If $f(x) = x + e^x$,then the area bounded by $f^{-1}(x)$,the ordinates $x = 1$ and $x = 1 + e$,and the $x$-axis is (in $sq. units$):

Area of the region bounded by the curve $y = x^3$,$x$-axis and the ordinates $x = -1$ and $x = 2$ is . . . . . . . (in $/4$)

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

The area of the region bounded by the curve $y = \cos x$ between $x = 0$ and $x = \frac{3\pi}{2}$ is . . . . . . square units.

$A$ quadratic polynomial $y = f(x)$ with absolute term $3$ neither touches nor intersects the abscissa axis and is symmetric about the line $x = 1$. The coefficient of the leading term of the polynomial is unity. $A$ point $A(x_1, y_1)$ with abscissa $x_1 = 1$ and a point $B(x_2, y_2)$ with ordinate $y_2 = 11$ are given in a Cartesian rectangular system of coordinates $OXY$ in the first quadrant on the curve $y = f(x)$,where $O$ is the origin. The area bounded by the curve $y = f(x)$ and the line $y = 3$ is: (in $/3$)

Let $q$ be the maximum integral value of $p$ in $[0, 10]$ for which the roots of the equation $x^2 - px + \frac{5}{4}p = 0$ are rational. Then the area of the region $\{(x, y): 0 \leq y \leq (x - q)^2, 0 \leq x \leq q\}$ is