Area of the region bounded by the curve `y=e^(x)` and lines x = 0 and y = e is :

Options

A. `e-1`
B. `int_(1)^(e )ln (e+1-y)dy`
C. `e-int_(0)^(1)e^(x)dx`
D. `int_(1)^(e )ln ydy`

Answer

To find the area of the region bounded by the curve \( y = e^x \) and the lines \( x = 0 \) and \( y = e \), we can follow these steps:

### Step 1: Identify the points of intersection
First, we need to find where the curve \( y = e^x \) intersects the line \( y = e \).

Set \( e^x = e \):
\[ e^x = e \implies x = 1 \]
Thus, the points of intersection are \( (0, 1) \) and \( (1, e) \).

### Step 2: Set up the integral for the area
The area \( A \) can be calculated using the integral of the function \( y = e^x \) from \( x = 0 \) to \( x = 1 \):
\[ A = \int_{0}^{1} e^x \, dx \]

### Step 3: Calculate the integral
Now, we compute the integral:
\[ A = \int e^x \, dx = e^x + C \]
Evaluating from \( 0 \) to \( 1 \):
\[ A = \left[ e^x \right]_{0}^{1} = e^1 - e^0 = e - 1 \]

### Step 4: Conclusion
The area of the region bounded by the curve \( y = e^x \) and the lines \( x = 0 \) and \( y = e \) is:
\[ A = e - 1 \]