The area bounded by the parabola $y^2=x$,the straight line $y=4$ and the $Y$-axis is

The area (in sq. units) of the region bounded by the curve $y = 2x - x^2$ and the $X$-axis 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 of the region bounded by the curve $y=\log x$,the $x$-axis,and the lines $x=1, x=e$ is

The area enclosed (in square units) by the curve $y=x^4-x^2$, the $x$-axis and the vertical lines passing through the two minimum points of the curve is

The area bounded by the curve $y = x^2 + 4x + 5$,the coordinate axes,and the minimum ordinate is: