$$\begin{gathered} \underset{0}{\overset{1}{\int }}dx\underset{0}{\overset{1}{\int }}(x+y)dy= \\ =\underset{0}{\overset{1}{\int }}dx\left({xy+\frac{y^2}{2}|}_0^1\right)=\underset{0}{\overset{1}{\int }}\left(x+\frac{1}{2}\right)dx={\left(\frac{x^2}{2}+\frac{x}{2}\right)|}_0^1=1 \end{gathered}$$