If the area of the region $\{(x, y): -1 \leq x \leq 1, 0 \leq y \leq a + e^{|x|} - e^{-x}, a > 0\}$ is $\frac{e^2 + 8e + 1}{e}$,then the value of $a$ is:

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

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

The straight line through the origin which divides the area formed by the curves $y=2x-x^2$, $y=0$, and $x=1$ into two equal halves is

The area of the region bounded by $y=\cos x$,$x=0$,$x=\pi$,and the $x$-axis is ... sq. units.