Question

If $f\left(x\right)={\left[x\right]}^2-\left[x^2\right]$ where $[\cdot ]$ denotes greatest integer function

then the value of ${\int }_{\sqrt{2}}^{\sqrt{3}}f\left(x\right)dx$ equals?

• A. $0$
• B. $\sqrt{2}-\sqrt{3}$
• C. $\sqrt{3}-\sqrt{2}$
• D. None of these

Answer 1

The correct option is B $\sqrt{2}-\sqrt{3}$

In the interval $[\sqrt{2},\sqrt{3})$, $[x{]}^2=1$ and $\left[x^2\right]=2$

$⟹f\left(x\right)=[x{]}^2-[x^2]=1-2=-1$

Hence, ${\int }_{\sqrt{2}}^{\sqrt{3}}f\left(x\right)dx$

$={\int }_{\sqrt{2}}^{\sqrt{3}}(-1)dx$

$=-(\sqrt{3}-\sqrt{2})$

$=\sqrt{2}-\sqrt{3}$

Hence, answer is option-(B).