6.1 Areas Between Curves/5

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

${\begin{aligned}\int _{-1}^{2}[(9-x^{2})-(x+1)]dx=[9x-{\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}-x]{\Bigg |}_{-1}^{2}\\[2ex]&=[9(2)-{\frac {(2)^{3}}{3}}-{\frac {(2)^{2}}{2}}-(2)-(9(-1)-{\frac {(-1)^{3}}{3}}-{\frac {(-1)^{2}}{2}}-(-1))]\\[2ex]&=18-{\frac {8}{3}}--2-2+9-{\frac {1}{3}}+{\frac {1}{2}}-1\\[2ex]&=19.5\\[2ex]\end{aligned}}$