Area Under The Curves

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 _____________.