The area of the region bounded by the curves $y = |x - 4|$, $x = 3$, $x = 5$, and the $X$-axis is

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

The area of the smaller part between the circle $x^2 + y^2 = 4$ and the line $x = 1$ is

The area enclosed by the curves $y=x|x|$,$x=-1$ and $x=1$ is.......... sq units.

The area between the curve $y = \sin^2 x$,the $x$-axis,and the ordinates $x = 0$ and $x = \frac{\pi}{2}$ is:

The area bounded by the curves $y = -\sqrt{-x}$ and $x = -\sqrt{-y}$ where $x, y \le 0$ is: