6. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Set

Set
$$
\begin{array}{l}
A=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 1\right\}, \\
B=\left\{(x, y) \mid[x]^{2}+[y]^{2} \leqslant 1\right\} .
\end{array}
$$

Then the area of the plane region represented by $A \cup B$ is
$\qquad$
$6.5+\frac{\pi}{4}$.
When $x \in[-1,0)$, $[x]=-1$. Thus,
$[y]=0, y \in[0,1)$;
When $x \in[0,1)$, $[x]=0$.
Thus, $[y]=-1$ or 0 or $1, y \in[-1,2)$;
When $x \in[1,2)$, $[x]=1$.
Thus, $[y]=0, y \in[0,1)$.
Therefore, the plane region represented by $A \cup B$ consists of five unit squares and one quarter of a unit circle in the third quadrant, with an area of $5+\frac{\pi}{4}$.