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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.

Use the definite integral to find the area between the x-axis and f(x) = e^x - 4 over the interval [-1, 3]. Check first to see if the graph crosses the x-axis in the given interval.

This problem demonstrates how to use a definite integral to calculate the area between the x-axis and the curve f(x) = e^x - 4 over the interval [-1, 3]. The function crosses the x-axis within the interval, so the integral must be split into two parts to account for negative and positive areas.

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