Let the function `: R to R and g: R to R` be defined by
`f=(x)^(x1)-e^(x|1) and g(x)=(1)/(2)(e^(x-1)+ e^(1x))`
Then the area f the region in the first quarant bounded by the curves `g=f(x),y=g(x)` and x=0n is

Options

A. `(2-sqrt(3))+(1)/(2)(e-e^(1))`
B. `(2+sqrt(3))+(1)/(2)(e-e^(1))`
C. `(2-sqrt(3))+(1)/(2)(e+e^(1))`
D. `(2+sqrt(3))+(1)/(2)(e+e^(1))`

Answer

To find the area of the region in the first quadrant bounded by the curves \( g = f(x) \), \( y = g(x) \), and \( x = 0 \), we first need to define the functions \( f(x) \) and \( g(x) \) properly and then determine the points of intersection.

### Step 1: Define the Functions
The functions are given as:
- \( f(x) = e^x - e^{|1-x|} \)
- \( g(x) = \frac{1}{2}(e^{x-1} + e^{1-x}) \)

### Step 2: Analyze the Function \( g(x) \)
The function \( g(x) \) is symmetric about \( x = 1 \). We can verify this by substituting \( 1 - x \) into \( g(x) \):
\[ g(1-x) = \frac{1}{2}(e^{(1-x)-1} + e^{1-(1-x)}) = \frac{1}{2}(e^{-x} + e^{x}) = g(x) \]
This confirms the symmetry.

### Step 3: Find the Points of Intersection
We need to find where \( f(x) = g(x) \). This requires solving the equation:
\[ e^x - e^{|1-x|} = \frac{1}{2}(e^{x-1} + e^{1-x}) \]

#### Case 1: \( x < 1 \)
For \( x < 1 \), \( |1-x| = 1-x \), so:
\[ f(x) = e^x - e^{1-x} \]
Setting this equal to \( g(x) \):
\[ e^x - e^{1-x} = \frac{1}{2}(e^{x-1} + e^{1-x}) \]
This simplifies to:
\[ e^x - e^{1-x} = \frac{1}{2}(e^{x-1} + e^{1-x}) \]
Multiplying through by 2:
\[ 2e^x - 2e^{1-x} = e^{x-1} + e^{1-x} \]
Rearranging gives:
\[ 2e^x - e^{x-1} = 2e^{1-x} + e^{1-x} \]
This can be solved to find the intersection point.

#### Case 2: \( x \geq 1 \)
For \( x \geq 1 \), \( |1-x| = x-1 \), so:
\[ f(x) = e^x - e^{x-1} \]
Setting this equal to \( g(x) \):
\[ e^x - e^{x-1} = \frac{1}{2}(e^{x-1} + e^{1-x}) \]
This can also be solved similarly.

### Step 4: Calculate the Area
The area \( A \) can be calculated as:
\[ A = \int_{0}^{1} g(x) \, dx + \int_{1}^{\alpha} (g(x) - f(x)) \, dx \]
Where \( \alpha \) is the point of intersection found in the previous steps.

### Step 5: Evaluate the Integrals
1. **For \( x \) from 0 to 1**:
\[ A_1 = \int_{0}^{1} g(x) \, dx \]
Substitute \( g(x) \) and compute the integral.

2. **For \( x \) from 1 to \( \alpha \)**:
\[ A_2 = \int_{1}^{\alpha} (g(x) - f(x)) \, dx \]
Substitute \( g(x) \) and \( f(x) \) and compute the integral.

### Final Step: Combine Areas
Combine the areas from both integrals to get the total area.