2. A certain function must be integrated on $[2,3].$ One can check that $f(x)\ge g(x)$ on that interval where $f$ and $g$ are respectively the functions.
So the integral which gives the expected area is $\int_{2}^{3}{\left( 5x-4-{{x}^{2}}-2 \right)\,dx}=\frac{1}{6},$ which is the same answer you got.