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This worksheet is asking for an approximation of the area under the curve, which is depicted as a shaded region between \(x = 0\) and \(x = 6\), and above the x-axis. This type of problem uses methods like Riemann sums to estimate the area. The worksheet asks students to divide the interval into subintervals and approximate the area using vertical rectangles, where the height of each rectangle depends on specific points chosen within each subinterval.

Each color (Red, Green, Blue, and Purple) represents a different rule for choosing the point \(x_i^*\) within each subinterval:

- **Red**: Always choose the left endpoint of each subinterval.
- **Green**: Always choose the right endpoint of each subinterval.
- **Blue**: Always choose the midpoint of each subinterval.
- **Purple**: You can choose any point within the subinterval.

For each color, the task is to compute the area approximation for two subintervals, three subintervals, and six subintervals.

Would you like help with calculating the approximations for each of these methods and intervals? I can walk you through one or all of them!

### 5 Related Questions:
1. What are Riemann sums and how do they work for approximating area?
2. Why does choosing the left, right, or midpoint of the subinterval affect the result?
3. How can increasing the number of subintervals improve the accuracy of the approximation?
4. What is the function of the curve, and how would you compute the function's value for the points \(x_i^*\)?
5. How do the different colors (Red, Green, Blue, Purple) compare in terms of the accuracy of approximation?

**Tip**: When approximating areas, the midpoint (Blue method) often gives a more accurate result because it balances overestimating and underestimating across subintervals.

Your group is going to attempt to approximate the area of the region below a curve by dividing the interval into subintervals and approximating the area with vertical rectangles. Different methods of choosing the points in each subinterval are defined by colors (Red, Green, Blue, Purple).

Learn how to approximate the area under a curve using Riemann sums, with different rules for choosing points in subintervals based on color (Red, Green, Blue, Purple). This problem demonstrates the use of left, right, and midpoint methods to calculate approximate areas for various numbers of subintervals.

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