# Filled glasses

Calculus Level 3

$f\left( x \right) =\sec ^{ 2 }{ x-\csc ^{ 2 }{ x } }$

The yellow region shown in the picture can be formed by bounding $$f\left( x \right)$$ with two horizontal lines at $$y=8\sqrt { 3 }$$ and $$y=-8\sqrt { 3 }$$.

Considering the periodic properties of $$f\left( x \right)$$, the area of any region like this one can be expressed in the form $$\frac { a\pi }{ \sqrt { b } }$$, with $$a$$ a highly composite number and $$b$$ a prime integer. Find $$\frac { a }{ b }$$.

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