The area of the region given by $\{(x, y): xy \leq 8, 1 \leq y \leq x^2\}$ is :

Let $y=p(x)$ be the parabola passing through the points $(-1,0), (0,1)$ and $(1,0)$. If the area of the region $\{(x, y) : (x+1)^2+(y-1)^2 \leq 1, y \leq p(x)\}$ is $A$,then $12(\pi-4A)$ is equal to $.........$.

The area enclosed by the curves $y = \sqrt{x}$ and $x = -\sqrt{y}$,and the circle $x^2 + y^2 = 2$ above the $x$-axis,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 ..... .

For $a>0$, let the curves $C_1: y^2=a x$ and $C _2: x ^2=$ ay intersect at origin O and a point P Let the line $x = b (0 < b < a )$ intersect the chord $O P$ and the x -axis at points Q and R , respectively. If the line $x=b$ bisects the area bounded by the curves, $C _1$ and $C _2$, and the area of $\Delta OQR =\frac{1}{2}$, then ' $a$ ' satisfies the equation

The area of the figure enclosed by $y = \cos^{-1}(\cos x)$, $x \in [2\pi, 4\pi]$, the $x$-axis, and $y = \tan^{-1} x + \tan^{-1} \frac{1}{x}$ is