\[\large \lim_{n\to\infty} \dfrac{1}{n^2} \sum\limits_{k=0}^{n-1} \left( k \int_{k}^{k+1} \sqrt{(x-k)(k+1-x)} \, dx \right) \]

The above expression can be represented as \( \dfrac{\pi}{a} \), then what is \(a\)?

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