The area (in square units) bounded by the line $y = x$, the $X$-axis and the lines $x = -2$ and $x = 4$ is...

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

The area of the region lying in the first quadrant bounded by $y=4x^2$,$x=0$,$y=2$,and $y=4$ is

The area enclosed by the ellipse $25x^2 + 16y^2 = 400$ is . . . . . . sq. units. (in $\pi$)

Let $A$ be the area bounded by the curve $y = \cos^{-1}\sqrt{1 - x^2}$,the tangent to the curve $y = \sin^{-1}x$ at $x = 0$,and the line $x = 1$. Then the value of $2(\{A\} + \text{sgn}(A))$ is (where $\{.\}$ is the fractional part function and $\text{sgn}(x)$ is the signum function).

The area bounded by the curve $y = f(x)$,the $x$-axis,and the ordinates $x = 1$ and $x = b$ is $(b - 1) \sin(3b + 4)$. Then $f(x)$ is: