Area of the region bounded by `[x] ^(2) =[y] ^(2), if x in [1,5]`, where [ ] denotes the greatest integer function is:

Answer

To find the area of the region bounded by the equations \([x]^2 = [y]^2\) for \(x \in [1, 5]\), where \([ ]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example:
- \([1] = 1\)
- \([1.5] = 1\)
- \([2] = 2\)
- \([3.9] = 3\)

### Step 2: Determine the Values of \([x]\) and \([y]\)
For \(x \in [1, 5]\), the possible values of \([x]\) are:
- \([1] = 1\) when \(1 \leq x < 2\)
- \([2] = 2\) when \(2 \leq x < 3\)
- \([3] = 3\) when \(3 \leq x < 4\)
- \([4] = 4\) when \(4 \leq x < 5\)
- \([5] = 5\) when \(x = 5\)

### Step 3: Set Up the Conditions for \([x]^2 = [y]^2\)
Since \([x]^2 = [y]^2\), we can derive the following conditions based on the values of \([x]\):
1. If \([x] = 1\), then \([y] = 1\) or \([y] = -1\)
2. If \([x] = 2\), then \([y] = 2\) or \([y] = -2\)
3. If \([x] = 3\), then \([y] = 3\) or \([y] = -3\)
4. If \([x] = 4\), then \([y] = 4\) or \([y] = -4\)

### Step 4: Identify the Bounded Regions
We need to find the area of the regions defined by these conditions:
- For \([x] = 1\) (i.e., \(1 \leq x < 2\)), \([y] = 1\) gives \(1 \leq y < 2\).
- For \([x] = 2\) (i.e., \(2 \leq x < 3\)), \([y] = 2\) gives \(2 \leq y < 3\).
- For \([x] = 3\) (i.e., \(3 \leq x < 4\)), \([y] = 3\) gives \(3 \leq y < 4\).
- For \([x] = 4\) (i.e., \(4 \leq x < 5\)), \([y] = 4\) gives \(4 \leq y < 5\).

### Step 5: Calculate the Area of Each Region
Each of these regions forms a square of side length 1 unit:
1. Area for \([x] = 1\) and \([y] = 1\): \(1 \times 1 = 1\)
2. Area for \([x] = 2\) and \([y] = 2\): \(1 \times 1 = 1\)
3. Area for \([x] = 3\) and \([y] = 3\): \(1 \times 1 = 1\)
4. Area for \([x] = 4\) and \([y] = 4\): \(1 \times 1 = 1\)

### Step 6: Total Area
The total area is the sum of the areas of all four regions:
\[ \text{Total Area} = 1 + 1 + 1 + 1 = 4 \text{ square units} \]

### Final Answer
The area of the region bounded by \([x]^2 = [y]^2\) for \(x \in [1, 5]\) is \(4\) square units.

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