The area (in sq. units) bounded by the curve $y=2x-x^2$ and the line $y=-x$ is

If the area bounded by the curve $2y^2 = 3x$,the lines $x+y=3$,$y=0$ and outside the circle $(x-3)^2 + y^2 = 2$ is $A$,then $4(\pi + 4A)$ is equal to $.........$.

For $x \in [0, 2\pi]$,the area of the region bounded by the curves $y = x + \sin x$ and $y = x$ is:

The area of the figure enclosed by $y = \cos^{-1}(\cos x)$, $x \in [2\pi, 4\pi]$, the $x$-axis, and $y = \tan^{-1} x + \tan^{-1} \frac{1}{x}$ is

The area enclosed by the curves $y^2+4x=4$ and $y-2x=2$ is:

If the area of the region $S=\{(x, y): 2y - y^2 \leq x^2 \leq 2y, x \geq y\}$ is equal to $\frac{n+2}{n+1} - \frac{\pi}{n-1}$,then the natural number $n$ is equal to $...............$.