The ratio in which the $x$-axis divides the area of the region bounded by the curves $y = x^2 - 4x$ and $y = 2x - x^2$ is:

The area (in sq units) bounded by the curves $y^2=4x$ and $x^2=4y$ is

Let $AB$ be the latus rectum of the parabola $y^2 = 4ax$ in the $XY$-plane. Let $T$ be the region bounded by the finite arc $AB$ of the parabola and the line segment $AB$. $A$ rectangle $PQRS$ of maximum possible area is inscribed in $T$ with $P, Q$ on line $AB$,and $R, S$ on arc $AB$. Then,$\frac{\text{area}(PQRS)}{\text{area}(T)}$ equals

The area of the shaded region bounded by the curves $y = \sin x$, $y = \cos x$ and the $x$-axis between $x = 2\pi$ and $x = \frac{5\pi}{2}$ is ... sq. units.

The area (in sq. units) of the region bounded by the lines $x=0, x=\frac{\pi}{2}$ and $f(x)=\sin x, g(x)=\cos x$ is

The area of the region above the $X$-axis included between the parabola $y^{2}=x$ and the circle $x^{2}+y^{2}=2x$ in square units is