The area of the region bounded by the curves $y=x^{3}$, $y=\frac{1}{x}$, and the line $x=2$ in the first quadrant is:

The area bounded by the parabola $y^2 = 4x$ and the line $2x - 3y + 4 = 0$,in square units,is

Let $f :[-3,1] \rightarrow R$ be given as
$f(x)=\begin{cases} \min \{(x+6), x^{2}\}, & -3 \leq x \leq 0 \\ \max \{\sqrt{x}, x^{2}\}, & 0 \leq x \leq 1 \end{cases}$
If the area bounded by $y = f(x)$ and the $x$-axis is $A$,then the value of $6A$ is equal to ....... .

The area (in sq units) bounded by the curves $x = -2y^2$ and $x = 1 - 3y^2$ is

The area of the region enclosed by the curves $y=x^2-4x+4$ and $y^2=16-8x$ is:

Let $f: R \to R$ be a function such that $f(x) + 3f(\frac{\pi}{2} - x) = \sin x, x \in R$. Let the maximum value of $f$ on $R$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3, \beta > 0$, is $\alpha^2$, then $30\beta^3$ is equal to ————