The graph shows the function $f(x) = \frac{3x^3}{x^3 + 1}$, and the task is to calculate the Riemann sum for $n = 65$ subintervals, choosing the left endpoint of each subinterval. The shaded region in the graph indicates the area under the curve from approximately $x = 5$ to $x = 8$.

Here’s how we can approach the problem:

Steps for calculating the Riemann Sum:

1. Interval Selection: First, we define the interval over which we will compute the Riemann sum. Based on the graph, the interval appears to be from $x = 5$ to $x = 8$.

2. Divide the interval: The number of subintervals $n = 65$ divides the interval into equal parts. Each subinterval width $\Delta x$ can be computed as: $\Delta x = \frac{8 - 5}{65} = \frac{3}{65}$

3. Left Endpoint Rule: For the left Riemann sum, we evaluate the function at the left endpoint of each subinterval. These points are given by: $x_i = 5 + i \cdot \Delta x, \quad i = 0, 1, 2, \dots, 64$

4. Function Evaluation: For each $x_i$, calculate the function value $f(x_i)$.

5. Riemann Sum: The sum is approximated by: $S = \sum_{i=0}^{64} f(x_i) \cdot \Delta x$ where $f(x_i) = \frac{3x_i^3}{x_i^3 + 1}$.

Let me calculate the Riemann sum for you:

The Riemann sum for $n = 65$ subintervals, using the left endpoint method for the function $f(x) = \frac{3x^3}{x^3 + 1}$ on the interval $[5, 8]$, is approximately $8.9632$, rounded to four decimal places.

Would you like more details about this calculation or have any questions?

Here are 5 related questions to expand on this topic:

1. What is the difference between left Riemann sum and right Riemann sum?
2. How does increasing the number of subintervals affect the accuracy of the Riemann sum?
3. Can we calculate the exact area under the curve using integration instead of Riemann sums?
4. What are some real-life applications of Riemann sums?
5. How would the result change if we used the midpoint rule instead of the left endpoint rule?

Tip: For better accuracy in approximating the area under a curve, consider using methods like the trapezoidal rule or Simpson's rule in addition to Riemann sums.