Given : $$f(x) = \left\{ {\matrix{ {x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr {{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr {1 - x\,\,\,,} & {{1 \over 2} < x \le 1} \cr } } \right.$$

and $$g(x) = \left( {x - {1 \over 2}} \right)^2,x \in R$$

Then the area (in sq. units) of the region bounded by the curves, y = ƒ(x) and y = g(x) between the lines, 2x = 1 and 2x = $$\sqrt 3 $$, is :

The area (in sq. units) of the region

{(x,y) $$ \in $$ R² : x² $$ \le $$ y $$ \le $$ 3 – 2x}, is :

For a > 0, let the curves C₁ : y² = ax and C₂ : x² = ay intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the chord OP and the x-axis at points Q and R, respectively. If the line x = b bisects the area bounded by the curves, C₁ and C₂, and the area of
$$\Delta $$OQR = $${1 \over 2}$$, then 'a' satisfies the equation :

The area (in sq. units) of the region
{(x, y) $$ \in $$ R² | 4x² $$ \le $$ y $$ \le $$ 8x + 12} is :