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

If $\int\limits_0^1 {\left( {4{x^3} - f(x)} \right)f(x)dx = \frac{4}{7}} $ ,then the area of region bounded by $y = f(x)$ , $x-$ axis and ordinate $x = 1$ and $x = 2$ , is 

If $A$ is the area in the first quadrant enclosed by the curve $C: 2 x^2-y+1=0$, the tangent to $C$ at the point $(1,3)$ and the line $x+y=1$, then the value of $60 A$ is

The area bounded by the $y-$ axis, $y=\cos x$ and $y=\sin x$ when $0 \leq x \leq \frac{\pi}{2}$ is

Smaller area enclosed by the circle $x^{2}+y^{2}=4$ and the line $x+y=2$ is

The area enclosed by $y ^{2}=8 x$ and $y=\sqrt{2} x$ that lies outside the triangle formed by $y=\sqrt{2} x, x=$ $1, y=2 \sqrt{2}$, is equal to