The area of the region bounded by the curve $y=|x-3|$,the $X$-axis,and the lines $x=0$ and $x=2$ is . . . . . . sq. units.

The area of the region bounded by the parabola $y^2 = 8x$ and its latus rectum is . . . . . . sq. units.

The area of the region bounded by the curve $y = x^3$,and the lines $y = 8$ and $x = 0$ is

Area of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ 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 bounded by the parabola $y^{2}=x$,the straight line $y=4$,and the $y$-axis in square units is: