Is the integral actually zero?

Calculus Level 5

The following shown here is a portion of the graph for \(f\left( x \right) =\sqrt { 1-\tan ^{ 2 }{ x } }\), in which all regions bound by the \(x\)-axis and \(f\left( x \right)\) are shaded.

Considering the periodic properties of \(f\left( x \right)\), the area of any shaded region can be expressed in the form \(\pi \tan { \frac { \pi }{ A } }\), with \({ A }\) being a positive, root-free integer. Find \({ A }\).


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

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