The area bounded by the curve $y = 4x - x^2$ and the $X$-axis in square units is:

The area of the figure bounded by the parabolas $x=-2y^{2}$ and $x=1-3y^{2}$ is

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 (in sq. units) of the region bounded by the curve $y = 2x - x^2$ and the $X$-axis is...

The area of the region bounded by the curve $x + 2y + 8 = 0$ and the lines $y = -3$ and $y = -1$ is . . . . . . sq. units.

The area bounded by the curve $xy = c$,the $x$-axis,and the lines $x = 1$ and $x = 4$ is: