IIT-JEE 1992Question 1
Sketch the region bounded by the curves $y=x^{2}$ and $y=\dfrac{2}{1+x^{2}}$. Find the area.
Application of Integrals
Answer$\left(\pi-\dfrac23\right)$ sq. units
Solution
$x^2=\frac2{1+x^2}\Rightarrow x^4+x^2-2=0\Rightarrow x=\pm1$. Area $=\int_{-1}^{1}\left(\frac2{1+x^2}-x^2\right)dx=4\cdot\frac\pi4-\frac23=\pi-\frac23$.