If $(\alpha, \beta)$ is the stationary point of the curve $y=2x-x^2$,then the area bounded by the curves $y=2^x, y=2x-x^2, x=0$ and $x=\alpha$ is

The area of the region,bounded by the parabola $y=x^2+2$ and the lines $y=x, x=0$ and $x=3$,is

The area (in sq units) bounded by the curves $y^2=4x$ and $x^2=4y$ 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 (in sq. units) bounded between the parabolas $x^2 = \frac{y}{4}$ and $x^2 = 9y$ and the line $y = 2$ is

If the area of the region bounded by the curves $y=4-\frac{x^2}{4}$ and $y=\frac{x-4}{2}$ is equal to $\alpha$,then $6 \alpha$ equals