Answer
The $a>0$ that gives the area of $2/3$ is
$$a=\frac{1}{2}.$$
Work Step by Step
The area bounded by these functions is given by the integral of the absolute value of their difference between the points of intersection:
$$A=\int_0^{1/a}|f(x)-g(x)|dx.$$
When $x<1/a$ i.e. when $a>\frac{1}{x}$ then $g(x)=ax^2$
