The area (in square units) in the first quadrant bounded by the curve $y=x^2+2$ and the lines $y=x+1, x=0$ and $x=3$,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 . . . . . . .

Area enclosed by the curves $y = \ln x$,$y = \ln|x|$,$y = |\ln x|$,and $y = |\ln|x||$ 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 (in square units) of the region bounded by the circle $x^2 + y^2 = 9$ and the parabola $y^2 = 8x$ is...

The area (in square units) of the region enclosed by the ellipse $x^2+3y^2=18$ in the first quadrant below the line $y=x$ is: