The area bounded by `y=||x|-1|` with the x - axis from `x =0` to `x=1` is k square units, then 4k is equal to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the area bounded by the curve \( y = ||x| - 1| \) with the x-axis from \( x = 0 \) to \( x = 1 \), we will follow these steps:

### Step 1: Understand the Function
The function given is \( y = ||x| - 1| \). We can break this down:
1. The inner modulus \( |x| \) is simply \( x \) for \( x \geq 0 \).
2. Therefore, for \( x \geq 0 \), the function simplifies to \( y = |x - 1| \).

### Step 2: Analyze the Function in the Given Range
For \( x \) in the interval \( [0, 1] \):
- When \( x < 1 \), \( |x - 1| = 1 - x \).
- Thus, in the range \( [0, 1] \), the function is \( y = 1 - x \).

### Step 3: Sketch the Graph
The graph of \( y = 1 - x \) is a straight line that starts at \( (0, 1) \) and ends at \( (1, 0) \). The area we need to find is the area under this line and above the x-axis from \( x = 0 \) to \( x = 1 \).

### Step 4: Calculate the Area
The area under the line from \( x = 0 \) to \( x = 1 \) forms a right triangle with:
- Base = 1 (from \( x = 0 \) to \( x = 1 \))
- Height = 1 (from \( y = 0 \) to \( y = 1 \))

The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \]

### Step 5: Find \( k \)
From the area calculation, we have \( k = \frac{1}{2} \).

### Step 6: Calculate \( 4k \)
Now, we need to find \( 4k \):
\[ 4k = 4 \times \frac{1}{2} = 2 \]

### Final Answer
Thus, \( 4k \) is equal to \( 2 \).

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