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

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

The area inside the parabola $y^2 = 4ax$,between the lines $x = a$ and $x = 4a$ is equal to

Let $S(\alpha) = \{(x,y) : y^2 \leq x, 0 \leq x \leq \alpha\}$ and $A(\alpha)$ be the area of the region $S(\alpha)$. If for a $\lambda, 0 < \lambda < 4, A(\lambda) : A(4) = 2 : 5$,then $\lambda$ equals:

The area of the shaded region in the given figure of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is . . . . . . sq. units.

The area of the region bounded by the curve $y = \cot x$,the lines $x = \frac{\pi}{4}$,$x = \frac{\pi}{2}$,and the $X$-axis is . . . . . . sq. units.