Let $S$ be the region bounded by the curves $y=x^{3}$ and $y^{2}=x$. The curve $y=2|x|$ divides $S$ into two regions of areas $R_{1}$ and $R_{2}$. If $\max \{R_{1}, R_{2}\}=R_{2}$,then $\frac{R_{2}}{R_{1}}$ is equal to

The area of the region $\{(x, y) \in R^2: y \geq \sqrt{|x+3|}, 5y \leq x+9 \leq 15\}$ is equal to

If the area of the region ${(x, y) : -2x + 1 \le y \le 4 - x^2, x \ge 0, y \ge 0}$ is $\frac{\alpha}{\beta}$, where $\alpha, \beta \in N$ and $\gcd(\alpha, \beta) = 1$, then the value of $(\alpha + \beta)$ is:

Let the function $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ a^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in R$,where $a > 1, b \in R$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}$,where $\alpha, \beta \in Z$,then the value of $\alpha + \beta$ is . . . . .

If $A$ is the area in the first quadrant enclosed by the curve $C: 2x^2 - y + 1 = 0$,the tangent to $C$ at the point $(1, 3)$ and the line $x + y = 1$,then the value of $60A$ is

The area bounded by the curve $y^2=2x+1$ and the line $x-y=1$ is