The area bounded by $y = x^2 + 2$ and $y = 2|x| - \cos(\pi x)$ is equal to

If the area of the bounded region $R=\{(x, y): \max \{0, \log _{e} x\} \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\}$ is $\alpha(\log _{e} 2)^{-1}+\beta(\log _{e} 2)+\gamma$,then the value of $(\alpha+\beta-2 \gamma)^{2}$ is equal to:

The area of the region included between the parabola $y^{2}=x$ and the line $x+y=2$ in the first quadrant is

The area of the figure enclosed by $y = \cos^{-1}(\cos x)$, $x \in [2\pi, 4\pi]$, the $x$-axis, and $y = \tan^{-1} x + \tan^{-1} \frac{1}{x}$ is

The area (in sq. units) of the region lying in the first quadrant and enclosed by the $X$-axis,the straight line $x - \sqrt{3}y = 0$ and the circle $x^2 + y^2 = 4$ is

The area of the region bounded by the curve $y = \max \{| x |, x | x - 2 |\}$,the $x$-axis,and the lines $x = -2$ and $x = 4$ is equal to . . . . . . .