What is the ratio of blue area to the grey area in the following image?

Question

Two parabolas with vertical axes of symmetry share each other's vertex, and are contained between horizontal lines that go through their vertices.

What is the ratio of blue area (intersection) to the grey area in the image?

Answer 1

The equation for a parabola is y= x (exp2)
Let us calculate the blue area, as follows
For x=a, the area under the curve and axis x is the integral of xExp 2 between 0 and a, which yields 1/3 a (exp 3)
Let us calculate the grey area,
Due to the symmetry of the two parabolas
We can calculate the area of a rectangle base ( -a to 2a) height a (exp2) which yields 3 a (exp 3) and substract three times the blue area , therefore the grey area is 2 a (exp3)
Then the ratio is 1/3 a (exp3) / 2 a (exp3) = 1/6