Find the area bounded by the curves $y = ||x - 1| - 2|$ and $y = 2$.

If the area enclosed by the parabolas $P_1: 2y = 5x^2$ and $P_2: x^2 - y + 6 = 0$ is equal to the area enclosed by $P_1$ and the line $y = \alpha x$,where $\alpha > 0$,then $\alpha^3$ is equal to $......$.

The area of the region enclosed by the curves $y = x$,$y = \frac{1}{x}$,$x = e$ and the positive $X$-axis is

$A$ line passing through the point $A(-2, 0)$ touches the parabola $P: y^2 = x - 2$ at the point $B$ in the first quadrant. The area of the region bounded by the line $AB$,the parabola $P$,and the $x$-axis is:

The area of the region between the curves $y=\sqrt{\frac{1+\sin x}{\cos x}}$ and $y=\sqrt{\frac{1-\sin x}{\cos x}}$ bounded by the lines $x=0$ and $x=\frac{\pi}{4}$ is

The area bounded by the curves $|x| + |y| \geq 1$ and $x^2 + y^2 \leq 1$ is