The image you provided contains a mathematical problem involving a curve and two lines. Specifically, the equation $y = e^x - 1$ represents the curve, and there are two vertical lines at $x = -1$ and $x = 1$. The question asks for the area of the region enclosed by the curve and the two vertical lines.

To calculate this area, we need to integrate the curve $y = e^x - 1$ between $x = -1$ and $x = 1$. The formula for the area between the curve and the x-axis is given by the integral:

$A = \int_{-1}^{1} (e^x - 1) \, dx$

Let me calculate the integral to find the area. sympy import symbols, exp, integrate

Define the variable and the function

x = symbols('x') f = exp(x) - 1

Calculate the definite integral from x = -1 to x = 1

area = integrate(f, (x, -1, 1)) areaThe area enclosed by the curve $y = e^x - 1$ and the vertical lines $x = -1$ and $x = 1$ is:

$A = -2 - \frac{1}{e} + e$

Now, let's simplify this expression:

$A = e - \frac{1}{e} - 2$

This result corresponds to one of the given choices. Let's compare it with the options:

1. $e + \frac{1}{e} - 2$
2. $e - \frac{1}{e} - 2$
3. $e + \frac{1}{e} - 1$
4. $e - \frac{1}{e} - 1$
5. $e + \frac{1}{e} + 2$

The correct answer is option 2, $e - \frac{1}{e} - 2$.

Would you like to go over any further details, or do you have any other questions related to this problem?

Here are some related questions:

1. What does the shape of the curve $y = e^x - 1$ look like, and why?
2. How does the area change if the lines are at different values of $x$?
3. What is the effect of changing the exponent on the function $e^x$?
4. How can the concept of integration help in finding areas between curves?
5. How do you compute areas between curves using integration for other types of functions?

Tip: Remember, the area under a curve can be found using integration, where the function to be integrated represents the height, and the limits of integration define the region's boundaries.