Can't solve double integral with Gauss mark

Question

evaluate the integral

$$\int \int_D [x+y]dA \qquad , D=[1,3]\times[2,5]$$

(let [x] denote the greatest integer in x )

Answer 1

Have a look at the following figure displaying surface with equation

$$z=[x+y]$$

Nice staircase, isn't it ? A projection is as follows :

It remains to compute the volumes of the different prismatic shapes using formula : volume = base area times height) and add them.

Hint : The computation can be eased by "piling up" some of these prismatic shapes. For example the two extreme ones, under the form $V=B\times h=1 \times (3+7)$.

Edit : in fact, I just saw a very similar question : Let $Q:=[0,2] \times [0,2]$ ; then how to evaluate ${\int\int}_Q\lfloor x+y\rfloor dxdy$ ?. A comment of Achille Hui was very enlightning, extending the idea above : just pile up a copy of the global given volume, upside down and reversed at 180°, on the first one in order to obtain a parallelepiped (a "brick") with volume $2 \times 3 \times 10=60$... and finally divide by $2$.