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Calculus Level 4

$\large f(x) = \lim_{n\to\infty} \dfrac{\lfloor x\rfloor+\lfloor 2x\rfloor+\lfloor 3x\rfloor+\cdots +\lfloor nx\rfloor}{n^2}$

Let $f(x)$ be defined as above and $A$ be the area bounded by $f(x)$ and $y=x^2$ . Then find $\displaystyle\int_{0}^{1/A} f(x) \, dx$ .