Find the area enclosed by the curves
`max(|x|,|y|)=1`

Answer

To find the area enclosed by the curves defined by the equation `max(|x|, |y|) = 1`, we can follow these steps:

### Step 1: Understand the equation
The equation `max(|x|, |y|) = 1` means that either `|x|` or `|y|` must be equal to 1. This gives us four cases to consider:

1. **Case 1**: \( |x| = 1 \) and \( |y| \leq 1 \)
2. **Case 2**: \( |y| = 1 \) and \( |x| \leq 1 \)

### Step 2: Identify the lines
From the above cases, we can derive the following lines:
- From Case 1: \( x = 1 \) and \( x = -1 \) (for \( |y| \leq 1 \))
- From Case 2: \( y = 1 \) and \( y = -1 \) (for \( |x| \leq 1 \))

### Step 3: Plot the lines
Now, we can plot these lines on the Cartesian plane:
- The vertical lines \( x = 1 \) and \( x = -1 \)
- The horizontal lines \( y = 1 \) and \( y = -1 \)

### Step 4: Identify the enclosed area
The intersection of these lines forms a square with vertices at the points:
- \( (1, 1) \)
- \( (1, -1) \)
- \( (-1, 1) \)
- \( (-1, -1) \)

### Step 5: Calculate the area of the square
The side length of the square can be calculated as:
- Side length = Distance between \( x = 1 \) and \( x = -1 \) = \( 1 - (-1) = 2 \)

Thus, the area \( A \) of the square is given by:
\[ A = \text{side}^2 = 2^2 = 4 \text{ square units} \]

### Final Answer
The area enclosed by the curves is \( 4 \) square units.

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