If the area of the region bounded by the curves $y^2-2y=-x$ and $x+y=0$ is $A$,then $8A$ is equal to

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

The area (in sq. units) of the region bounded by $y-x=2$ and $x^2=y$ is equal to

Let $S$ be the region bounded by the curves $y=x^{3}$ and $y^{2}=x$. The curve $y=2|x|$ divides $S$ into two regions of areas $R_{1}$ and $R_{2}$. If $\max \{R_{1}, R_{2}\}=R_{2}$,then $\frac{R_{2}}{R_{1}}$ is equal to

If the area lying in the first quadrant and bounded by the circle $x^2+y^2-4x=0$, the parabola $y^2=x$ and the $X$-axis is $A$, then $6A-9\sqrt{3}=$

The area (in sq. units) of the region enclosed between the parabola $y^{2}=2x$ and the line $x+y=4$ is