If the area enclosed between the curves $y^2 = 4kx$ and $y = kx$ for $k > 0$ is $\frac{2}{3}$ sq. units, then $k =$ ?

The area (in sq. units) of the region enclosed by the curves $y=x^{2}-1$ and $y=1-x^{2}$ is equal to

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

The area bounded by the curves $y = \cos x$ and $y = \sin x$ between the ordinates $x = 0$ and $x = \frac{3\pi}{2}$ is:

Area of the region $\{(x, y): x^2+(y-2)^2 \leq 4, x^2 \geq 2y\}$ is