Question 1083342
The function y = ax - ax^2 intersects the x-axis at the points 0 and 1, regardless of the value of a. 
The area enclosed by this curve and the x-axis is given by: {{{int((ax-ax^2), dx, 0,1))}}}.  Let us call this area A1.
The curve y = x^2 divides this area into two parts
The area of the second part is the sum of {{{int( x^2, dx, 0, c )}}} and {{{int((ax-ax^2), dx, c,1))}}},
where c is the intersection point of the two curves.  Let's call this area A2.
So we need to find the value of a for which A1 - A2 = A2, or A1 = 2A2.
The intersection point, c, of the two curves is ax - ax^2 = x^2 -> x = a/(a+1)
Performing all the integrations and simplifying, you should be left with the following equation:
a^2 - 2a - 1 = 0
This has solutions a = -0.4142 and a = 2.4142.  Since a>0, we take the positive solution, a = 2.4142
The exact value for a is a = 1 + sqrt(2) (Check for yourself)