Let's calculate the Left Endpoint Riemann Sum for the function \( y = -\frac{x^2}{5} + 2x \) over the interval \([4, 8]\).

### Step 1: Determine the Subintervals
Assume the interval \([4, 8]\) is divided into \( n \) subintervals. For simplicity, let's consider \( n = 5 \) subintervals, as this matches the number of rectangles seen in the image.

The width of each subinterval (denoted by \( \Delta x \)) is:
\[
\Delta x = \frac{8 - 4}{5} = \frac{4}{5} = 0.8
\]

### Step 2: Identify the Left Endpoints
The left endpoints of the subintervals are:
- \( x_0 = 4 \)
- \( x_1 = 4.8 \)
- \( x_2 = 5.6 \)
- \( x_3 = 6.4 \)
- \( x_4 = 7.2 \)

### Step 3: Evaluate the Function at Each Left Endpoint
Calculate the value of \( y \) at each left endpoint:
\[
f(x) = -\frac{x^2}{5} + 2x
\]

- \( f(x_0) = f(4) = -\frac{4^2}{5} + 2(4) = -\frac{16}{5} + 8 = -3.2 + 8 = 4.8 \)
- \( f(x_1) = f(4.8) = -\frac{4.8^2}{5} + 2(4.8) = -\frac{23.04}{5} + 9.6 = -4.608 + 9.6 = 4.992 \)
- \( f(x_2) = f(5.6) = -\frac{5.6^2}{5} + 2(5.6) = -\frac{31.36}{5} + 11.2 = -6.272 + 11.2 = 4.928 \)
- \( f(x_3) = f(6.4) = -\frac{6.4^2}{5} + 2(6.4) = -\frac{40.96}{5} + 12.8 = -8.192 + 12.8 = 4.608 \)
- \( f(x_4) = f(7.2) = -\frac{7.2^2}{5} + 2(7.2) = -\frac{51.84}{5} + 14.4 = -10.368 + 14.4 = 4.032 \)

### Step 4: Calculate the Area of Each Rectangle
Each rectangle's area is given by:
\[
\text{Area of each rectangle} = f(x_i) \times \Delta x
\]
Thus, the areas are:
- \( \text{Area}_1 = 4.8 \times 0.8 = 3.84 \)
- \( \text{Area}_2 = 4.992 \times 0.8 = 3.9936 \)
- \( \text{Area}_3 = 4.928 \times 0.8 = 3.9424 \)
- \( \text{Area}_4 = 4.608 \times 0.8 = 3.6864 \)
- \( \text{Area}_5 = 4.032 \times 0.8 = 3.2256 \)

### Step 5: Sum the Areas to Find the Total Riemann Sum
The Left Endpoint Riemann Sum \( S \) is:
\[
S = \sum_{i=1}^{5} f(x_{i-1}) \times \Delta x = 3.84 + 3.9936 + 3.9424 + 3.6864 + 3.2256 = 18.688
\]

### Final Answer
The Left Endpoint Riemann Sum for the function \( y = -\frac{x^2}{5} + 2x \) over the interval \([4, 8]\) with 5 subintervals is approximately **18.688**.

Do you have any more questions or need further details?

Here are some related questions:
1. How would the Riemann sum change if we used the Right Endpoint method instead?
2. How can we interpret the error in the Riemann sum approximation?
3. What happens to the Riemann sum if the number of subintervals is doubled?
4. How does the function \( y = -\frac{x^2}{5} + 2x \) behave over other intervals?
5. What is the exact value of the integral of \( y = -\frac{x^2}{5} + 2x \) over \([4, 8]\)?

**Tip:** Riemann sums are just one way to approximate the area under a curve. Other methods, like the Trapezoidal Rule or Simpson's Rule, can often provide more accurate results with fewer subintervals.

Learn how to calculate the Left Endpoint Riemann Sum for the function y = -x^2/5 + 2x over the interval [4, 8] with 5 subintervals. Detailed step-by-step solution provided. Suitable for advanced high school or college-level students.

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