6.1 Areas Between Curves/25

${\begin{aligned}&\color {red}\mathbf {y=x^{2}} &\color {royalblue}\mathbf {y={\frac {2}{({x^{2}}+1)}}} \\\end{aligned}}$

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