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The problem asks for two things:

1. **Set up the definite integral for the area** between the curve \( f(x) = \sqrt{x} \) and the x-axis over the interval \([6, 18]\).
2. **Approximate the area using the left sum** with the given partition \( P = \{6, 10, 14, 18\} \).

### Step 1: Set up the definite integral

The function given is \( f(x) = \sqrt{x} \). The definite integral for the area between the graph of \( f(x) \) and the x-axis over the interval \([6, 18]\) is:

\[
A = \int_6^{18} \sqrt{x} \, dx
\]

### Step 2: Approximate the area using the left sum

The left sum approximation uses the left endpoints of each subinterval defined by the partition \( P = \{6, 10, 14, 18\} \). These subintervals are:

- From \( x = 6 \) to \( x = 10 \)
- From \( x = 10 \) to \( x = 14 \)
- From \( x = 14 \) to \( x = 18 \)

For each subinterval, the left endpoint is used to evaluate \( f(x) \) (i.e., \( f(6), f(10), f(14) \)) and these values are multiplied by the width of the corresponding subinterval.

#### Step-by-step calculation of the left sum:

1. **Width of each subinterval**:
   - \( \Delta x_1 = 10 - 6 = 4 \)
   - \( \Delta x_2 = 14 - 10 = 4 \)
   - \( \Delta x_3 = 18 - 14 = 4 \)

2. **Evaluating the function at the left endpoints**:
   - \( f(6) = \sqrt{6} \)
   - \( f(10) = \sqrt{10} \)
   - \( f(14) = \sqrt{14} \)

3. **Left sum approximation**:
   \[
   A_{\text{left sum}} = \sum \left( f(x_i) \cdot \Delta x_i \right)
   \]
   Substituting the values:
   \[
   A_{\text{left sum}} = \sqrt{6} \cdot 4 + \sqrt{10} \cdot 4 + \sqrt{14} \cdot 4
   \]
   Factoring out the 4:
   \[
   A_{\text{left sum}} = 4 \left( \sqrt{6} + \sqrt{10} + \sqrt{14} \right)
   \]

Let me calculate this approximation.The left sum approximation for the area is:

\[
A_{\text{left sum}} \approx 37.41
\]

Would you like more details about the process, or do you have any further questions?

Here are five related questions to explore this topic further:

1. How would the right sum approximation differ from the left sum?
2. Can you use the midpoint Riemann sum to approximate the area?
3. How does increasing the number of partitions affect the accuracy of the approximation?
4. What is the exact value of the definite integral \( \int_6^{18} \sqrt{x} \, dx \)?
5. How does the trapezoidal rule compare to the left sum for approximating integrals?

**Tip**: When using a left sum approximation, the more partitions you have, the closer your approximation will be to the actual area under the curve.

Set up the definite integral for the area A of the region between the graph of f(x) = √x and the x-axis on [6, 18] and then approximate A by using the left sum with the partition P = {6, 10, 14, 18}.

Learn how to set up the definite integral for the area under the curve of f(x) = √x between [6, 18] and approximate it using the left Riemann sum with partition P = {6, 10, 14, 18}. This problem involves definite integrals, square root functions, and Riemann sums. Suitable for advanced high school or early college-level students.

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