Area enclosed by two curves

Calculus Level 5

The parabola \(f(x)=x^2\) is tangent to the graph of \( g(x) = x^4 + ax^3 + x^2+ bx +1\) at two distinct points.

Given that the area enclosed by these two curves is \(\frac{p\sqrt{6}}{q}\), where \(p\) and \(q\) are coprime positive integers, find the value of \(p+q\).

Remark: The image above shows for the case \(a<0\). The area is the same regardless the parity of \(a\).

Bonus: If the parabola \(f(x)=x^2\) is tangent to the graph of \( g(x) = x^4 + ax^3 + \color{red}cx^2+ bx +1\) at two distinct points, what is the area enclosed by \(f\) and \(g\)?


This problem is part of Curves... cut or touch?

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