Let the function, f(x)= {-3 a x^2-2, x<1; a^2+b x, x ≥ 1} be differentiable for all x ∈ R, where a >1, b ∈ R. If the area of the region enclosed by…

Let the function, $$f(x)= \begin{cases}-3 a x^2-2, & x<1 \\ a^2+b x, & x \geqslant 1\end{cases} $$ be differentiable for all $\displaystyle x \in \mathbf{R}$, where $\displaystyle \mathrm{a}>1, \mathrm{~b} \in \mathbf{R}$. If the area of the region enclosed by $\displaystyle y=f(x)$ and the line $\displaystyle y=-20$ is $\displaystyle \alpha+\beta \sqrt{3}, \alpha, \beta \in \mathrm{Z}$, then the value of $\displaystyle \alpha+\beta$ is $\displaystyle \_\_\_\_$.