This problem involves evaluating the definite integral of $\int_{-3}^{3} (2 - |x|) dx$ by interpreting it geometrically. Here's the step-by-step explanation:

Step 1: Understand the function $2 - |x|$

The function $2 - |x|$ represents a "V"-shaped graph because of the absolute value term. Specifically:

• For $x \geq 0$, $|x| = x$, so $2 - |x| = 2 - x$.
• For $x < 0$, $|x| = -x$, so $2 - |x| = 2 + x$.

Step 2: Sketch the graph of $2 - |x|$

The function $2 - |x|$ is a straight line that forms a triangle:

1. At $x = 0$, the value is $2$.
2. At $x = 2$, the value becomes $0$ (since $2 - |2| = 0$).
3. Similarly, at $x = -2$, the value becomes $0$.

Step 3: Break the integral into parts

The interval $[-3, 3]$ can be divided into:

1. From $x = -3$ to $x = -2$: The function is negative, but geometric interpretation will simplify this.
2. From $x = -2$ to ( final