The area (in sq. units) of the region bounded by the curves $y=4|\cos x|$ and $y=-|\cos x|$ from $x=-\frac{\pi}{2}$ to $x=\frac{\pi}{2}$ is

If the area enclosed between the curves $y^2 = 4kx$ and $y = kx$ for $k > 0$ is $\frac{2}{3}$ sq. units, then $k =$ ?

The area of the region described by $\{(x, y) \mid x^2+y^2 \leq 1\}$ and $\{y^2 \leq 1-x\}$ is

The area of the region, inside the ellipse $x^{2}+4y^{2}=4$ and outside the region bounded by the curves $y=|x|-1$ and $y=1-|x|$, is:

Let $\Delta$ be the area of the region $\left\{( x , y ) \in \mathbb{R} ^2: x ^2+ y ^2 \leq 21, y ^2 \leq 4 x , x \geq 1\right\}$. Then $\frac{1}{2}\left(\Delta-21 \sin ^{-1} \frac{2}{\sqrt{7}}\right)$ is equal to

If the area enclosed by the parabolas $P_1: 2y = 5x^2$ and $P_2: x^2 - y + 6 = 0$ is equal to the area enclosed by $P_1$ and the line $y = \alpha x$,where $\alpha > 0$,then $\alpha^3$ is equal to $......$.