The area (in square units) enclosed between the curves $y=\sin x$ and $y=\cos x$ for $\frac{\pi}{4} \leq x \leq \frac{5 \pi}{4}$ is

The area of the region $\{(x, y) : 0 \le y \le 6 - x, y^2 \ge 4x - 3, x \ge 0\}$ is:

The area (in sq. units) of the region $\{(x, y) \in R^{2} | 4 x^{2} \leq y \leq 8 x+12\}$ is:

The area of the region lying above the $X$-axis and included between the circle $x^{2}+y^{2}=2ax$ and the parabola $y^{2}=ax$, $a>0$ is

The polynomial $f(x)$ satisfies the condition $f(x + 1) = x^2 + 4x$. The area enclosed by $y = f(x - 1)$ and the curve $x^2 + y = 0$ is

Consider the functions $f, g: R \rightarrow R$ defined by
$f(x)=x^2+\frac{5}{12}$ and $g(x)=\begin{cases} 2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4} \end{cases}$
If $\alpha$ is the area of the region
$\{( x , y ) \in R \times R :| x | \leq \frac{3}{4}, 0 \leq y \leq \min \{f( x ), g( x )\}\}$,
then the value of $9 \alpha$ is.