The area (in sq. units) bounded by `[|x|]+[|y|]=2` in the first and third quardant is (where `[.]` is the greatest integer function).

Options

A. 4
B. 3
C. 6
D. 10

Answer

To find the area bounded by the equation \([|x|] + [|y|] = 2\) in the first and third quadrants, we will follow these steps:

### Step 1: Understand the Equation
The equation \([|x|] + [|y|] = 2\) involves the greatest integer function (also known as the floor function). This means we need to consider the integer values of \(|x|\) and \(|y|\).

### Step 2: Analyze the First Quadrant
In the first quadrant, both \(x\) and \(y\) are positive, so we can rewrite the equation as:
\[ [x] + [y] = 2 \]
This equation can be satisfied by different combinations of \([x]\) and \([y]\):
1. \([x] = 2\) and \([y] = 0\)
2. \([x] = 1\) and \([y] = 1\)
3. \([x] = 0\) and \([y] = 2\)

### Step 3: Determine the Ranges for Each Case
1. **Case 1**: \([x] = 2\) and \([y] = 0\)
- This implies \(2 \leq x < 3\) and \(0 \leq y < 1\).
- The area for this case is a rectangle with width \(1\) (from \(2\) to \(3\)) and height \(1\) (from \(0\) to \(1\)).
- Area = \(1 \times 1 = 1\).

2. **Case 2**: \([x] = 1\) and \([y] = 1\)
- This implies \(1 \leq x < 2\) and \(1 \leq y < 2\).
- The area for this case is a square with side length \(1\).
- Area = \(1 \times 1 = 1\).

3. **Case 3**: \([x] = 0\) and \([y] = 2\)
- This implies \(0 \leq x < 1\) and \(2 \leq y < 3\).
- The area for this case is a rectangle with width \(1\) (from \(0\) to \(1\)) and height \(1\) (from \(2\) to \(3\)).
- Area = \(1 \times 1 = 1\).

### Step 4: Calculate Total Area in the First Quadrant
Adding the areas from all three cases:
\[ \text{Total Area (First Quadrant)} = 1 + 1 + 1 = 3 \]

### Step 5: Consider the Third Quadrant
Due to symmetry, the area in the third quadrant will be the same as in the first quadrant. Therefore:
\[ \text{Total Area (Third Quadrant)} = 3 \]

### Step 6: Calculate Total Area
The total area bounded by the equation in both the first and third quadrants is:
\[ \text{Total Area} = \text{Area (First Quadrant)} + \text{Area (Third Quadrant)} = 3 + 3 = 6 \]

### Final Answer
The area bounded by \([|x|] + [|y|] = 2\) in the first and third quadrants is \(6\) square units.

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