The area defined by `|y| le e ^(|x|)-1/2` in cartesian co-ordinate system, is :

Options

A. `(2-2 ln 2)`
B. `(4-ln 2)`
C. `(2-ln 2)`
D. `(2-2 ln 2)`

Answer

To find the area defined by the inequality \(|y| \leq e^{|x|} - \frac{1}{2}\) in the Cartesian coordinate system, we can break this down into steps.

### Step-by-Step Solution:

1. **Understanding the Inequality**:
The inequality \(|y| \leq e^{|x|} - \frac{1}{2}\) implies that \(y\) is bounded between two curves:
\[ -\left(e^{|x|} - \frac{1}{2}\right) \leq y \leq e^{|x|} - \frac{1}{2} \]

2. **Identifying Cases**:
We will consider two cases based on the absolute value of \(x\):
- Case 1: \(x \geq 0\)
- Case 2: \(x < 0\)

For both cases, we will derive the equations for \(y\):
- For \(x \geq 0\):
\[ y \leq e^{x} - \frac{1}{2} \quad \text{and} \quad y \geq -\left(e^{x} - \frac{1}{2}\right) = -e^{x} + \frac{1}{2} \]
- For \(x < 0\):
\[ y \leq e^{-x} - \frac{1}{2} \quad \text{and} \quad y \geq -\left(e^{-x} - \frac{1}{2}\right) = -e^{-x} + \frac{1}{2} \]

3. **Finding Intersection Points**:
We need to find the points where the curves intersect the x-axis. Setting \(y = 0\):
- For \(x \geq 0\):
\[ e^{x} - \frac{1}{2} = 0 \implies e^{x} = \frac{1}{2} \implies x = \ln(2) \]
- For \(x < 0\):
\[ e^{-x} - \frac{1}{2} = 0 \implies e^{-x} = \frac{1}{2} \implies -x = \ln(2) \implies x = -\ln(2) \]

4. **Calculating the Area**:
The area can be calculated by integrating the upper curve minus the lower curve. The total area can be calculated for \(x \geq 0\) and then multiplied by 2 (due to symmetry):
\[ \text{Area} = 2 \int_{0}^{\ln(2)} \left(e^{x} - \frac{1}{2} - (-e^{x} + \frac{1}{2})\right) \, dx \]
Simplifying the integrand:
\[ = 2 \int_{0}^{\ln(2)} \left(2e^{x} - 1\right) \, dx \]

5. **Evaluating the Integral**:
\[ = 2 \left[ 2e^{x} - x \right]_{0}^{\ln(2)} \]
Evaluating at the limits:
\[ = 2 \left[ 2e^{\ln(2)} - \ln(2) - (2e^{0} - 0) \right] \]
\[ = 2 \left[ 2 \cdot 2 - \ln(2) - 2 \right] = 2 \left[ 4 - \ln(2) - 2 \right] = 2(2 - \ln(2)) = 4 - 2\ln(2) \]

6. **Final Area Calculation**:
Since we calculated the area for \(x \geq 0\), we multiply by 2 for the total area:
\[ \text{Total Area} = 2(4 - 2\ln(2)) = 8 - 4\ln(2) \]

### Final Answer:
The area defined by \(|y| \leq e^{|x|} - \frac{1}{2}\) is:
\[ \text{Area} = 8 - 4\ln(2) \]