To calculate the left endpoint Riemann sum for the function $y = \frac{x^2}{9}$ over the interval $[3, 7]$, we’ll follow these steps:

1. Identify the interval and subintervals:

   • Interval: $[3, 7]$
   • Subintervals: Based on the graph, there are five rectangles.

2. Determine the width of each rectangle ($\Delta x$):

   • The interval length is $7 - 3 = 4$.
   • With 5 rectangles, $\Delta x = \frac{4}{5} = 0.8$.

3. Calculate the left endpoint values:

   • For each rectangle, we use the left endpoint to calculate the height. The left endpoints will be:
     • $x_1 = 3$
     • $x_2 = 3.8$
     • $x_3 = 4.6$
     • $x_4 = 5.4$
     • $x_5 = 6.2$

4. Evaluate the function at each left endpoint:

   • $f(x) = \frac{x^2}{9}$
   • $f(3) = \frac{3^2}{9} = 1$
   • $f(3.8) = \frac{3.8^2}{9} \approx 1.60$
   • $f(4.6) = \frac{4.6^2}{9} \approx 2.35$
   • $f(5.4) = \frac{5.4^2}{9} \approx 3.24$
   • $f(6.2) = \frac{6.2^2}{9} \approx 4.27$

5. Calculate the Riemann sum:

   • Riemann Sum $= \sum f(x_i) \cdot \Delta x$
   • $= (1 + 1.60 + 2.35 + 3.24 + 4.27) \times 0.8$
   • $= 12.46 \times 0.8 \approx 9.97$

The left endpoint Riemann sum for $y = \frac{x^2}{9}$ on $[3, 7]$ is approximately 9.97.

Would you like a detailed breakdown of any specific steps or have further questions?

Here are some related questions for further exploration:

1. How does the result change with a right endpoint Riemann sum?
2. How would a midpoint Riemann sum affect the result?
3. What happens if we increase the number of rectangles (subintervals)?
4. Can we calculate the exact integral of $y = \frac{x^2}{9}$ over $[3, 7]$ and compare it?
5. How does the Riemann sum relate to the area under the curve?

Tip: Increasing the number of rectangles in a Riemann sum generally improves the approximation of the integral.