Given that \(\displaystyle \int_0^4 x^3\sqrt{9+x^2} dx = a\), what is the value of \(\lfloor a \rfloor\)?

**Details and assumptions**

**Greatest Integer Function:** \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer less than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -3.4 \rfloor = -4\).

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