Find the area bounded by the curves `x+2|y|=1` and `x=0` .

Explanation

To find the area bounded by the curves \(x + 2|y| = 1\) and \(x = 0\), we will follow these steps:

### Step 1: Understand the equation \(x + 2|y| = 1\)

This equation can be split into two cases based on the definition of absolute value:

1. When \(y \geq 0\), we have:
\[ x + 2y = 1 \quad \Rightarrow \quad y = \frac{1 - x}{2} \]

2. When \(y < 0\), we have:
\[ x - 2y = 1 \quad \Rightarrow \quad y = \frac{x - 1}{2} \]

### Step 2: Identify the lines

From the above cases, we can identify two lines:
1. \(y = \frac{1 - x}{2}\) (for \(y \geq 0\))
2. \(y = \frac{x - 1}{2}\) (for \(y < 0\))

### Step 3: Find the intersection points with the line \(x = 0\)

Now, we will find the points where these lines intersect the line \(x = 0\).

1. For \(y = \frac{1 - x}{2}\):
\[ y = \frac{1 - 0}{2} = \frac{1}{2} \quad \text{(Point A: (0, 0.5))} \]

2. For \(y = \frac{x - 1}{2}\):
\[ y = \frac{0 - 1}{2} = -\frac{1}{2} \quad \text{(Point B: (0, -0.5))} \]

### Step 4: Find the x-intercept of the lines

Next, we find the x-intercepts of both lines by setting \(y = 0\):

1. For \(y = \frac{1 - x}{2}\):
\[ 0 = \frac{1 - x}{2} \quad \Rightarrow \quad 1 - x = 0 \quad \Rightarrow \quad x = 1 \quad \text{(Point C: (1, 0))} \]

2. For \(y = \frac{x - 1}{2}\):
\[ 0 = \frac{x - 1}{2} \quad \Rightarrow \quad x - 1 = 0 \quad \Rightarrow \quad x = 1 \quad \text{(Point C: (1, 0))} \]

### Step 5: Plot the points and form a triangle

We have the points:
- A: \((0, 0.5)\)
- B: \((0, -0.5)\)
- C: \((1, 0)\)

These points form a triangle in the coordinate plane.

### Step 6: Calculate the area of the triangle

The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Here, the base \(BC\) is the distance between points B and C, which is \(1\) (from \(y = -0.5\) to \(y = 0\)), and the height is the distance from point A to the x-axis, which is \(0.5\).

Thus, the area is:
\[ A = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \]

### Final Answer

The area bounded by the curves \(x + 2|y| = 1\) and \(x = 0\) is \(\frac{1}{2}\).

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