2. Let the vectors $\overrightarrow{O A}=(1,0), \overrightarrow{O B}=(1,1)$, and $O$ be the origin. A moving point $P(x, y)$ satisfies
$$
\left\{\begin{array}{l}
0 \leqslant \overrightarrow{O P} \cdot \overrightarrow{O A} \leqslant 1, \\
0 \leqslant \overrightarrow{O P} \cdot \overrightarrow{O B} \leqslant 2
\end{array}\right.
$$

Then the area of the figure formed by the point $Q(x+y, y)$ is
2. 2 .

From the problem, the point $P(x, y)$ satisfies
$$
\begin{array}{l}
\left\{\begin{array}{l}
0 \leqslant x \leqslant 1, \\
0 \leqslant x+y \leqslant 2 .
\end{array}\right. \\
\text { Let }\left\{\begin{array}{l}
x+y=u, \\
y=v .
\end{array}\right.
\end{array}
$$

Then the point $Q(u, v)$ satisfies
$$
\left\{\begin{array}{l}
0 \leqslant u-v \leqslant 1, \\
0 \leqslant u \leqslant 2 .
\end{array}\right.
$$

In the $u O v$ plane, draw
the plane region formed by
the point $Q(u, v)$ (the shaded part in Figure 2), its area equals 2.