Given that $y=\tan x$, use the quotient rule to show that

$$\frac{dy}{dx}={\sec }^{2}x$$

The cross-section of a high-precision optical lens is modeled by the region bounded by the curve $y=3{\tan }^{2}x+2$, the $x$-axis, and the vertical boundaries $x=\frac{\pi }{6}$ and $x=\frac{\pi }{3}$.

Show that the area of this cross-section is

$$3\sqrt{3}-\sqrt{3}-\frac{\pi }{6}$$

which simplifies to

$$2\sqrt{3}-\frac{\pi }{6}$$

Fully justify your answer.