The area covered by the curve `y=max{2-x,2,1+x}` with `x`-axis from `x=-1` to `x=1` is `(alpha)/(beta)`, then `alpha-beta=` (where `alpha, beta` are coprime to each other)

Explanation

To solve the problem of finding the area covered by the curve \( y = \max\{2-x, 2, 1+x\} \) with the x-axis from \( x = -1 \) to \( x = 1 \), we will follow these steps:

### Step 1: Identify the functions
We have three functions:
1. \( y = 2 - x \)
2. \( y = 2 \)
3. \( y = 1 + x \)

### Step 2: Determine the points of intersection
We need to find the points where these functions intersect within the interval \( x = -1 \) to \( x = 1 \).

1. Set \( 2 - x = 2 \):
\[ 2 - x = 2 \implies x = 0 \]
So, the intersection point is \( (0, 2) \).

2. Set \( 2 - x = 1 + x \):
\[ 2 - x = 1 + x \implies 2 - 1 = 2x \implies x = \frac{1}{2} \]
So, the intersection point is \( \left(\frac{1}{2}, \frac{3}{2}\right) \).

3. Set \( 1 + x = 2 \):
\[ 1 + x = 2 \implies x = 1 \]
So, the intersection point is \( (1, 2) \).

### Step 3: Determine the maximum function in the interval
Now we need to evaluate the maximum of these functions in the interval \( x = -1 \) to \( x = 1 \):

- At \( x = -1 \):
\[ y = 2 - (-1) = 3, \quad y = 2, \quad y = 1 + (-1) = 0 \quad \Rightarrow \quad \max = 3 \]

- At \( x = 0 \):
\[ y = 2 - 0 = 2, \quad y = 2, \quad y = 1 + 0 = 1 \quad \Rightarrow \quad \max = 2 \]

- At \( x = \frac{1}{2} \):
\[ y = 2 - \frac{1}{2} = \frac{3}{2}, \quad y = 2, \quad y = 1 + \frac{1}{2} = \frac{3}{2} \quad \Rightarrow \quad \max = 2 \]

- At \( x = 1 \):
\[ y = 2 - 1 = 1, \quad y = 2, \quad y = 1 + 1 = 2 \quad \Rightarrow \quad \max = 2 \]

### Step 4: Define the piecewise function
From the evaluations, we can define the maximum function:
\[ y = \begin{cases} 3 & \text{for } x = -1 \\ 2 & \text{for } 0 \leq x \leq 1 \end{cases} \]

### Step 5: Calculate the area
The area under the curve from \( x = -1 \) to \( x = 1 \) consists of two parts:
1. From \( x = -1 \) to \( x = 0 \) (area of a rectangle):
\[ \text{Area}_1 = \text{Base} \times \text{Height} = 1 \times 3 = 3 \]

2. From \( x = 0 \) to \( x = 1 \) (area of a rectangle):
\[ \text{Area}_2 = \text{Base} \times \text{Height} = 1 \times 2 = 2 \]

Total area:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 3 + 2 = 5 \]

### Step 6: Express the area in the form \( \frac{\alpha}{\beta} \)
We can express the total area as:
\[ \frac{5}{1} \]
Here, \( \alpha = 5 \) and \( \beta = 1 \).

### Step 7: Find \( \alpha - \beta \)
Since \( \alpha = 5 \) and \( \beta = 1 \):
\[ \alpha - \beta = 5 - 1 = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]