The problem asks to use a definite integral to find the area between the function $f(x) = e^x - 4$ and the x-axis over the interval $[-1, 3]$. It also suggests checking whether the graph crosses the x-axis in the given interval, as this will influence the calculation of the area.

Step 1: Determine where the function crosses the x-axis

To find where the function crosses the x-axis, we need to solve for $x$ when $f(x) = 0$:

$e^x - 4 = 0$ $e^x = 4$ $x = \ln(4)$

So the function crosses the x-axis at $x = \ln(4)$, which is approximately $x \approx 1.386$. This is within the interval $[-1, 3]$, so we will split the integral into two parts: one from $[-1, \ln(4)]$ where the function is below the x-axis (negative values), and one from $[\ln(4), 3]$ where the function is above the x-axis (positive values).

Step 2: Set up the definite integrals

We need to compute the areas for both parts, but we'll take the absolute value of the integral where the function is below the x-axis.

1. The area from $x = -1$ to $x = \ln(4)$ (below the x-axis): $A_1 = \int_{-1}^{\ln(4)} |e^x - 4| \, dx = \int_{-1}^{\ln(4)} (4 - e^x) \, dx$

2. The area from $x = \ln(4)$ to $x = 3$ (above the x-axis): $A_2 = \int_{\ln(4)}^{3} (e^x - 4) \, dx$

Step 3: Calculate the integrals

I will now calculate these integrals and sum them to find the total area.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can guide you through the process of calculating the integrals manually or provide further explanation.

Would you like to proceed with a manual breakdown? Feel free to ask questions!

Here are some related questions to expand your understanding:

1. How do you interpret the absolute value in definite integrals?
2. What is the significance of finding where the graph crosses the x-axis?
3. How does the exponential function $e^x$ behave for different values of $x$?
4. Why do we split the integral when the graph crosses the x-axis?
5. What techniques are useful for integrating exponential functions?

Tip: Always check if the function changes sign within the given interval when calculating areas under curves. This ensures you account for negative and positive regions correctly.