Will you integrate?(1)

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 $n$ .

Notations :

• $\lfloor \cdot \rfloor$ denotes the floor function .

• $| \cdot |$ denotes the absolute value function .