We need to compute the Left Riemann Sum for the function:

$f(x) = \frac{6}{x}$

on the interval $[1,10]$ using three subintervals of equal length.

Step 1: Determine the Subinterval Width

The formula for the width of each subinterval ($\Delta x$) is:

$\Delta x = \frac{b-a}{n}$

where:

• $a = 1$ (left endpoint),
• $b = 10$ (right endpoint),
• $n = 3$ (number of subintervals).

$\Delta x = \frac{10 - 1}{3} = \frac{9}{3} = 3$

Step 2: Identify Left Endpoints

For the left Riemann sum, we use the function values at the left endpoints of each subinterval.

The subintervals are:

1. $[1,4]$, left endpoint: 1
2. $[4,7]$, left endpoint: 4
3. $[7,10]$, left endpoint: 7

Step 3: Evaluate Function at Left Endpoints

We evaluate $f(x) = \frac{6}{x}$ at $x = 1, 4, 7$:

$f(1) = \frac{6}{1} = 6$

$f(4) = \frac{6}{4} = 1.5$

$f(7) = \frac{6}{7} \approx 0.857$

Step 4: Compute the Left Riemann Sum

The Left Riemann Sum is given by:

$L_n = \sum f(x_i) \Delta x$

$L_3 = (6 + 1.5 + 0.857) \times 3$

$= 8.357 \times 3$

$\approx 25.071$

Final Answer

$\mathbf{25.071}$

Let me know if you want more details!

Related Questions:

1. How does the Right Riemann Sum compare to the Left Riemann Sum?
2. What happens if we increase the number of subintervals?
3. How can we approximate this integral using the Midpoint Rule?
4. How does this approximation compare to the exact integral result?
5. What is the Trapezoidal Rule approximation for this integral?

Tip:

The Left Riemann Sum tends to underestimate decreasing functions and overestimate increasing functions.