that the area of \(R_k\) is \(\dfrac{k^3}{6}\).[3]
2. Find \(k\) given that the area of \(R_k\) is \(36\) square units.[1]
3. For this value of \(k\), the line \(y=mx\), where \(0<m<k\), divides \(R_k\) into two regions of equal area. Find \(m\) exactly.[3]