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 can 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 that \(|x|\) and \(|y|\) can take.

### Step 2: Analyze in the First Quadrant
In the first quadrant, both \(x\) and \(y\) are non-negative. Therefore, we can rewrite the equation as:
\[ [x] + [y] = 2 \]
Here, \([x]\) and \([y]\) can take the following pairs of values that satisfy the equation:
- \([x] = 0\) and \([y] = 2\)
- \([x] = 1\) and \([y] = 1\)
- \([x] = 2\) and \([y] = 0\)

### Step 3: Determine the Ranges for Each Case
1. **For \([x] = 0\) and \([y] = 2\)**:
- This means \(0 \leq x < 1\) and \(2 \leq y < 3\).
- This forms a rectangle with vertices at \((0, 2)\), \((1, 2)\), \((1, 3)\), and \((0, 3)\).

2. **For \([x] = 1\) and \([y] = 1\)**:
- This means \(1 \leq x < 2\) and \(1 \leq y < 2\).
- This forms a square with vertices at \((1, 1)\), \((2, 1)\), \((2, 2)\), and \((1, 2)\).

3. **For \([x] = 2\) and \([y] = 0\)**:
- This means \(2 \leq x < 3\) and \(0 \leq y < 1\).
- This forms a rectangle with vertices at \((2, 0)\), \((3, 0)\), \((3, 1)\), and \((2, 1)\).

### Step 4: Calculate the Area in the First Quadrant
Now, we calculate the area of each region:
1. Area of the rectangle from case 1:
\[ \text{Area} = \text{length} \times \text{width} = 1 \times 1 = 1 \text{ sq. unit} \]
2. Area of the square from case 2:
\[ \text{Area} = 1 \times 1 = 1 \text{ sq. unit} \]
3. Area of the rectangle from case 3:
\[ \text{Area} = 1 \times 1 = 1 \text{ sq. unit} \]

Total area in the first quadrant:
\[ \text{Total Area} = 1 + 1 + 1 = 3 \text{ sq. units} \]

### Step 5: Analyze in the Third Quadrant
In the third quadrant, both \(x\) and \(y\) are negative. The analysis is similar to the first quadrant:
- The same pairs of values apply, but we consider \(-x\) and \(-y\):
1. \([x] = 0\) and \([y] = 2\) corresponds to \(-1 < x \leq 0\) and \(-3 < y \leq -2\).
2. \([x] = 1\) and \([y] = 1\) corresponds to \(-2 < x \leq -1\) and \(-2 < y \leq -1\).
3. \([x] = 2\) and \([y] = 0\) corresponds to \(-3 < x \leq -2\) and \(-1 < y \leq 0\).

The areas calculated in the third quadrant will also be the same as in the first quadrant, giving us another total of 3 sq. units.

### Step 6: Combine Areas from Both Quadrants
Finally, the total area bounded by the equation \([|x|] + [|y|] = 2\) in the first and third quadrants is:
\[ \text{Total Area} = 3 + 3 = 6 \text{ sq. units} \]

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