The value of $\tan ^{-1}\left\{\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right\}+\frac{1}{2} \cos ^{-1} x$ is

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be three non-zero vectors such that no two of them are collinear and $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=\frac{1}{3}|\overline{\mathrm{~b}}||\overline{\mathrm{c}}| \overline{\mathrm{a}}$. If ' $\theta$ ' is the angle between the vectors $\bar{b}$ and $\bar{c}$, then value of $\sin \theta$ is

In an experiment with 15 observations for $x$, the following results were available $\sum x^2=2830, \sum x=170$. One observation 20 was found to be wrong and was replaced by the correct value 30 . Then the corrected variance is

The area of the region lying in the first quadrant by $y=4 x^2, x=0, y=2, y=4$ is