Which of the following is not the area of the region bounded by `y= e^(x)` and x=0 and y= e?

Options

A. e-1
B. `int_1^(e)In (e+1-y)dy`
C. `e-int_0^(1)e^(x)dx`
D. `int_0^(e)In y dy`

Answer

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

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

\[ e^x = e \implies x = 1 \]

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: Evaluate the integral
To evaluate the integral, we find the antiderivative of \( e^x \):

\[ \int e^x \, dx = e^x + C \]

Now we can evaluate the definite integral:

\[ A = \left[ e^x \right]_{0}^{1} = e^1 - e^0 = e - 1 \]

### Step 4: Determine the area bounded by the given lines
The area we are interested in is bounded by \( y = e^x \), \( x = 0 \), and \( y = e \). The area can also be calculated using the horizontal strip method, which involves integrating with respect to \( y \).

### Step 5: Change the variable
To use the horizontal strip method, we need to express \( x \) in terms of \( y \):

\[ y = e^x \implies x = \ln(y) \]

### Step 6: Set up the new integral
The area can now be expressed as:

\[ A = \int_{1}^{e} \ln(y) \, dy \]

### Step 7: Evaluate the new integral
Using integration by parts, where \( u = \ln(y) \) and \( dv = dy \):

\[ du = \frac{1}{y} \, dy, \quad v = y \]

Applying integration by parts:

\[ \int \ln(y) \, dy = y \ln(y) - \int y \cdot \frac{1}{y} \, dy = y \ln(y) - y + C \]

Now we evaluate the definite integral:

\[ A = \left[ y \ln(y) - y \right]_{1}^{e} \]

Calculating it:

\[ = \left[ e \cdot 1 - e \right] - \left[ 1 \cdot 0 - 1 \right] = (e - e) - (0 - 1) = 1 \]

### Conclusion
The area of the region bounded by the specified curves is \( 1 \).