If the area of the region $\{(x, y): \frac{a}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2, 0 < a < 1\}$ is $(\log_e 2) - \frac{1}{7}$,then the value of $7a - 3$ is equal to:

The area bounded by the curve $x=\log (|y|)$,the lines $x=-1$ and $x=0$ is

The area of the region bounded by the curve $y=9-x^2$,the $X$-axis,and the lines $x=0$ and $x=3$ is . . . . . . sq. units.

The volume of a spherical cap of height $h$ cut off from a sphere of radius $a$ is equal to

The area of the region bounded by the curve $y = 2^{kx}$ and the lines $x = 0$ and $x = 2$ in the first quadrant is $\frac{2}{\ln 2}$. Find the value of $k$.

If $(a, 0); a > 0$ is the point where the curve $y = \sin 2x - \sqrt{3} \sin x$ cuts the $x$-axis first,and $A$ is the area bounded by this part of the curve,the origin,and the positive $x$-axis,then: