Past exam of the mathematical methods II course of the University of Oxford / 2015 / 6 / Solution

The boundaries intersect where $x^2=2x+8$, giving $x=-2$ and $x=4$. Between these values, the straight line lies above the parabola. ThereforeEvaluation givesA useful independent check uses the region's symmetry about $x=1$: its vertical width is $9-(x-1{)}^2$. Its area is ${\int }_{-3}^3(9-u^2)du=36$, and its centroid has $x$ coordinate one, so its first moment $\iint xdA$ is again $36$.