The relation between a and b
Using y=x2 and a general cubic polynomial, form the diagram, and then set up an integral for area, and find the relation.
The general form of cubic polynomial can be written as px3+qx2+rx+s
where p, q, r, and s are reals.
The cubic polynomial passes through the origin. Therefore, the equation becomes
At intersection of parabola an polynomial,
p x3+q-1x2+rx=0 Let the curve fx = px3+qx2+rx-x2
That is, fx=px3+q-1x2+rx
Since a and b are points of intersection for both curves,
f(0)=fa=fb=0 That is a,b and 0 are the roots of cubic polynomial fx.
So, the above function can be written as
This is the same as
The two areas are equal.
For the second region, there is a change in position of the upper and lower curves. So,
Calculate the antiderivative of f(x).
So, F0=F(b) gives
So either b=0 and / or b=2a
But b=0 is not possible, so, b=2a
Hence, the relation between a and b is that b is twice of a.