How to show that $\int_{a}^{b}\left \lfloor x \right \rfloor dx+ \int_{a}^{b}\left \lfloor -x \right \rfloor dx=a-b$? How to show that
$$
\int_{a}^{b}\left \lfloor x \right \rfloor dx+ \int_{a}^{b}\left \lfloor -x \right \rfloor dx=a-b
$$
Where $\left \lfloor x \right \rfloor$ means greatest integer $\leqslant x$.

What happens when you add $\lfloor x \rfloor$ and $\lfloor -x \rfloor$? Can you see how this relates to the bound of the integral and the answer you want?
