Find the area under the curve $y=x^{4}$ bounded by the lines $x=1$,$x=5$ and the $x$-axis.

The area enclosed by the curves $y=x|x|$,$x=-1$ and $x=1$ is.......... sq units.

Area of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ 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 bounded by the curve $y=x^3-3x^2+2x$ and the $X$-axis is (in square units)

The area bounded by the parabola $y=x^2$ and the line $y=x$ is