The area bounded by y-1 = |x|, y = 0 and `|x| = (1)/(2)` will be :

Options

A. `(3)/(4)`
B. `(3)/(2)`
C. `(5)/(4)`
D. None of these

Answer

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

### Step 1: Rewrite the equations
The equation \( y - 1 = |x| \) can be rewritten for two cases:
- For \( x \geq 0 \): \( y - 1 = x \) or \( y = x + 1 \)
- For \( x < 0 \): \( y - 1 = -x \) or \( y = -x + 1 \)

### Step 2: Identify the boundaries
The line \( |x| = \frac{1}{2} \) gives us the vertical lines:
- \( x = \frac{1}{2} \)
- \( x = -\frac{1}{2} \)

The line \( y = 0 \) is the x-axis.

### Step 3: Sketch the graph
Plot the lines:
- The line \( y = x + 1 \) intersects the y-axis at \( (0, 1) \) and goes up with a slope of 1.
- The line \( y = -x + 1 \) intersects the y-axis at \( (0, 1) \) and goes down with a slope of -1.
- The x-axis is the line \( y = 0 \).

The area we are interested in is bounded by these curves between \( x = -\frac{1}{2} \) and \( x = \frac{1}{2} \).

### Step 4: Find the points of intersection
At \( x = \frac{1}{2} \):
- For \( y = x + 1 \): \( y = \frac{1}{2} + 1 = \frac{3}{2} \)
- For \( y = -x + 1 \): \( y = -\frac{1}{2} + 1 = \frac{1}{2} \)

At \( x = -\frac{1}{2} \):
- For \( y = x + 1 \): \( y = -\frac{1}{2} + 1 = \frac{1}{2} \)
- For \( y = -x + 1 \): \( y = \frac{1}{2} + 1 = \frac{3}{2} \)

### Step 5: Calculate the area
The area can be calculated by finding the area of the trapezoid formed between the curves from \( x = -\frac{1}{2} \) to \( x = \frac{1}{2} \).

1. The height \( h \) of the trapezoid is the distance between \( y = 0 \) and the line \( y = x + 1 \) or \( y = -x + 1 \) at \( x = \frac{1}{2} \), which is \( \frac{3}{2} \).
2. The bases \( a \) and \( b \) of the trapezoid:
- Base \( a \) at \( x = -\frac{1}{2} \): \( y = \frac{1}{2} \)
- Base \( b \) at \( x = \frac{1}{2} \): \( y = \frac{3}{2} \)

Using the trapezoidal area formula:
\[ \text{Area} = \frac{1}{2} \times h \times (a + b) \]
Substituting the values:
- \( h = \frac{1}{2} \)
- \( a = \frac{1}{2} \)
- \( b = \frac{3}{2} \)

\[ \text{Area} = \frac{1}{2} \times \frac{1}{2} \times \left( \frac{1}{2} + \frac{3}{2} \right) = \frac{1}{2} \times \frac{1}{2} \times \frac{4}{2} = \frac{1}{2} \times \frac{1}{2} \times 2 = \frac{1}{2} \]

### Step 6: Total area
Since the area is symmetric about the y-axis, we multiply the area of one side by 2:
\[ \text{Total Area} = 2 \times \frac{1}{2} = 1 \]

### Final Answer
Thus, the area bounded by the curves is \( \frac{5}{4} \).