Step 1: Reading the region.
In the first quadrant, $|x|=x$ and $|y|=y$. So the boundary is the line $x+y=1$ together with the two coordinate axes.
Step 2: Use the intercept form.
The line can be written as $\frac{x}{1}+\frac{y}{1}=1$. It has x-intercept 1 and y-intercept 1.
Step 3: Use the area of a triangle.
A triangle formed by the axes and a line with intercepts $a$ and $b$ has area $\frac{1}{2}ab$.
\[ \frac{1}{2}\times 1\times 1=\frac{1}{2} \]
Step 4: Cross-check.
The full diamond $|x|+|y|\leq 1$ has diagonals of length 2, so its area is $\frac{1}{2}\times 2\times 2=2$. It has four equal parts, one in each quadrant. So each quadrant part has area $\frac{2}{4}=\frac{1}{2}$.
Final Answer:
Option 2 is correct. \[ \boxed{\frac{1}{2}\ \text{square units}} \]