1. **State the problem:** We need to find the area between the curve $f(x) = e^x - 6$ and the x-axis from the zero of the function to $x=4$. 2. **Find the zero of the function:** Solve $e^x - 6 = 0$. $$e^x = 6$$ Take the natural logarithm of both sides: $$x = \ln(6)$$ Calculate $x$ to 4 decimal places: $$x \approx 1.7918$$ 3. **Set up the integral for the area:** The area between the curve and the x-axis from $x=1.7918$ to $x=4$ is $$\text{Area} = \int_{1.7918}^{4} (e^x - 6) \, dx$$ 4. **Integrate the function:** $$\int (e^x - 6) \, dx = e^x - 6x + C$$ 5. **Evaluate the definite integral:** $$\text{Area} = \left[ e^x - 6x \right]_{1.7918}^{4} = (e^4 - 6 \times 4) - (e^{1.7918} - 6 \times 1.7918)$$ Calculate each term: $$e^4 \approx 54.5982$$ $$6 \times 4 = 24$$ $$e^{1.7918} \approx 6$$ $$6 \times 1.7918 = 10.7508$$ So, $$\text{Area} = (54.5982 - 24) - (6 - 10.7508) = 30.5982 - (-4.7508) = 30.5982 + 4.7508 = 35.349$$ 6. **Round the final answer:** $$\boxed{35.349}$$ This is the area of the red shaded region rounded to the nearest thousandth.