The area (in sq. units) of the region $\left\{(x, y): 0 \leq y \leq x^{2}+1,0 \leq y \leq x+1\right.$ $\left.\frac{1}{2} \leq x \leq 2\right\}$ is 

The area (in sq. units) of the region described by $\left\{(\mathrm{x}, \mathrm{y}): \mathrm{y}^2 \leq 2 \mathrm{x}\right.$, and $\left.\mathrm{y} \geq 4 \mathrm{x}-1\right\}$ is

The line $y = mx$ bisects the area enclosed by the curve $y = 1 + 4x - x^2\,\, \&$ the lines $x = 0, x = \frac{3}{2} \& \,\,y = 0.$ Then the value of $m$ is:

The area bounded by the curves $y = {\log _e}x$ and $y = {({\log _e}x)^2}$ is

The area bounded by the curves ${y^2} - x = 0$ and $y - {x^2} = 0$ is

If the area bounded by the curve $2 y^2=3 x$, lines $x+y=3, y=0$ and outside the circle $(x-3)^2+y^2=2$ is $A$, then $4(\pi+4 A )$ is equal to $.........$.