#### To determine

**To find:**

The relation between a and b

#### Answer

b=2a

#### Explanation

**1) Concept:**

Using y=x2 and a general cubic polynomial, form the diagram, and then set up an integral for area, and find the relation.

**2) Calculation:**

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

y=px3+qx2+rx

At intersection of parabola an polynomial,

px3+qx2+rx=x2

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

fx=x x-ax-b

This is the same as

fx=xx2-a+bx+ab=[x3-a+bx2+abx]

The two areas are equal.

For the second region, there is a change in position of the upper and lower curves. So,

∫0afxdx=∫ab-fxdx

∫0afxdx=-∫abfxdx

Fa-F0=-[Fb-Fa]

Fa-F0= -Fb+F(a)

-F0= -Fb

F0=F(b)

Calculate the antiderivative of f(x).

∫fxdx=Fx= x44-b+ax33+abx22

So, F0=F(b) gives

0=b44-b+ab33+ab32

3b4-4b4-4ab3+6ab312=0

-b4+2ab3=0

b3-b+2a=0

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.

**Conclusion:**

b=2a