6.1 Areas Between Curves/18

${\begin{aligned}&\color {red}\mathbf {y=8-x^{2}} &\color {royalblue}\mathbf {y=x^{2}} \\&x=-3&x=3\\\end{aligned}}$

${\begin{aligned}8-x^{2}&=x^{2}\\8&=2x^{2}\\4&=x^{2}\\\end{aligned}}$

$x=\pm 2$

${\begin{aligned}\int _{-2}^{2}\left((8-x^{2})-(x^{2})\right)dx&=\int _{-2}^{2}\left(8-2x^{2}\right)dx\\[2ex]&=\left(8x-2{\frac {x^{3}}{3}}\right){\bigg |}_{-2}^{2}\end{aligned}}$