The area under the curve $y = x^2 - 4x$ within the $x$-axis and the line $x = 2$ is:

The smaller area enclosed by the circle $x^{2}+y^{2}=4$ and the line $x+y=2$ is

The volume of the solid formed by rotating the area enclosed between the curve $y^{2}=4x$, $x=4$, and $x=5$ about the $x$-axis is (in cubic units): (in $\pi$)

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

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 (in sq unit) of the region bounded by the curves $2x = y^2 - 1$ and $x = 0$ is