The area of the region bounded by the curve $y=|x-3|$,the $X$-axis,and the lines $x=0$ and $x=2$ is . . . . . . sq. units.

If the line $x=\alpha$ divides the area of region $R=\{(x, y) \in \mathbb{R}^2: x^3 \leq y \leq x, 0 \leq x \leq 1\}$ into two equal parts,then which of the following is true?
$[A] \ 0 < \alpha \leq \frac{1}{2}$
$[B] \ \frac{1}{2} < \alpha < 1$
$[C] \ 2 \alpha^4 - 4 \alpha^2 + 1 = 0$
$[D] \ \alpha^4 + 4 \alpha^2 - 1 = 0$

The area of the region $\{(x, y) : y \le x - |x|, y \le |x \sin x|, y \ge 0\}$ is:

The area of the region bounded by the curve $y^2 = 4x$ and the line $x = 3$ is . . . . . . sq. units.

The area of the region bounded by the curve $x+2y=8$,the $X$-axis,and the lines $x=1$ and $x=5$ using integration is . . . . . . sq. units.

The area bounded by the curves $y = -\sqrt{-x}$ and $x = -\sqrt{-y}$ where $x, y \le 0$ is: