The area (in sq. units) of the region bounded by the curves $y=2-x^2$ and $y=\left|x\right|$ is $k$, then the value of $3k$ is

Solution

$2-x^2=x$
$x^2+x-2=0$
$x\ne -2$
$x=1$
Required area $=2{\int }_0^1\left[\left(2-x^2\right)-\left(x\right)\right]dx$
$\Rightarrow k=2{\left[2x-\frac{x^3}{3}-\frac{x^2}{2}\right]}_0^1$
$\Rightarrow k=2\left[2-\frac{1}{3}-\frac{1}{2}\right]=\frac{7}{3}\mathrm{s}\mathrm{q}.\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}$
$\Rightarrow 3k=7$

Hence, the correct answer is 7.