The area of the region included between the parabolas $y^2=8x$ and $x^2=8y$ is

If the area of the region enclosed by the curve $ay = x^2$ and the line $x + y = 2a$ is $ka^2$,then $k =$

The area of the region $\{(x, y): xy \leq 8, 1 \leq y \leq x^2\}$ is

If the area of the region $\{(x, y): |4-x^2| \leq y \leq x^2, y \leq 4, x \geq 0\}$ is $\left(\frac{80 \sqrt{2}}{\alpha}-\beta\right)$,where $\alpha, \beta \in \mathbb{N}$,then $\alpha+\beta$ is equal to . . . . . . .

For $a>0$, let the curves $C_1: y^2=a x$ and $C _2: x ^2=$ ay intersect at origin O and a point P Let the line $x = b (0 < b < a )$ intersect the chord $O P$ and the x -axis at points Q and R , respectively. If the line $x=b$ bisects the area bounded by the curves, $C _1$ and $C _2$, and the area of $\Delta OQR =\frac{1}{2}$, then ' $a$ ' satisfies the equation

The area bounded by the curves $y = (x + 1)^2$,$y = (x - 1)^2$ and the line $y = \frac{1}{4}$ is