The area enclosed between the curves $y = x^3$ and $y = \sqrt{x}$ is,(in square units)

Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f(x)=2 x^{3}-3 x^{2}-12 x$. If $A$ is the total area of the region bounded by $y=f(x)$,the $x$-axis and the lines $x=a$ and $x=b$,then $4 A$ is equal to ..... .

The area of the region bounded by the curves $y^{2}=8x$ and $y=x$ is

Let $C_{1}$ be the curve obtained by the solution of the differential equation $2xy \frac{dy}{dx} = y^{2} - x^{2}, x > 0$. Let the curve $C_{2}$ be the solution of $\frac{2xy}{x^{2} - y^{2}} = \frac{dy}{dx}$. If both the curves pass through $(1, 1)$,then the area enclosed by the curves $C_{1}$ and $C_{2}$ is equal to:

The area (in sq. units) of the region bounded by $y-x=2$ and $x^2=y$ is equal to

The area (in sq. units) of the region $\{(x, y) : x \geq 0, x+y \leq 3, x^2 \leq 4y \text{ and } y \leq 1+\sqrt{x}\}$ is