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To set up the definite integral that represents the area of the shaded region, we need to find the points where the function \( f(x) = 4 - 2x \) intersects the x-axis, as these will be the bounds of integration.

1. **Find the x-intercept**: Set \( f(x) = 0 \) and solve for \( x \):
   \[
   4 - 2x = 0
   \]
   \[
   2x = 4
   \]
   \[
   x = 2
   \]

So, the function intersects the x-axis at \( x = 2 \). This gives us the interval over which we’ll integrate to find the area, from \( x = 0 \) to \( x = 2 \).

2. **Set up the integral**: The area can be represented by the following definite integral:
   \[
   \int_{0}^{2} (4 - 2x) \, dx
   \]

This integral represents the area under the curve \( f(x) = 4 - 2x \) from \( x = 0 \) to \( x = 2 \).

Would you like more details or have any questions?

Here are some related questions to consider:
1. How would you evaluate the integral if required?
2. What is the geometric interpretation of this area under a linear function?
3. How would the integral setup change if the limits were different?
4. Can we confirm this area by calculating it as a triangle instead?
5. How does changing the function affect the area calculated by the integral?

**Tip**: When setting up an integral for area, always check if the region lies above or below the x-axis to ensure correct interpretation.

Set up a definite integral that yields the area of the region. f(x) = 4 - 2x from x = 0.

Learn how to set up a definite integral to find the area of the region under the function f(x) = 4 - 2x from x = 0 to x = 2. This problem uses key calculus concepts and is suitable for high school students in Grades 11-12.

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