Question

${\int }_0^{1.5}\left[x^2\right]dx,$ where $[,]$ denotes the greatest integer function, is equal to :

• A. $\sqrt{2}-2$
• B. $2-\sqrt{2}$
• C. $2+\sqrt{2}$
• D. None of these

Answer 1

The correct option is B $2-\sqrt{2}$

${\int }_0^{1.5}\left[x^2\right]dx={\int }_0^1\left[x^2\right]dx+{\int }_1^{\sqrt{2}}\left[x^2\right]dx+{\int }_{\sqrt{2}}^{1.5}\left[x^2\right]dx$
$={\int }_0^{1}0\cdot dx+{\int }_1^{\sqrt{2}}\cdot dx+{\int }_{\sqrt{2}}^{1.5}2\cdot dx$
$=0+(\sqrt{2}-1)+2(1.5-\sqrt{2})=2-\sqrt{2}$