$\underset{0}{\overset{25}{\int }}\left[\sqrt{x}\right]dx=$

$\underset{0}{\overset{25}{\int }}\left[\sqrt{x}\right]dx$

$=\underset{0}{\overset{1}{\int }}\left[\sqrt{x}\right]dx+\underset{1}{\overset{4}{\int }}\left[\sqrt{x}\right]dx+\underset{4}{\overset{9}{\int }}\left[\sqrt{x}\right]dx+\underset{9}{\overset{16}{\int }}\left[\sqrt{x}\right]dx+\underset{16}{\overset{25}{\int }}\left[\sqrt{x}\right]dx$

$=\underset{0}{\overset{1}{\int }}0dx+\underset{1}{\overset{4}{\int }}dx+\underset{4}{\overset{9}{\int }}2dx+\underset{9}{\overset{16}{\int }}3dx+\underset{16}{\overset{25}{\int }}4dx$

$=0+{\left[x\right]}_1^4+{\left[2x\right]}_4^9+{\left[3x\right]}_9^{16}+{\left[4x\right]}_{16}^{25}$

$=3+10+21+36$

$=70$