$\text{ If }\left[\mathrm{x}\right]\text{ is greatest integer function, then }{\int }_0^3 x^2\left[x\right]dx=$

$\begin{array}{l}{\int }_0^3 x^2\left[x\right]dx={\int }_0^1 0dx+{\int }_1^2 x^2\left(1\right)dx+{\int }_2^3 x^2\left(2\right)dx\\ =0+{\left[\frac{x^3}{3}\right]}_1^2+2{\left[\frac{x^3}{3}\right]}_2^3=\frac{8}{3}-\frac{1}{3}+\frac{54}{3}-\frac{16}{3}=\frac{62-17}{3}=15=0015.00\end{array}$