Determine the value of a at which the area, S(a), of the region enclosed by the curves y=x^2(x-2) and y=ax(x-2), is at a minimum. Assume that 0<a<2.

First of all, I found the points of interdections, which are 0,a,2.

And then, I did S(a), and then S'(a). Is it correct?