Will you integrate?(1)

Calculus Level 5

Let \({A}_{1}\) be the area of polygon with vertices \(P (x,y)\) such that \(\left\lfloor \left| x \right| \right\rfloor +\left\lfloor \left| y \right| \right\rfloor =n\); and \({A}_{2}\) be the area bounded by the curve \(\left| x \right| +\left| y \right| =n+1\). Find value of \(n\) such that \(\left|{A}_{1}-{A}_{2}\right|=96 \text{ unit}^2 \).

Bonus: Generalize for all positive integers \(n\).

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