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