We are tasked with finding the area between the x-axis and the function $f(x) = 2e^x - 3$ over the interval $[-2, 4]$ using a definite integral. To solve this, we need to follow these steps:

Step 1: Check for x-axis crossings

First, we determine if the graph crosses the x-axis within the interval $[-2, 4]$. This happens when $f(x) = 0$:

$2e^x - 3 = 0$ $2e^x = 3$ $e^x = \frac{3}{2}$ $x = \ln\left(\frac{3}{2}\right) \approx 0.405$

Since $x = 0.405$ lies within the interval $[-2, 4]$, the graph crosses the x-axis at this point. This means we need to break the integral into two parts: one from $[-2, 0.405]$ where the function is below the x-axis, and one from $[0.405, 4]$ where the function is above the x-axis.

Step 2: Set up the integrals

Now we can set up the definite integrals to calculate the area:

1. From $[-2, 0.405]$, the function is negative, so we take the absolute value of the integral: $A_1 = - \int_{-2}^{0.405} (2e^x - 3) \, dx$

2. From $[0.405, 4]$, the function is positive: $A_2 = \int_{0.405}^{4} (2e^x - 3) \, dx$

Step 3: Calculate the integrals

Let me compute these integrals.The total area between the function $f(x) = 2e^x - 3$ and the x-axis over the interval $[-2, 4]$ can be expressed as:

$\text{Total Area} = -9.5672 + 2e^{-2} + 2e^4$

This is the exact expression for the area. We can further simplify or approximate it if needed by substituting numerical values for the exponential terms. Let me now compute the approximate value.It seems like I can’t do more advanced data analysis right now. Please try again later. However, the exact expression for the total area is:

$\text{Total Area} = -9.5672 + 2e^{-2} + 2e^4$

If you substitute the values of $e^{-2}$ and $e^4$, you will get an approximate numerical result for the total area.

Let me know if you need further clarification or details!

Here are some related questions to explore:

1. How does the exponential function influence the area calculations?
2. What happens to the area if the interval is extended beyond $[-2, 4]$?
3. How would the calculation differ if the function were $f(x) = 2e^x - 4$?
4. How can you generalize this process for any function that crosses the x-axis?
5. How can we interpret negative areas in integral calculus?

Tip: When working with definite integrals over regions where the function crosses the x-axis, remember to split the intervals to account for positive and negative values correctly.