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

If the area bounded by the curve $y = x^3 + ax$ (where $a > 0$), the $X$-axis, and the lines $x = -2$ and $x = 1$ is $\frac{37}{4} \text{ square units}$, find the value of $a$.

The area bounded by the curve $xy - 3x - 2y - 10 = 0$,the $x$-axis,and the lines $x = 3$ and $x = 4$ is:

The area (in square units) bounded by the line $y = x$, the $X$-axis and the lines $x = -2$ and $x = 4$ is...

If a straight line $y - x = 2$ divides the region $x^2 + y^2 \le 4$ into two parts,then the ratio of the area of the smaller part to the area of the greater part 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?
$[A] \ 0 < \alpha \leq \frac{1}{2}$
$[B] \ \frac{1}{2} < \alpha < 1$
$[C] \ 2 \alpha^4 - 4 \alpha^2 + 1 = 0$
$[D] \ \alpha^4 + 4 \alpha^2 - 1 = 0$