The area of the region bounded by the curve $y=4x^{3}-6x^{2}+4x+1$ and the lines $x=1, x=5$ and the $x$-axis is

Area inside the parabola ${y^2} = 4ax$ between the lines $x = a$ and $x = 4a$ is equal to

The area of the smaller region bounded by the circle $x^2 + y^2 = 4$ and the line $x = 1$ 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 of the circle $x^{2}+y^{2}=16$ exterior to the parabola $y^{2}=6x$ is

The area of the figure bounded by the parabolas $x=-2y^{2}$ and $x=1-3y^{2}$ is