The area of the region above the $x-$axis bounded by the curve $y = \tan x$,$0 \leq x \leq \frac{\pi}{2}$ and the tangent to the curve at $x = \frac{\pi}{4}$ is

The area (in $sq. units$) of the region $A = \{(x,y) : \frac{y^2}{2} \le x \le y + 4\}$ is

The area of the region (in square units) bounded by the curves $y=x^3$, $y=x$ and $-1 \leq x \leq 1$ is:

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

Let $A_{1}=\{(x, y):|x| \leq y^{2},|x|+2 y \leq 8\}$ and $A_{2}=\{(x, y):|x|+|y| \leq k\}$. If $27 \times \text{Area}(A_{1}) = 5 \times \text{Area}(A_{2})$,then $k$ 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