The area (in sq. units) bounded by the curve $x^2+2x+y-3=0$,the $X$-axis and the tangent at the point where the curve meets the $Y$-axis is

The area of the shorter region bounded by $|y| = 4 - x^2$ and $|y| = 3x$ is given by $\left( 3K + \frac{1}{3} \right)$ sq-unit,where $K$ is equal to:

Let $f: R \to R$ be a function such that $f(x) + 3f(\frac{\pi}{2} - x) = \sin x, x \in R$. Let the maximum value of $f$ on $R$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3, \beta > 0$, is $\alpha^2$, then $30\beta^3$ is equal to ————

The area (in sq. units) of the region described by $A = \{ (x,y) | y \ge x^2 - 5x + 4, x + y \ge 1, y \le 0 \}$ is:

The area of the region, inside the ellipse $x^{2}+4y^{2}=4$ and outside the region bounded by the curves $y=|x|-1$ and $y=1-|x|$, is:

The volume of the solid generated by revolving about the $y-$axis the figure bounded by the parabolas $y = x^2$ and $x = y^2$ is