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 \(\frac { a }{ b }\).


This problem is original. The picture of the graph was produced from Desmos.
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