The area (in sq. units) of the region $\{(x,y):y^2 \geq 2x\,and\,x^2+y^2 \leq 4x,x \geq 0,y \leq 0 \}$ is

The area of region enclosed by the parabolas  ${y^2} = 4x$ and ${x^2} = 4y$ is

The value of $\int \limits_0^{2 \pi} \min \left\{|x-\pi|, \cos ^{-1}(\cos x)\right\} d x$ is

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

If the area enclosed between the curves $y = kx^2$ and $x = ky^2, (k > 0)$, is $1$ square unit. Then $k$ is

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y= f(x)$, $x-$ axis and the ordinates $x = \frac{\pi }{4}$ and $x = \beta  > \frac{\pi }{4}$ is $\left( {\beta \sin \beta  + \frac{\pi }{4}\cos \beta  + \sqrt 2 \beta } \right)$.Then $f\left( {\frac{\pi }{2}} \right)$ is