The area bounded by the curves x + 2|y| = 1 and x = 0 is

Options

A. `1/4`
B. `1/2`
C. 1
D. 2

Answer

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

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

First, we need to rewrite the equation \( x + 2|y| = 1 \) in terms of \( y \).

1. Rearranging gives us:
\[ 2|y| = 1 - x \]
\[ |y| = \frac{1 - x}{2} \]

2. This leads to two cases for \( y \):
- Case 1: \( y = \frac{1 - x}{2} \)
- Case 2: \( y = -\frac{1 - x}{2} \)

### Step 2: Determine the points of intersection with \( x = 0 \)

Next, we find the points where these curves intersect the line \( x = 0 \).

1. Substitute \( x = 0 \) into the equations:
- For Case 1:
\[ y = \frac{1 - 0}{2} = \frac{1}{2} \]
- For Case 2:
\[ y = -\frac{1 - 0}{2} = -\frac{1}{2} \]

So, the points of intersection are \( (0, \frac{1}{2}) \) and \( (0, -\frac{1}{2}) \).

### Step 3: Sketch the region

Now we can sketch the curves:
- The line \( y = \frac{1 - x}{2} \) is a straight line that decreases from \( (1, 0) \) to \( (0, \frac{1}{2}) \).
- The line \( y = -\frac{1 - x}{2} \) is a straight line that increases from \( (1, 0) \) to \( (0, -\frac{1}{2}) \).
- The area bounded by these lines and the line \( x = 0 \) forms a vertical strip between \( y = \frac{1}{2} \) and \( y = -\frac{1}{2} \).

### Step 4: Calculate the area

The area can be calculated using the formula for the area of a rectangle:
\[ \text{Area} = \text{Base} \times \text{Height} \]

1. The base is the distance along the x-axis from \( x = 0 \) to \( x = 1 \), which is \( 1 \).
2. The height is the distance from \( y = -\frac{1}{2} \) to \( y = \frac{1}{2} \), which is:
\[ \frac{1}{2} - \left(-\frac{1}{2}\right) = 1 \]

Thus, the area is:
\[ \text{Area} = 1 \times 1 = 1 \]

### Final Answer

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

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