To solve the problem, we calculate the left Riemann sum for the given integral $\int_{3}^{14} f(x) \, dx$ using 5 subintervals as indicated by the table. Here's the step-by-step process:

Steps:

1. Identify subintervals: The subintervals are determined by the given $x$-values in the table: $[3, 5], [5, 6], [6, 9], [9, 11], [11, 14]$.

2. Determine the left endpoint values: For a left Riemann sum, we use the $f(x)$-values corresponding to the left endpoints of each subinterval. From the table, these are: $f(3) = 17, \, f(5) = 12, \, f(6) = 11, \, f(9) = 10, \, f(11) = 8.$

3. Compute the width of each subinterval ($\Delta x$):

   • For $[3, 5]$, $\Delta x = 5 - 3 = 2$,
   • For $[5, 6]$, $\Delta x = 6 - 5 = 1$,
   • For $[6, 9]$, $\Delta x = 9 - 6 = 3$,
   • For $[9, 11]$, $\Delta x = 11 - 9 = 2$,
   • For $[11, 14]$, $\Delta x = 14 - 11 = 3$.

4. Calculate the Riemann sum: Multiply each $f(x)$ value by the corresponding $\Delta x$ and sum: $\text{Left Riemann Sum} = f(3) \cdot \Delta x_1 + f(5) \cdot \Delta x_2 + f(6) \cdot \Delta x_3 + f(9) \cdot \Delta x_4 + f(11) \cdot \Delta x_5.$ Substituting the values: $\text{Left Riemann Sum} = (17 \cdot 2) + (12 \cdot 1) + (11 \cdot 3) + (10 \cdot 2) + (8 \cdot 3).$

5. Simplify: $\text{Left Riemann Sum} = 34 + 12 + 33 + 20 + 24 = 123.$

Final Answer:

The left Riemann sum is 123.

Would you like a detailed explanation of any step or more examples? 😊

5 Related Questions:

1. How does a left Riemann sum differ from a right Riemann sum?
2. How do you compute a Riemann sum when the subinterval widths ($\Delta x$) are uniform?
3. Can you explain the connection between Riemann sums and definite integrals?
4. What happens if we increase the number of subintervals in a Riemann sum?
5. How would the result change if we used the midpoint Riemann sum for this problem?

Tip:

For irregular subintervals (different $\Delta x$), always double-check the widths before applying the sum formula.