The area (in square units) of the region bounded by the parabola $y=x^2+2$ and the lines $y=x+1$,$x=0$,and $x=3$ is:

The area of the region common to the parabolas $4y^2 = 9x$ and $3x^2 = 16y$ is...

If the area of the region bounded by $y=\cos x$, $y=\sin x$, $x=\frac{\pi}{4}$ and $x=\pi$ is bisected by the line $x=a$, then $\sin \left(a+\frac{\pi}{4}\right)=$

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

Area of the region $\{(x, y): x^2+(y-2)^2 \leq 4, x^2 \geq 2y\}$ is

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