The curve \(C\) has equation \(y=x^{\frac{3}{2}}\). Find the coordinates of the centroid of the region bounded by \(C\), the lines \(x=1, x=4\) and the \(x\)-axis.

Show that the length of the arc of \(C\) from the point where \(x=5\) to the point where \(x=28\) is 139 .