6.1 Areas Between Curves/25

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

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} \int_{1}^{-1}\left(\frac{2}{{x^2}+1} - x^{2}\right) dx &= \int_{1}^{-1}\left(2\cdot\frac{1}{x^{2}+1}-\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{align} }