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

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

$[step of \sqrt{x}]$

$make \sqrt{x}=0,1,2,3,4,5$

$x=0,1^2,2^2,3^2,4^2,5^2$

$x=0,1,4,9,16,25$

split the integral at the above points

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

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

$=0+1\left(3\right)+2\left(5\right)+3\left(7\right)+4\left(9\right)$

$=70$