5. Let $\Gamma$ be the region formed by the points $(x, y)$ satisfying
$$
\left\{\begin{array}{l}
x \geqslant 0, \\
y \geqslant 0, \\
x+y+[x]+[y] \leqslant 5
\end{array}\right.
$$

The area of the region $\Gamma$ is
$\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$).
5. $\frac{9}{2}$.

When $x+y>3$,
$$
[x]+[y]+\{x\}+\{y\}>3 \text {. }
$$

and $\{x\}+\{y\}\leq1 \Rightarrow[x]+[y] \geqslant 2} \\
\Rightarrow x+y+[x]+[y]>5,
\end{array}
$$

which contradicts the condition.
Therefore, the area of region $\Gamma$ is $\frac{9}{2}$.