To solve this problem using the left Riemann sum ($L_6$), we need to approximate the integral $\int_0^{12} f(x) \, dx$ by dividing the interval $[0, 12]$ into 6 equal subintervals. Since the graph shows a function $y = f(x)$, we will use the left endpoints of these subintervals to calculate the height of each rectangle.

Step-by-Step Approach:

1. Subinterval Width: The interval $[0, 12]$ is divided into 6 equal subintervals. The width of each subinterval is: $\Delta x = \frac{12 - 0}{6} = 2$

2. Left Endpoints: The left endpoints of the six subintervals are: $x = 0, 2, 4, 6, 8, 10$.

3. Function Values: From the graph, we estimate the function values $f(x)$ at these left endpoints:

   • $f(0) \approx 8$
   • $f(2) \approx 7.5$
   • $f(4) \approx 6.5$
   • $f(6) \approx 5$
   • $f(8) \approx 3.5$
   • $f(10) \approx 2$

4. Left Riemann Sum: Now, we compute the left Riemann sum: $L_6 = \Delta x \times \left[ f(0) + f(2) + f(4) + f(6) + f(8) + f(10) \right]$ Substituting the values: $L_6 = 2 \times \left[ 8 + 7.5 + 6.5 + 5 + 3.5 + 2 \right]$ $L_6 = 2 \times (32.5) = 65$

Approximation:

Thus, the approximation of the integral $\int_0^{12} f(x) \, dx$ using six rectangles with left endpoints is approximately 65.

Would you like more details, or do you have any questions?

Here are five related questions:

1. How would the result change if we used right endpoints instead of left endpoints?
2. How does the accuracy of a Riemann sum improve with more subintervals?
3. Can you explain the relationship between the Riemann sum and the definite integral?
4. How would you compute the exact value of the integral $\int_0^{12} f(x) \, dx$ if the function $f(x)$ were given explicitly?
5. How does the trapezoidal rule compare to the left Riemann sum for this problem?

Tip: Using more rectangles (i.e., increasing the number of subintervals) in a Riemann sum provides a better approximation of the integral.