The area of the region bounded by the curves and is : [2025]

Question

The area of the region bounded by the curves $x (1 + y^{2}) = 1$ and $y^{2} = 2 x$ is : [2025]

Smart Achievers · Accepted Answer

(3)

We have, $x (1 + y^{2}) = 1$ ... (i)

and $y^{2} = 2 x$ ... (ii)

From equation (i) and (ii), we get

$x (1 + 2 x) = 1 \Rightarrow 2 x^{2} + x - 1 = 0$

$\Rightarrow x = \frac{1}{2}, x = - 1$ (Reject)

From (ii), $y^{2} = 2 (\frac{1}{2}) \Rightarrow y = \pm 1$

$∴$ Required area $= \int_{- 1}^{1} (\frac{1}{1 + y^{2}} - \frac{y^{2}}{2}) d y$

$= (\tan^{- 1} y - \frac{y^{3}}{6}) |_{- 1}^{1} = \frac{π}{2} - \frac{1}{3}$ .

Want Us to Email you About Special Offers & Updates?