The area of the curve x + |y| = 1 and the y-axis is

Options

A. 1 sq. units
B. 2 sq. units
C. 1/2 sq. units
D. gt 1/2 sq. unit

Answer

To find the area of the region bounded by the curve \( x + |y| = 1 \) and the y-axis, we can follow these steps:

### Step 1: Understand the Equation
The equation \( x + |y| = 1 \) can be split into two cases based on the definition of the absolute value.

- **Case 1**: When \( y \geq 0 \), we have:
\[ x + y = 1 \quad \Rightarrow \quad y = 1 - x \]

- **Case 2**: When \( y < 0 \), we have:
\[ x - y = 1 \quad \Rightarrow \quad y = x - 1 \]

### Step 2: Identify Points of Intersection
Next, we find the points where these lines intersect the axes.

- For \( y = 1 - x \):
- When \( x = 0 \), \( y = 1 \) (point \( (0, 1) \))
- When \( y = 0 \), \( x = 1 \) (point \( (1, 0) \))

- For \( y = x - 1 \):
- When \( x = 0 \), \( y = -1 \) (point \( (0, -1) \))
- When \( y = 0 \), \( x = 1 \) (point \( (1, 0) \))

### Step 3: Sketch the Region
The lines \( y = 1 - x \) and \( y = x - 1 \) form a V-shape. The region bounded by these lines and the y-axis is symmetric about the x-axis. The area we need to calculate is between the y-axis and the lines from \( (0, 1) \) to \( (1, 0) \) and from \( (0, -1) \) to \( (1, 0) \).

### Step 4: Set Up the Integral
To find the area, we can calculate the area above the x-axis and double it (due to symmetry). The area above the x-axis can be calculated using the integral:
\[ \text{Area} = 2 \int_{0}^{1} (1 - x) \, dx \]

### Step 5: Evaluate the Integral
Now, we evaluate the integral:
\[ \int_{0}^{1} (1 - x) \, dx = \left[ x - \frac{x^2}{2} \right]_{0}^{1} = \left( 1 - \frac{1}{2} \right) - (0 - 0) = \frac{1}{2} \]
Thus, the area above the x-axis is \( \frac{1}{2} \).

### Step 6: Calculate the Total Area
Since we have symmetry, the total area is:
\[ \text{Total Area} = 2 \times \frac{1}{2} = 1 \]

### Final Answer
The area of the region bounded by the curve \( x + |y| = 1 \) and the y-axis is \( 1 \) square unit.

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