The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 27,1 \leq y \leq x^2\right\}$ is equal to :

The area of the region bounded by the curves $x+3 y^2=0$ and $x+4 y^2=1$ is equal to :

The area of the region $\{(x, y): y \leq \pi-|x|, y \leq|x \sin x|, y \geq 0\}$ is:

Let $P_1 : y = 4x^2$ and $P_2 : y = x^2 + 27$ be two parabolas. If the area of the bounded region enclosed between $P_1$ and $P_2$ is six times the area of the bounded region enclosed between the line $y = \alpha x$, $\alpha > 0$ and $P_1$, then $\alpha$ is equal to :