To find the area of the region enclosed by the circle \( x^2 + y^2 = 1 \) that is not common to the region bounded by \( |x + y| \leq 1 \) and \( |x - y| \leq 1 \), we can follow these steps:
### Step 1: Understand the Circle
The equation \( x^2 + y^2 = 1 \) represents a circle with:
- Center at the origin (0, 0)
- Radius \( r = 1 \)
### Step 2: Understand the Bounded Region
The inequalities \( |x + y| \leq 1 \) and \( |x - y| \leq 1 \) define a square in the coordinate plane.
1. **For \( |x + y| \leq 1 \)**:
- This can be split into two inequalities:
- \( x + y \leq 1 \)
- \( x + y \geq -1 \)
- The lines \( x + y = 1 \) and \( x + y = -1 \) intersect the axes at (1, 0), (0, 1) and (-1, 0), (0, -1) respectively.
2. **For \( |x - y| \leq 1 \)**:
- This can also be split into two inequalities:
- \( x - y \leq 1 \)
- \( x - y \geq -1 \)
- The lines \( x - y = 1 \) and \( x - y = -1 \) intersect the axes at (1, 0), (0, -1) and (-1, 0), (0, 1) respectively.
### Step 3: Graph the Regions
- The intersection of these inequalities forms a square with vertices at (1, 0), (0, 1), (-1, 0), and (0, -1).
- The area of this square can be calculated as follows:
- The side length of the square is \( \sqrt{2} \) (distance between (1, 0) and (0, 1)).
- Area of the square = side length² = \( (\sqrt{2})^2 = 2 \).
### Step 4: Calculate the Area of the Circle
- The area of the circle is given by the formula:
\[
\text{Area}_{\text{circle}} = \pi r^2 = \pi \cdot 1^2 = \pi.
\]
### Step 5: Find the Required Area
- The area of the region enclosed by the circle that is not common to the square is given by:
\[
\text{Required Area} = \text{Area}_{\text{circle}} - \text{Area}_{\text{square}} = \pi - 2.
\]
### Final Answer
The area of the region enclosed by the circle \( x^2 + y^2 = 1 \) that is not common to the region bounded by \( |x + y| \leq 1 \) and \( |x - y| \leq 1 \) is:
\[
\boxed{\pi - 2}.
\]
