6.1 Areas Between Curves/5

${\begin{aligned}&\color {red}\mathbf {y=x+1} &\color {royalblue}\mathbf {y=9-x^{2}} \\&x=-1&x=2\\\end{aligned}}$

${\begin{aligned}\int _{-1}^{2}\left((9-x^{2})-(x+1)\right)dx&=\left[9x-{\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-x\right]{\Bigg |}_{-1}^{2}\\[2ex]&=\left[9(2)-{\frac {(2)^{3}}{3}}-{\frac {(2)^{2}}{2}}-(2)\right]-\left[9(-1)-{\frac {(-1)^{3}}{3}}-{\frac {(-1)^{2}}{2}}-(-1)\right]\\[2ex]&=\left[18-{\frac {8}{3}}-2-2\right]-\left[-9+{\frac {1}{3}}-{\frac {1}{2}}+1\right]\\[2ex]&={\frac {117}{6}}\\[2ex]\end{aligned}}$