6.1 Areas Between Curves/25

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

${\begin{aligned}\int _{1}^{-1}\left({\frac {2}{{x^{2}}+1}}-x^{2}\right)dx&=\int _{1}^{-1}\left(2\cdot {\frac {1}{(x^{2}+1)}}\right)-\left(x^{2}\right)dx\\[2ex]&=\left[2\arctan(x)-{\frac {x^{3}}{3}}\right]{\Bigg |}_{-1}^{1}\\[2ex]&=\left[2\arctan(1)-{\frac {(1)^{3}}{3}}\right]-\left[2\arctan(-1)-{\frac {(-1)^{3}}{3}}\right]\\[2ex]&=\left[{\frac {2\pi }{4}}-{\frac {1}{3}}\right]-\left[-{\frac {2\pi }{4}}-\left(-{\frac {1}{3}}\right)\right]\\[2ex]&={\frac {\pi }{2}}-{\frac {1}{3}}+{\frac {\pi }{2}}-{\frac {1}{3}}\\[2ex]&=\pi -{\frac {2}{3}}\end{aligned}}$