Expert: Abe Mantell - 12/26/2007

Question
Let R be the region bounded by the parabola y=x-x^2 and the x-axis. Find the equation of the line y=mx that divided this region into two regions of equal area.

Answer
The parabola intersects the x-axis at x=0 and x=1.
The area is integral(x-x^2,x=0..1)=1/6
Now we seek a line of the form y=mx so that
integral(x-x^2-mx,x=0..a)=1/12, where x=a is where the
parabola and line intersect again after x=0...which is
where x-x^2=mx ==> x=1-m.  Thus, we need to solve for 'm'
so that integral(x-x^2-mx,x=0..1-m)=1/12...which yields
(1/6)(1-m)^3=1/12 ==> m=1-2^(-1/3)