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

The line $x=\frac{\pi}{4}$ divides the area of the region bounded by $y=\sin x$,$y=\cos x$ and the $x$-axis $\left(0 \leq x \leq \frac{\pi}{2}\right)$ into two regions of areas $A_1$ and $A_2$. Then $A_1 : A_2$ equals (in $: 1$)

The area bounded by the lines $y = x$,$x = -1$,$x = 2$ and the $x$-axis is

The area enclosed (in square units) by the curve $y=x^4-x^2$,the $x$-axis and the vertical lines passing through the two minimum points of the curve is

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.

If the area of the region enclosed by the curve $x^2+y^2=16$ and the lines $x=2$ and $x=3$ is $\left(3 \sqrt{7}-4 \sqrt{3}-\frac{8 \pi}{3}+k\right)$ sq units,then $k$ equals