The area of the region bounded by the curves x(1+y^2)=1 and y^2=2x is:

Question

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

Accepted Answer

$$\begin{aligned}
& x\left(1+y^2\right)=1 \quad\text{..... (1)}\
& y^2=2 x\quad\text{..... (2)}
\end{aligned}$$

$$\begin{aligned}
&\text { From equation (1) & (2) }\
&\begin{aligned}
x(1+2 x)=1 & \Rightarrow 2 x^2+x-1=0 \
& \Rightarrow x=\frac{1}{2}, x=-1 \text { (Reject) } \
& \Rightarrow y^2=2\left(\frac{1}{2}\right) \
& \Rightarrow y= \pm 1
\end{aligned}
\end{aligned}$$

$$\begin{aligned}
\text { Area bounded } & =\int_{-1}^1\left(\frac{1}{1+y^2}-\frac{y^2}{2}\right) d y \
& =\left.\left(\tan ^{-1} y-\frac{y^3}{6}\right)\right|_{-1} ^1 \
& =\frac{\pi}{2}-\frac{1}{3}
\end{aligned}$$

Solutions · 1