The area (in sq. units) bounded by `y = 2 - |x - 2|` and the x-axis is

Answer

To find the area bounded by the curve \( y = 2 - |x - 2| \) and the x-axis, we can follow these steps:

### Step 1: Understand the Function
The function \( y = 2 - |x - 2| \) can be broken down into two cases based on the definition of the absolute value.

- For \( x < 2 \):
\[ y = 2 - (2 - x) = x \]

- For \( x \geq 2 \):
\[ y = 2 - (x - 2) = 4 - x \]

### Step 2: Identify the Points of Intersection with the x-axis
To find the area bounded by the curve and the x-axis, we need to find where the curve intersects the x-axis (where \( y = 0 \)).

1. For \( x < 2 \):
\[ x = 0 \quad \text{(intersects at the origin)} \]

2. For \( x \geq 2 \):
\[ 4 - x = 0 \implies x = 4 \]

Thus, the points of intersection with the x-axis are \( x = 0 \) and \( x = 4 \).

### Step 3: Sketch the Graph
The graph of the function will consist of two line segments:
- From \( (0, 0) \) to \( (2, 2) \) (line \( y = x \))
- From \( (2, 2) \) to \( (4, 0) \) (line \( y = 4 - x \))

### Step 4: Calculate the Area
The area can be calculated by finding the area of the two triangles formed by the segments of the graph.

1. **Area of Triangle 1** (from \( (0, 0) \) to \( (2, 2) \)):
- Base = 2 (from \( x = 0 \) to \( x = 2 \))
- Height = 2 (at \( x = 2 \))
\[ \text{Area}_1 = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 2 \times 2 = 2 \]

2. **Area of Triangle 2** (from \( (2, 2) \) to \( (4, 0) \)):
- Base = 2 (from \( x = 2 \) to \( x = 4 \))
- Height = 2 (at \( x = 2 \))
\[ \text{Area}_2 = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 2 \times 2 = 2 \]

### Step 5: Total Area
The total area bounded by the curve and the x-axis is:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 2 + 2 = 4 \text{ square units} \]

### Final Answer
The area bounded by \( y = 2 - |x - 2| \) and the x-axis is \( \boxed{4} \) square units.