In this problem, we are asked to evaluate a left endpoint Riemann sum for the function $f(x) = \frac{x^2}{9}$ over the interval $[3, 7]$. The graph shows rectangles representing the left endpoint Riemann sum, which approximates the area under the curve of $f(x)$ from $x = 3$ to $x = 7$.

Step 1: Set up the Riemann Sum

To calculate the Riemann sum, we’ll divide the interval $[3, 7]$ into equal subintervals. Let $n$ be the number of rectangles (subintervals). The width of each rectangle (denoted as $\Delta x$) is: $\Delta x = \frac{7 - 3}{n} = \frac{4}{n}$

Step 2: Left Endpoint Evaluation

For a left endpoint Riemann sum, we evaluate the function at the left endpoint of each subinterval. If we let $x_i$ represent the left endpoint of each subinterval, then: $x_i = 3 + (i - 1) \cdot \Delta x$ The function value at each left endpoint is $f(x_i) = \frac{(x_i)^2}{9}$.

Step 3: Sum the Areas of the Rectangles

The Riemann sum is the sum of the areas of each rectangle, given by: $\text{Riemann sum} = \sum_{i=1}^{n} f(x_i) \cdot \Delta x = \sum_{i=1}^{n} \frac{(x_i)^2}{9} \cdot \Delta x$

Observing the Graph and Answering the Questions

From the graph, we can observe that the rectangles lie beneath the curve of $f(x) = \frac{x^2}{9}$, meaning this left endpoint Riemann sum is an underestimate of the true area.

To complete the problem:

• Fill in the blank for the value of the Riemann sum by calculating it directly if given the number of subintervals $n$.
• Note that since the function is increasing, the left endpoint sum underestimates the actual area under the curve.

Would you like to proceed with a specific calculation based on the number of rectangles, or do you have further questions?

Related Questions

1. What would change in the Riemann sum if we used the right endpoints instead of the left?
2. How does increasing the number of rectangles affect the accuracy of the Riemann sum?
3. How can we calculate the exact area under the curve using integration?
4. What is the significance of using the midpoint instead of left or right endpoints for the Riemann sum?
5. How does this method of approximation apply to real-world problems?

Tip:

For a smoother approximation, increase the number of subintervals, which will reduce the gap between the Riemann sum and the true integral value.