The area bounded by the curves $y = 1 - \cos(\pi x)$,$y = -x^2$ and the lines $x = \frac{1}{2}$ and $x = -\frac{1}{2}$ is:

The curve $y = ax^2 + bx + c$ passes through the point $(1, 2)$ and its tangent at the origin is the line $y = x$. The area bounded by the curve,the ordinate of the curve at its minima,and the tangent line is

Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f(x)=2 x^{3}-3 x^{2}-12 x$. If $A$ is the total area of the region bounded by $y=f(x)$,the $x$-axis and the lines $x=a$ and $x=b$,then $4 A$ is equal to ..... .

Let $S = \{(x, y) \in R \times R : x \geq 0, y \geq 0, y^2 \leq 4x, y^2 \leq 12 - 2x \text{ and } 3y + \sqrt{8}x \leq 5\sqrt{8}\}$. If the area of the region $S$ is $\alpha \sqrt{2}$,then $\alpha$ is equal to

The area of the region enclosed between the circles $x^{2}+y^{2}=4$ and $x^{2}+(y-2)^{2}=4$ is:

The area between the parabolas $y^2 = 4ax$ and $x^2 = 8ay$ is