Area bounded by the curve `[|x|] + [|y|] = 3,` where` [.]` denotes the greatest integer function

Explanation

To find the area bounded by the curve \([|x|] + [|y|] = 3\), where \([.]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Understanding the Equation
The equation \([|x|] + [|y|] = 3\) implies that the sum of the greatest integer values of \(|x|\) and \(|y|\) must equal 3. This means we need to consider different cases based on the values of \(|x|\) and \(|y|\).

### Step 2: Analyze Cases
We will analyze the possible integer pairs \((a, b)\) such that \(a + b = 3\), where \(a = [|x|]\) and \(b = [|y|]\). The possible pairs are:
- \( (0, 3) \)
- \( (1, 2) \)
- \( (2, 1) \)
- \( (3, 0) \)

### Step 3: Determine the Regions
For each pair, we can determine the corresponding regions in the coordinate plane:

1. **For (0, 3)**:
- \(0 \leq |x| < 1\) and \(3 \leq |y| < 4\).
- This corresponds to the rectangle with vertices at \((0, 3)\), \((0, 4)\), \((1, 3)\), and \((1, 4)\).

2. **For (1, 2)**:
- \(1 \leq |x| < 2\) and \(2 \leq |y| < 3\).
- This corresponds to the rectangle with vertices at \((1, 2)\), \((1, 3)\), \((2, 2)\), and \((2, 3)\).

3. **For (2, 1)**:
- \(2 \leq |x| < 3\) and \(1 \leq |y| < 2\).
- This corresponds to the rectangle with vertices at \((2, 1)\), \((2, 2)\), \((3, 1)\), and \((3, 2)\).

4. **For (3, 0)**:
- \(3 \leq |x| < 4\) and \(0 \leq |y| < 1\).
- This corresponds to the rectangle with vertices at \((3, 0)\), \((3, 1)\), \((4, 0)\), and \((4, 1)\).

### Step 4: Calculate the Area of Each Region
Each of the rectangles has a width of 1 and a height of 1, thus the area of each rectangle is:
\[ \text{Area} = \text{width} \times \text{height} = 1 \times 1 = 1 \text{ square unit} \]

### Step 5: Total Area
Since there are 4 such rectangles (one for each case), the total area bounded by the curve is:
\[ \text{Total Area} = 4 \times 1 = 4 \text{ square units} \]

### Final Answer
The area bounded by the curve \([|x|] + [|y|] = 3\) is \(4\) square units.

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