Question

If area enclosed by the simultaneous inequations $x-2y^2\ge 0$ and $1-x-\left|y\right|\ge 0$ is $A$ sq. units, then the value of $12A$ is

Answer 1

Solving $x=2y^2$ and $\left|y\right|=1-x,$ we get
$x=\frac{1}{2}$ and $y=\frac{1}{2}$
or $x=\frac{1}{2}$ and $y=-\frac{1}{2}$

$A=\frac{2}{\sqrt{2}}\underset{0}{\overset{1/2}{\int }}\sqrt{x}dx+2\times \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}$
$=\frac{1}{3}+\frac{1}{4}=\frac{7}{12}$