### Double integral involving greatest integer function and change of variables

In order to evaluate $\displaystyle\int_{0}^{2}\int_{0}^{2}\lfloor x+y \rfloor dx \,dy$, I transformed co-ordinates using $x= u-v$ and $y=v$. Then the Jacobian becomes $1$ and $0 \leq u \leq 4$ and $0 \leq v \leq 2$ implying that$\displaystyle\int_{0}^{2}\int_{0}^{2}\lfloor x+y \rfloor dx \,dy = \displaystyle\int_{0}^{2}\int_{0}^{4}\lfloor u \rfloor du\,dv = \displaystyle\int_{0}^{2}6\, dv = 12$However the answer on Wolfram alpha is $6$. Can someone point out where I went wrong?...Read more