The area of the region bounded by the curve $y=\log x$,the $x$-axis,and the lines $x=1, x=e$ is

The area of the region (in square units) bounded by the curve ${x^2} = 4y$,the line $x = 2$,and the $x$-axis is:

The area of the region bounded by the parabola $y = x^2 + 2$,the $X$-axis,and the lines $x = 1$ 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 bounded by the curve $y^2 = 4x$,the $Y$-axis,and the line $y = 3$ is . . . . . . sq. units.

Area of the region bounded by the curve $|x| + y = 1$ is . . . . . . sq. units.