To find the area enclosed by the curve \( x + |y| = 1 \) and the y-axis, we can break this down into steps:
### Step 1: Analyze the equation
The equation \( x + |y| = 1 \) can be split into two cases based on the definition of the absolute value.
- **Case 1:** When \( y \geq 0 \), \( |y| = y \). Thus, the equation becomes:
\[
x + y = 1 \quad \text{(1)}
\]
- **Case 2:** When \( y < 0 \), \( |y| = -y \). Thus, the equation becomes:
\[
x - y = 1 \quad \text{(2)}
\]
### Step 2: Find the intercepts
We will find the intercepts for both equations to understand the shape of the curves.
- **For equation (1)** \( x + y = 1 \):
- When \( x = 0 \): \( y = 1 \) (Point A: \( (0, 1) \))
- When \( y = 0 \): \( x = 1 \) (Point B: \( (1, 0) \))
- **For equation (2)** \( x - y = 1 \):
- When \( x = 0 \): \( -y = 1 \) or \( y = -1 \) (Point C: \( (0, -1) \))
- When \( y = 0 \): \( x = 1 \) (Point B: \( (1, 0) \))
### Step 3: Sketch the curves
The lines from the equations:
- The line from equation (1) will be in the first and second quadrants.
- The line from equation (2) will be in the third and fourth quadrants.
### Step 4: Identify the area
The area we need to calculate is bounded by the y-axis and the two lines. The area can be divided into two triangles:
1. Triangle OAB (formed by points O(0,0), A(0,1), B(1,0))
2. Triangle OCB (formed by points O(0,0), C(0,-1), B(1,0))
### Step 5: Calculate the area of triangle OAB
The area \( A \) of triangle OAB can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base \( OB = 1 \) and the height \( OA = 1 \):
\[
A_{OAB} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}
\]
### Step 6: Calculate the total area
Since triangle OCB is symmetric to triangle OAB, its area will also be \( \frac{1}{2} \). Therefore, the total area \( A \) is:
\[
\text{Total Area} = A_{OAB} + A_{OCB} = \frac{1}{2} + \frac{1}{2} = 1
\]
### Final Answer
The area of the region bounded by the curve \( x + |y| = 1 \) and the y-axis is \( 1 \) square unit.
