The area of the region defined by `| |x|-|y| | le1 and x^(2)+y^(2)le1` in the xy plane is

Options

A. `pi-2`
B. `2 pi-1`
C. `3 pi`
D. 1

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 xy-plane, we can follow these steps:

### Step 1: Understand the inequalities

1. The inequality \( ||x| - |y|| \leq 1 \) describes a region in the xy-plane where the absolute difference between the absolute values of \( x \) and \( y \) is at most 1.
2. The inequality \( x^2 + y^2 \leq 1 \) describes a circle of radius 1 centered at the origin.

### Step 2: Graph the inequalities

1. **Graph the first inequality** \( ||x| - |y|| \leq 1 \):
- This can be broken down into four cases:
- \( |x| - |y| \leq 1 \) and \( |y| - |x| \leq 1 \).
- The lines \( |x| - |y| = 1 \) and \( |y| - |x| = 1 \) will create a diamond shape (or rhombus) centered at the origin with vertices at (1, 0), (0, 1), (-1, 0), and (0, -1).

2. **Graph the second inequality** \( x^2 + y^2 \leq 1 \):
- This is a circle with radius 1 centered at the origin.

### Step 3: Identify the area of intersection

1. The area we are interested in is the region where the diamond intersects with the circle.
2. The vertices of the diamond are at (1, 0), (0, 1), (-1, 0), and (0, -1), and the circle intersects these points.

### Step 4: Calculate the area of the region

1. The area of the diamond can be calculated using the formula for the area of a rhombus:
\[ \text{Area of diamond} = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. Here, both diagonals are 2 (from -1 to 1), so:
\[ \text{Area of diamond} = \frac{1}{2} \times 2 \times 2 = 2 \]

2. The area of the circle is:
\[ \text{Area of circle} = \pi r^2 = \pi \times 1^2 = \pi \]

3. The area of the intersection can be found by subtracting the area of the triangle formed by the vertices of the diamond from the area of the quarter circle:
- The area of one triangle formed by two vertices of the diamond and the origin is:
\[ \text{Area of triangle} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \]
- Since there are four such triangles, the total area of the triangles is:
\[ \text{Total area of triangles} = 4 \times \frac{1}{2} = 2 \]

4. The area of the region defined by the inequalities is:
\[ \text{Area of intersection} = \text{Area of circle} - \text{Total area of triangles} = \pi - 2 \]

### Final Answer

The area of the region defined by \( ||x| - |y|| \leq 1 \) and \( x^2 + y^2 \leq 1 \) is:
\[ \text{Area} = \pi - 2 \]