The area of the region bounded by the curve $y = x^3$,and the lines $y = 8$ and $x = 0$ is

The area bounded by $y = -x^2 + 2x + 3$ and $y = 0$ is

Area of the region bounded by the curve $x^2 = 4y$ and the line $y = 3$ is . . . . . . . (in $\sqrt{3}$)

Find the area of the region lying in the first quadrant and bounded by $y=4x^2$,$x=0$,$y=1$,and $y=4$.

The area of the region (in sq. units) enclosed by the curves $y=8x^3-1$,$y=0$,$x=-1$ and $x=1$ is

$A$ farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0,0)$,$Q(1,1)$,and $R(2,0)$. From this land,a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y = x^n$ $(n > 1)$. If the area of the region taken away by the farmer $F_2$ is exactly $30\%$ of the area of $\triangle PQR$,then the value of $n$ is: