The area of the region bounded by the line $y = 3 - x$,the $X$-axis,and the lines $x = 2$ and $x = 5$ is . . . . . . .

The area bounded by the curve $x(x^2 + p) = y - 1$ and the line $y = 1$ is:

The area bounded by the curve $y^2 = 4ax$,the $x$-axis,and the ordinates $x = 0$ and $x = a$ is

The area of the region bounded by the curve $y=2x-x^2$ and the $x$-axis is

Let $n \geq 2$ be a natural number and $f:[0,1] \rightarrow \mathbb{R}$ be the function defined by
$f(x)= \begin{cases} n(1-2nx) & \text{if } 0 \leq x \leq \frac{1}{2n} \\ 2n(2nx-1) & \text{if } \frac{1}{2n} \leq x \leq \frac{3}{4n} \\ 4n(1-nx) & \text{if } \frac{3}{4n} \leq x \leq \frac{1}{n} \\ \frac{n}{n-1}(nx-1) & \text{if } \frac{1}{n} \leq x \leq 1 \end{cases}$
If $n$ is such that the area of the region bounded by the curves $x=0, x=1, y=0$ and $y=f(x)$ is $4$,then the maximum value of the function $f$ is

The area of the region $\{(x, y) : y \le x - |x|, y \le |x \sin x|, y \ge 0\}$ is: