# 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?

×