The area enclosed by the curves $y^2+4x=4$ and $y-2x=2$ is:

The area (in sq. units) of the region described by $A = \{(x, y) : x^2 + y^2 \leq 1 - x\}$ is

The area of the region included between the parabola $y^{2}=x$ and the line $x+y=2$ in the first quadrant is

If the area of the region $S=\{(x, y): 2y - y^2 \leq x^2 \leq 2y, x \geq y\}$ is equal to $\frac{n+2}{n+1} - \frac{\pi}{n-1}$,then the natural number $n$ is equal to $...............$.

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:

If $(\alpha, \beta)$ is the stationary point of the curve $y=2x-x^2$,then the area bounded by the curves $y=2^x, y=2x-x^2, x=0$ and $x=\alpha$ is