Question

The area bounded by the curve $y=x^{2}$ and $y=\frac{2}{1+x^{2}}$ is $\lambda$ sq. units, then the value of $[\lambda]$ is
[$[k]$ denotes greatest integer less than or equal to $k$.]

Answer 1

Required area $=\int\limits_{-1}^{1}\left(\frac{2}{1+x^{2}}-x^{2}\right) d x$
$=2 \int\limits_{0}^{1}\left(\frac{2}{1+x^{2}}-x^{2}\right) d x$
$=2\left(2 \tan ^{-1} x-\frac{x^{3}}{3}\right)_{0}^{1}=2\left(\frac{\pi}{2}-\frac{1}{3}\right)=\pi-\frac{2}{3}$ sq. units

$\therefore\left[\pi-\frac{2}{3}\right]=2$