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To solve this problem, let’s break it down step by step.

### Given:
The function \( f(x) = \frac{3}{4 + x^2} \) needs to be integrated approximately using a Riemann Sum with left endpoints over the interval \([2, 4]\) and split into four intervals of equal width.

### Steps:
1. **Calculate the Interval Width:**
   Since we are dividing \([2, 4]\) into four equal intervals, the width \( \Delta x \) is:
   \[
   \Delta x = \frac{4 - 2}{4} = 0.5
   \]

2. **Determine the Endpoints:**
   Starting at \(x = 2\) and using a width of \(0.5\), the left endpoints for the intervals will be:
   - \( x_0 = 2 \)
   - \( x_1 = 2.5 \)
   - \( x_2 = 3 \)
   - \( x_3 = 3.5 \)

3. **Evaluate \( f(x) \) at Each Left Endpoint:**
   Substitute each of the \( x \)-values into \( f(x) \) and round to three decimal places:
   - \( f(x_0) = f(2) = \frac{3}{4 + 2^2} = \frac{3}{4 + 4} = \frac{3}{8} = 0.375 \)
   - \( f(x_1) = f(2.5) = \frac{3}{4 + (2.5)^2} = \frac{3}{4 + 6.25} = \frac{3}{10.25} \approx 0.293 \)
   - \( f(x_2) = f(3) = \frac{3}{4 + 3^2} = \frac{3}{4 + 9} = \frac{3}{13} \approx 0.231 \)
   - \( f(x_3) = f(3.5) = \frac{3}{4 + (3.5)^2} = \frac{3}{4 + 12.25} = \frac{3}{16.25} \approx 0.185 \)

4. **Calculate the Riemann Sum \( L_4 \):**
   Now, apply the formula for the left Riemann sum:
   \[
   L_4 = \Delta x \cdot (f(x_0) + f(x_1) + f(x_2) + f(x_3))
   \]
   Substitute the values:
   \[
   L_4 = 0.5 \cdot (0.375 + 0.293 + 0.231 + 0.185) = 0.5 \cdot 1.084 = 0.542
   \]

### Final Answers:
- (a) The left endpoints are: \( x_0 = 2 \), \( x_1 = 2.5 \), \( x_2 = 3 \), \( x_3 = 3.5 \).
- (b) Values of \( f(x) \) at each point:
  - \( f(x_0) = 0.375 \)
  - \( f(x_1) = 0.293 \)
  - \( f(x_2) = 0.231 \)
  - \( f(x_3) = 0.185 \)
- (c) The Riemann sum \( L_4 = 0.542 \).

Would you like further details on any part of this solution?

### Related Questions:
1. How would the result change if a right Riemann sum was used?
2. What is the effect on accuracy if the number of rectangles is increased?
3. How can we use midpoint Riemann sums for this function?
4. How do you calculate the exact area under the curve for \( f(x) \) using integration?
5. How would this Riemann sum differ if the interval was \([1, 4]\) instead?

#### Tip:
For improved accuracy, increase the number of intervals in the Riemann sum approximation, making \(\Delta x\) smaller.

Consider \( f(x) = \frac{3}{4 + x^2} \). For this question, we want to estimate the area under the curve between \( x = 2 \) and \( x = 4 \) using a Riemann Sum with left endpoints and four rectangles of equal width.

Calculate the area under the curve for f(x) = 3 / (4 + x^2) from x = 2 to x = 4 using a left Riemann sum with four intervals of equal width. Detailed step-by-step solution included, covering concepts of Riemann Sum and definite integral approximation. Suitable for Grades 11-12.

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