Let f*f* be the function defined by f, of, x, equals, start fraction, 6, divided by, x, end fraction, .*f*(*x*)=*x*6​. If three subintervals of equal length are used, what is the value of the left Riemann sum approximation for integral, from, 1, to, 10, of, start fraction, 6, divided by, x, end fraction, d, x, question mark∫110​*x*6​*dx*? Round to the *nearest thousandth* if necessary.

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.

Let f(x) = 6/x. If three subintervals of equal length are used, what is the value of the left Riemann sum approximation for the integral from 1 to 10 of 6/x dx? Round to the nearest thousandth if necessary.

Learn how to compute the Left Riemann Sum approximation for the integral of the function 6/x over the interval from 1 to 10 using three subintervals. This problem covers the concept of Riemann sums and integral approximation methods, with a detailed solution and rounding to the nearest thousandth. Suitable for high school students (Grades 11-12).

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