Area lying between the curves `y= abs x`, x=1 and x=-1 is

Options

A. 1
B. 0
C. 3
D. `-1`

Answer

To find the area lying between the curves \( y = |x| \), \( x = 1 \), and \( x = -1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Function**:
The function \( y = |x| \) can be defined as:
\[ y = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]
This means for \( x \) from -1 to 1, the function will be \( y = x \) for \( x \geq 0 \) and \( y = -x \) for \( x < 0 \).

2. **Sketch the Graph**:
Draw the graph of \( y = |x| \) which forms a V-shape. The points of interest are at \( x = -1 \) and \( x = 1 \). At both these points, \( y = 1 \).

3. **Determine the Area**:
The area between the curve and the x-axis from \( x = -1 \) to \( x = 1 \) can be calculated by integrating the function \( y = |x| \).

4. **Set Up the Integral**:
Since the function is symmetric about the y-axis, we can calculate the area from \( 0 \) to \( 1 \) and then double it:
\[ \text{Area} = 2 \int_{0}^{1} x \, dx \]

5. **Calculate the Integral**:
Now, we compute the integral:
\[ \int_{0}^{1} x \, dx = \left[ \frac{x^2}{2} \right]_{0}^{1} = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2} \]

6. **Multiply by 2**:
Since we only calculated the area for \( x \) from \( 0 \) to \( 1 \), we multiply by 2:
\[ \text{Total Area} = 2 \times \frac{1}{2} = 1 \]

### Final Answer:
The area lying between the curves \( y = |x| \), \( x = 1 \), and \( x = -1 \) is \( 1 \) square unit.

---