Area bounded by the curves `y=abs(x-1),y=0 and abs(x)=2 is 5`. True or False

Answer

To determine whether the area bounded by the curves \( y = |x - 1| \), \( y = 0 \), and \( |x| = 2 \) is equal to 5, we will analyze the curves and calculate the area step by step.

### Step 1: Understand the curves
1. **Curve 1:** \( y = |x - 1| \)
- This can be expressed as:
- \( y = x - 1 \) for \( x \geq 1 \)
- \( y = -x + 1 \) for \( x < 1 \)

2. **Curve 2:** \( y = 0 \)
- This is simply the x-axis.

3. **Curve 3:** \( |x| = 2 \)
- This gives us two vertical lines:
- \( x = 2 \)
- \( x = -2 \)

### Step 2: Identify the points of intersection
- The curves intersect at the following points:
- For \( y = |x - 1| \) and \( y = 0 \):
- \( |x - 1| = 0 \) gives \( x = 1 \).
- For \( |x| = 2 \):
- The vertical lines are at \( x = -2 \) and \( x = 2 \).

### Step 3: Sketch the graph
- The graph will consist of:
- A V-shaped graph for \( y = |x - 1| \) with a vertex at (1, 0).
- The x-axis (y = 0).
- Two vertical lines at \( x = -2 \) and \( x = 2 \).

### Step 4: Determine the bounded area
- The area bounded by these curves can be divided into two triangles:
1. **Triangle A1:** Bounded by \( x = -2 \), \( x = 1 \), and the x-axis.
2. **Triangle A2:** Bounded by \( x = 1 \), \( x = 2 \), and the line \( y = |x - 1| \).

#### Area of Triangle A1:
- Base = distance from \( -2 \) to \( 1 \) = \( 1 - (-2) = 3 \).
- Height = value of \( y \) at \( x = -2 \):
- \( y = |-2 - 1| = 3 \).
- Area \( A1 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 3 = \frac{9}{2} \).

#### Area of Triangle A2:
- Base = distance from \( 1 \) to \( 2 \) = \( 2 - 1 = 1 \).
- Height = value of \( y \) at \( x = 2 \):
- \( y = |2 - 1| = 1 \).
- Area \( A2 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \).

### Step 5: Total area
- Total Area \( = A1 + A2 = \frac{9}{2} + \frac{1}{2} = \frac{10}{2} = 5 \).

### Conclusion
The area bounded by the curves \( y = |x - 1| \), \( y = 0 \), and \( |x| = 2 \) is indeed \( 5 \). Therefore, the statement is **True**.