Find the area of curve enclosed by `|x+y|+|x-y|le4,|x|le1, y ge sqrt(x^(2)-2x+1)`.

Answer

To find the area of the region enclosed by the inequalities given in the problem, we will follow these steps:

### Step 1: Analyze the inequalities
We have three inequalities to consider:
1. \( |x+y| + |x-y| \leq 4 \)
2. \( |x| \leq 1 \)
3. \( y \geq \sqrt{x^2 - 2x + 1} \)

### Step 2: Break down the first inequality
The first inequality \( |x+y| + |x-y| \leq 4 \) can be analyzed by considering different cases based on the signs of \( x+y \) and \( x-y \).

#### Case 1: \( x+y \geq 0 \) and \( x-y \geq 0 \)
In this case, we have:
\[ x+y + x-y \leq 4 \implies 2x \leq 4 \implies x \leq 2 \]

#### Case 2: \( x+y \geq 0 \) and \( x-y < 0 \)
Here, we get:
\[ x+y - (x-y) \leq 4 \implies 2y \leq 4 \implies y \leq 2 \]

#### Case 3: \( x+y < 0 \) and \( x-y \geq 0 \)
This gives us:
\[ -(x+y) + (x-y) \leq 4 \implies -2y \leq 4 \implies y \geq -2 \]

#### Case 4: \( x+y < 0 \) and \( x-y < 0 \)
We find:
\[ -(x+y) - (x-y) \leq 4 \implies -2x \leq 4 \implies x \geq -2 \]

### Step 3: Combine results from the first inequality
From the analysis, we have:
- \( x \leq 2 \)
- \( y \leq 2 \)
- \( y \geq -2 \)
- \( x \geq -2 \)

### Step 4: Analyze the second inequality
The second inequality \( |x| \leq 1 \) restricts \( x \) to the interval:
\[ -1 \leq x \leq 1 \]

### Step 5: Analyze the third inequality
The third inequality \( y \geq \sqrt{x^2 - 2x + 1} \) can be simplified:
\[ y \geq \sqrt{(x-1)^2} \]
This means:
\[ y \geq |x-1| \]

### Step 6: Graph the inequalities
We now need to graph the regions defined by:
1. \( -1 \leq x \leq 1 \)
2. \( y \leq 2 \)
3. \( y \geq |x-1| \)

The line \( y = |x-1| \) forms a V-shape with its vertex at \( (1, 0) \). The lines \( y = 2 \) and the vertical lines \( x = -1 \) and \( x = 1 \) will help us find the bounded region.

### Step 7: Identify the bounded region
The bounded region is where all these inequalities overlap. The area of interest is a triangle formed by the points:
- \( (-1, 2) \)
- \( (1, 2) \)
- \( (1, 0) \)

### Step 8: Calculate the area of the triangle
The base of the triangle is the distance between \( (-1, 2) \) and \( (1, 2) \), which is \( 2 - (-1) = 2 \).
The height of the triangle is the distance from the line \( y = 2 \) down to the point \( (1, 0) \), which is \( 2 - 0 = 2 \).

Using the area formula for a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2 \]

### Final Answer:
The area of the region enclosed by the given inequalities is \( 2 \) square units.