The area of the region enclosed by the lines $y = 2x$, $2y = x$ and $x = 2$ (in sq. units) is...

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

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).

If $x^{2}+y^{2}=a^{2}$, then $\int_{0}^{a} \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx=$

The ratio of the areas bounded by the curves $y = \cos x$ and $y = \cos 2x$ between $x = 0$ and $x = \frac{\pi}{3}$ with the $X$-axis is:

The area of the region $A = \{ (x,y) : 0 \le y \le x|x| + 1, -1 \le x \le 1 \}$ in square units is: