The area included between the parabolas $y^{2} = 5x$ and $x^{2} = 5y$ is

The area (in sq. units) of the region lying in the first quadrant and enclosed by the $X$-axis,the straight line $x - \sqrt{3}y = 0$ and the circle $x^2 + y^2 = 4$ is

The area of the smaller region enclosed by the curves $y^{2}=8x+4$ and $x^{2}+y^{2}+4\sqrt{3}x-4=0$ 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 region bounded by $y^2=x$ and $y=|x|$ 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 $\sin \left(a+\frac{\pi}{4}\right)=$