Given the revised values:

- \( f(-7) = 4 \)
- \( f(-4) = 4 \)
- \( f(1) = 2 \)
- \( f(3) = 1 \)

We can now recalculate the left-end approximation \( L_n \).

### Step 1: Recalculate the Left Riemann Sum \( L_n \)
Using the partition points \(\{-7, -4, 1, 3, 7\}\) and the given function values:

\[
L_n = f(-7) \cdot \Delta x_1 + f(-4) \cdot \Delta x_2 + f(1) \cdot \Delta x_3 + f(3) \cdot \Delta x_4
\]

Substituting the function values and the widths of the subintervals:

- \( \Delta x_1 = 3 \)
- \( \Delta x_2 = 5 \)
- \( \Delta x_3 = 2 \)
- \( \Delta x_4 = 4 \)

### Step 2: Substitute the Values into the Formula
\[
L_n = (4) \cdot 3 + (4) \cdot 5 + (2) \cdot 2 + (1) \cdot 4
\]

### Step 3: Perform the Calculations
\[
L_n = 12 + 20 + 4 + 4 = 40
\]

### Final Answer
The left-end approximation \( L_n \) of the area under the curve is **40**.

Would you like more details or have any other questions?

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**Related Questions:**

1. How does the Riemann sum approximation change with different partitions?
2. If you had a finer partition, how would \( L_n \) be affected?
3. How can you verify the accuracy of this approximation?
4. How does the shape of \( f(x) \) affect the Riemann sum approximation?
5. What are the advantages of using a left Riemann sum over other types?
6. How would you graphically interpret the sum \( L_n = 40 \)?
7. Can the Riemann sum be applied to functions with more complex shapes?
8. How would \( L_n \) change if the function values were negative?

**Tip:** Always check the consistency of your partition widths and function values before calculating the Riemann sum to avoid errors in approximation.

Learn how to calculate the left-end Riemann sum using specified function values at partition points. This problem involves using the Left Riemann Sum formula to approximate the area under a curve with detailed step-by-step calculations.

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