The area of the region bounded by the curve $y = \sin(\pi x)$ and the $X$-axis for $x \in [0, 2]$ is . . . . . . sq. units.

The area bounded by the curve $y=x^3-3x^2+2x$ and the $X$-axis is (in square units)

Consider the function $f(x) = x^3 - 8x^2 + 20x - 13$. The area enclosed by $y = f(x)$ and the coordinate axes is:

The area of the figure bounded by the parabolas $x=-2y^{2}$ and $x=1-3y^{2}$ is

Area of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ 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