Question

Let $\left[k\right]$ denote the greatest integer less than or equal to $k.$ If $S$ is the area enclosed by the curves $f\left(x\right)=4|x|-|x{|}^3$ and $g\left(x\right)+\sqrt{4-x^2}=0,$ then the value of $\left[S\right]$ is

Answer 1

$f\left(x\right)$ is symmetric about $y-$axis.
$g\left(x\right)=-\sqrt{4-x^2}$ represents the semi-circular part of the circle $x^2+y^2=4$ below $x-$axis.
$\therefore S=2\left(\underset{0}{\overset{2}{\int }}(4x-x^3)dx+\pi \right)$
$=2({(2x^2-\frac{x^4}{4})}_0^2+\pi )=8+2\pi$
$\left[S\right]=14$