The area of one region included between the $sine$ and $cosine$ curves is given by

The area enclosed by the graph of the function $y = \ln^2 x - 1$ lying in the $4^{th}$ quadrant is:

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

The slope of the tangent to a curve $y=f(x)$ at $(x, f(x))$ is $2x+1$. If the curve passes through the point $(1,2)$,then the area (in sq. units),bounded by the curve,the $X$-axis and the line $x=1$,is

The area of the region bounded by the curve $y=x^2$ and the line $y=16$ is

The area bounded by the curve $y = x |\sin(\frac{x}{2})|$ and the $X$-axis between the lines $x = 0$ and $x = 4\pi$ (in square units) is: (in $\pi$)