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

Let $g\left( x \right) = \cos {x^2},f\left( x \right) = \sqrt x $ and $\alpha ,\beta (\alpha < \beta )$ be the roots of the quadratic equation $18{x^2} - 9\pi x + {\pi ^2} = 0$. Then the area (in sq. units) bounded by the curve $y = \left( {gof} \right)\left( x \right)$ and the lines $x = \alpha ,x = \beta $ and $y = 0$ is :

The polynomial $f (x)$ satisfies the condition $f (x + 1) = x^2 + 4x.$ The area enclosed by $y = f (x - 1)$ and the curve $x^2 + y = 0,$ is

If the area of the region

$\left\{(\mathrm{x}, \mathrm{y}): \frac{\mathrm{a}}{\mathrm{x}^2} \leq \mathrm{y} \leq \frac{1}{\mathrm{x}}, 1 \leq \mathrm{x} \leq 2,0<\mathrm{a}<1\right\}$ is

$\left(\log _e 2\right)-\frac{1}{7}$ then the value of $7 a-3$ is equal to:

If the area bounded by $y = ax^2$ and $x = ay^2, a > 0$, is $1$ then $a =$

Find the area of the region bounded by the curves $y=x^{2}+2, \,y=x,\, x=0$ and $x=3$.