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The problem involves approximating the area under the curve of the function \( y = -2x^2 + 6x \) using a method of partitioning the interval \([0, 3]\) into equal sub-intervals. You are asked to approximate the area using 3 sub-intervals, applying the concept of Riemann sums.

### Steps to Solve:

1. **Determine the width of each sub-interval:**
   - The interval \([0, 3]\) is divided into 3 sub-intervals.
   - The width of each sub-interval, \(\Delta x\), is calculated as:
   \[
   \Delta x = \frac{3 - 0}{3} = 1
   \]

2. **Identify the points for evaluation:**
   - The sub-intervals will be: \([0, 1]\), \([1, 2]\), and \([2, 3]\).
   - The function \(y = -2x^2 + 6x\) will be evaluated at these points \(x_0 = 0\), \(x_1 = 1\), \(x_2 = 2\), and \(x_3 = 3\).

3. **Calculate the function values:**
   - \(f(x_0) = f(0) = -2(0)^2 + 6(0) = 0\)
   - \(f(x_1) = f(1) = -2(1)^2 + 6(1) = 4\)
   - \(f(x_2) = f(2) = -2(2)^2 + 6(2) = 4\)
   - \(f(x_3) = f(3) = -2(3)^2 + 6(3) = 0\)

4. **Use the trapezoidal rule to approximate the area:**
   - The trapezoidal rule formula for the area is:
   \[
   A \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3) \right]
   \]
   - Substituting the values:
   \[
   A \approx \frac{1}{2} \left[ 0 + 2(4) + 2(4) + 0 \right] = \frac{1}{2} \times 16 = 8
   \]
   So, the approximate area under the curve using the trapezoidal rule with 3 sub-intervals is **8 square units**.

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:
1. How does increasing the number of sub-intervals affect the accuracy of the approximation?
2. What is the difference between using the trapezoidal rule and the midpoint rule for area approximation?
3. How would you apply Simpson's rule to this problem, and how would the result compare?
4. What does the actual area under the curve (obtained via integration) compare with the approximated area?
5. How would you approach finding the area if the function had a more complex shape?

**Tip:** The trapezoidal rule is a good method for approximating the area under a curve, but for better accuracy, especially for curved functions, Simpson's rule is often preferred.

Rechercher une approximation de l'aire située entre la courbe d'équation y = -2x^2 + 6x et l'axe des X sur l'intervalle [0, 3]. Utiliser une partition régulière de 3 sous-intervalles.

Learn how to approximate the area under the curve for the quadratic function y = -2x^2 + 6x using the trapezoidal rule. This problem covers key mathematical concepts such as Riemann sums and area approximation techniques. Suitable for students in Grades 10-12.

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