Let the area enclosed by the lines $x+y=2, \mathrm{y}=0, x=0$ and the curve $f(x)=\min \left\{x^{2}+\frac{3}{4}, 1+[x]\right\}$ where $[x]$ denotes the greatest integer $\leq x$, be $\mathrm{A}$. Then the value of $12 \mathrm{~A}$ is _____________.