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 $sq. units$):

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 $y=x^3$, the $x$-axis, $x=-2$, and $x=4$ is

The area bounded by the curve $y = k \sin x$ between $x = \pi$ and $x = 2\pi$ is:

The area of the region bounded by the parabola $y^2 = 12x$ and its latus rectum is . . . . . . sq. units.

If $S$ be the area of the region enclosed by $y=e^{-x^2}, y=0, x=0$,and $x=1$. Then
$(A) S \geq \frac{1}{e}$
$(B) S \geq 1-\frac{1}{e}$
$(C) S \leq \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$
$(D) S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$