6.1 Areas Between Curves/5

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \color{red}\mathbf{y=x+1} & \color{royalblue}\mathbf{y=9-x^{2}} \\ & x=-1 & x=2 \\ \end{align} }

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