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Set up ( and do not evaluate) the internals for a.) net area b.) area ( based on the following)

To set up the integrals based on the provided graph, we first identify the regions specified for the net area and the absolute area.

### a) Net Area
The net area accounts for the areas above and below the x-axis, with areas below the x-axis being negative.

The net area can be expressed as:
\[
\text{Net Area} = \int_{-4}^{4} f(x) \, dx + \int_{4}^{8} f(x) \, dx
\]

### b) Absolute Area
The absolute area accounts for the absolute value of the area, treating all areas as positive regardless of their position relative to the x-axis.

The absolute area can be expressed as:
\[
\text{Absolute Area} = \int_{-4}^{4} |f(x)| \, dx + \int_{4}^{8} |f(x)| \, dx
\]

These integrals consider the areas labeled as 8, 17, and 9 on the graph. The regions are:
1. From \(x = -4\) to \(x = 0\) with an area of 8 (above the x-axis).
2. From \(x = 0\) to \(x = 4\) with an area of 17 (above the x-axis).
3. From \(x = 4\) to \(x = 8\) with an area of 9 (below the x-axis).

Is there anything specific you would like to explore or need further explanation on? Here are some related questions to consider:

1. What does it mean for an area to be "net" versus "absolute" in calculus?
2. How would the integral change if the function \( f(x) \) had discontinuities?
3. Why is the area below the x-axis considered negative in the calculation of the net area?
4. How does the concept of absolute value apply when calculating absolute areas?
5. What is the significance of the limits of integration in these calculations?
6. How can you determine if the function \( f(x) \) crosses the x-axis between given intervals?
7. How does one interpret the areas labeled on the graph in terms of definite integrals?
8. What are some applications of calculating the net area under a curve in real-world scenarios?

**Tip:** When setting up integrals for areas under a curve, always consider whether the question asks for net area or absolute area, as this affects the sign and value of the integral.

Explore how to set up integrals for calculating net and absolute areas under a curve with this detailed guide. Learn the difference between net area and absolute area, and how to apply integrals to find these areas. Suitable for advanced high school students and calculus enthusiasts.

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