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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 \(\displaystyle\int_{0}^{1/A} f(x) \, dx\).