The area of the region bounded by $y-x=2$ and $x^{2}=y$ is equal to:

The area (in sq. units) of the region $A = \{(x, y) : x^2 \le y \le x + 2\}$ is

The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to

The area of the region bounded by the curve $y = \sin(\pi x)$ and the $X$-axis for $x \in [-1, 2]$ is . . . . . . sq. units.

Let $P_{1}: y = 4x^{2}$ and $P_{2}: y = x^{2} + 27$ be two parabolas. If the area of the bounded region enclosed between $P_{1}$ and $P_{2}$ is six times the area of the bounded region enclosed between the line $y = \alpha x, \alpha > 0$ and $P_{1}$,then $\alpha$ is equal to:

The following figure shows the graph of a continuous function $y=f(x)$ on the interval $[1,3]$. The points $A, B, C$ have coordinates $(1,1), (3,2), (2,3)$ respectively,and the lines $l_1$ and $l_2$ are parallel,with $l_1$ being tangent to the curve at $C$. If the area under the graph of $y=f(x)$ from $x=1$ to $x=3$ is $4$ sq units,then the area of the shaded region is