The area of the region defined by ||x| - |y|| `le` 1 and `x^(2) + y^(2) le 1` in the x-y coordinate plane is

Options

A. 1
B. 2
C. `pi`
D. none of these

Answer

To find the area of the region defined by the inequalities \( ||x| - |y|| \leq 1 \) and \( x^2 + y^2 \leq 1 \) in the x-y coordinate plane, we can follow these steps:

### Step 1: Understand the inequalities

1. The inequality \( x^2 + y^2 \leq 1 \) represents the area inside (and including) a circle centered at the origin (0,0) with a radius of 1.
2. The inequality \( ||x| - |y|| \leq 1 \) can be broken down into four linear inequalities:
- \( |x| - |y| \leq 1 \) implies two cases:
- \( x - y \leq 1 \) and \( x + y \leq 1 \) (when \( x \geq 0 \) and \( y \geq 0 \))
- \( -x + y \leq 1 \) and \( -x - y \leq 1 \) (when \( x < 0 \) and \( y < 0 \))
- This leads to the following four lines:
1. \( x - y \leq 1 \)
2. \( x + y \leq 1 \)
3. \( -x + y \leq 1 \)
4. \( -x - y \leq 1 \)

### Step 2: Graph the inequalities

1. **Graph the circle**: The circle \( x^2 + y^2 = 1 \) is drawn with a radius of 1 centered at the origin.
2. **Graph the lines**:
- For \( x - y = 1 \): This line intersects the y-axis at (0, -1) and the x-axis at (1, 0).
- For \( x + y = 1 \): This line intersects the y-axis at (0, 1) and the x-axis at (1, 0).
- For \( -x + y = 1 \): This line intersects the y-axis at (0, 1) and the x-axis at (-1, 0).
- For \( -x - y = 1 \): This line intersects the y-axis at (0, -1) and the x-axis at (-1, 0).

### Step 3: Identify the feasible region

1. The intersection of the lines and the circle will form a square-like shape in the first quadrant, and similar shapes in the other quadrants.
2. The vertices of the feasible region can be found at the intersections of the lines and the circle.

### Step 4: Calculate the area

1. The area of the region can be calculated by finding the area of one of the triangles formed in each quadrant.
2. Each triangle has a base and height of 1 (from the points of intersection).
3. The area of one triangle is given by:
\[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \]
4. Since there are four such triangles (one in each quadrant), the total area is:
\[ \text{Total Area} = 4 \times \frac{1}{2} = 2 \]

### Final Answer

The area of the region defined by the inequalities \( ||x| - |y|| \leq 1 \) and \( x^2 + y^2 \leq 1 \) is **2**.

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