The area of the region bounded by the lines $2x - y + 1 = 0$, $y = -1$, $y = 3$ and the y-axis is

The area bounded by the curve $y = x(1 - \ln x)$,the line $x = e^{-1}$,and the positive $X$-axis between $x = e^{-1}$ and $x = e$ is:

The area of the region bounded by the curve $y = \cos x$,$x = 0$,and $x = \pi$ is

The area bounded by the parabola $y^{2}=16x$ and its latus rectum in the first quadrant is

The area of the region bounded by the curves $y = \sin x$, $y = \cos x$ and the lines $x = 0$ and $x = \frac{\pi}{4}$ 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?
$[A] \ 0 < \alpha \leq \frac{1}{2}$
$[B] \ \frac{1}{2} < \alpha < 1$
$[C] \ 2 \alpha^4 - 4 \alpha^2 + 1 = 0$
$[D] \ \alpha^4 + 4 \alpha^2 - 1 = 0$