The area of the figure bounded by `y=e^(x-1),y=0,x=0 and x=2` is

Options

A. `lt2`
B. `gt2`
C. `=2`
D. none of these

Answer

To find the area of the figure bounded by the curve \( y = e^{x-1} \), the x-axis \( y = 0 \), and the vertical lines \( x = 0 \) and \( x = 2 \), we will follow these steps:

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

\[ A = \int_{0}^{2} e^{x-1} \, dx \]

### Step 2: Simplify the integrand
We can rewrite the integrand \( e^{x-1} \) as:

\[ e^{x-1} = \frac{e^x}{e} = \frac{1}{e} e^x \]

Thus, the integral becomes:

\[ A = \int_{0}^{2} \frac{1}{e} e^x \, dx \]

### Step 3: Factor out the constant
Since \( \frac{1}{e} \) is a constant, we can factor it out of the integral:

\[ A = \frac{1}{e} \int_{0}^{2} e^x \, dx \]

### Step 4: Evaluate the integral
The integral of \( e^x \) is \( e^x \). Therefore:

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

Now, we evaluate the definite integral:

\[ \int_{0}^{2} e^x \, dx = e^x \bigg|_{0}^{2} = e^2 - e^0 = e^2 - 1 \]

### Step 5: Substitute back into the area formula
Now we substitute back into the area formula:

\[ A = \frac{1}{e} (e^2 - 1) = \frac{e^2 - 1}{e} \]

### Step 6: Simplify the expression
This can be simplified to:

\[ A = e - \frac{1}{e} \]

### Step 7: Calculate the numerical value
Using the approximate value of \( e \approx 2.718 \):

\[ A \approx 2.718 - \frac{1}{2.718} \approx 2.718 - 0.368 \approx 2.350 \]

### Final Result
Thus, the area of the figure bounded by the given curves is:

\[ A \approx 2.350 \]