$\underset{0}{\overset{2}{\int }}\left[x^2\right]dx=$

$\begin{array}{l}\text{ G.I }={\int }_0^1 \left[x^2\right]dx+{\int }_1^{\sqrt{2}} \left[x^2\right]dx+{\int }_{\sqrt{2}}^{\sqrt{3}} \left[x^2\right]dx+{\int }_{\sqrt{3}}^2 \left[x^2\right]dx\\ ={\int }_0^1 odx+{\int }_1^{\sqrt{2}} \left(1\right)dx+{\int }_{\sqrt{2}}^{\sqrt{3}} \left(2\right)dx+{\int }_{\sqrt{3}}^2 \left(3\right)dx\\ =0+(x{)}_1^{\sqrt{2}}+2(x{)}_{\sqrt{2}}^{\sqrt{3}}+3(x{)}_{\sqrt{3}}^2\\ =\sqrt{2}-1+2\sqrt{3}-2\sqrt{2}+6-3\sqrt{3}\\ =5-\sqrt{2}-\sqrt{3}=5-1.4142-1.7320\\ =5-3.1462\\ =1.8538\end{array}$