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To calculate \( L_4 \) and \( R_4 \) for the function \( f(x) = 7x^2 \) over the interval \([0, 7]\), we’ll use the left and right Riemann sums with 4 subintervals.

### Step-by-Step Solution

1. **Determine the Width of Each Subinterval**:
   \[
   \Delta x = \frac{7 - 0}{4} = 1.75
   \]

2. **Calculate \( L_4 \) (Left Riemann Sum)**:
   For \( L_4 \), we use the function values at the left endpoints of each interval.

   The left endpoints for each subinterval over \([0, 7]\) with width \( \Delta x = 1.75 \) are:
   \[
   x_0 = 0, \quad x_1 = 1.75, \quad x_2 = 3.5, \quad x_3 = 5.25
   \]

   Now we calculate \( f(x) = 7x^2 \) at each left endpoint:
   \[
   f(x_0) = 7(0)^2 = 0
   \]
   \[
   f(x_1) = 7(1.75)^2 = 7 \cdot 3.0625 = 21.4375
   \]
   \[
   f(x_2) = 7(3.5)^2 = 7 \cdot 12.25 = 85.75
   \]
   \[
   f(x_3) = 7(5.25)^2 = 7 \cdot 27.5625 = 192.9375
   \]

   Then, \( L_4 \) is:
   \[
   L_4 = \Delta x \cdot (f(x_0) + f(x_1) + f(x_2) + f(x_3))
   \]
   \[
   = 1.75 \cdot (0 + 21.4375 + 85.75 + 192.9375)
   \]
   \[
   = 1.75 \cdot 300.125 = 525.219
   \]

3. **Calculate \( R_4 \) (Right Riemann Sum)**:
   For \( R_4 \), we use the function values at the right endpoints of each interval.

   The right endpoints for each subinterval over \([0, 7]\) are:
   \[
   x_1 = 1.75, \quad x_2 = 3.5, \quad x_3 = 5.25, \quad x_4 = 7
   \]

   Now we calculate \( f(x) = 7x^2 \) at each right endpoint:
   \[
   f(x_1) = 7(1.75)^2 = 21.4375
   \]
   \[
   f(x_2) = 7(3.5)^2 = 85.75
   \]
   \[
   f(x_3) = 7(5.25)^2 = 192.9375
   \]
   \[
   f(x_4) = 7(7)^2 = 7 \cdot 49 = 343
   \]

   Then, \( R_4 \) is:
   \[
   R_4 = \Delta x \cdot (f(x_1) + f(x_2) + f(x_3) + f(x_4))
   \]
   \[
   = 1.75 \cdot (21.4375 + 85.75 + 192.9375 + 343)
   \]
   \[
   = 1.75 \cdot 643.125 = 1125.469
   \]

### Final Answers
- \( L_4 = 525.219 \)
- \( R_4 = 1125.469 \)

Would you like a deeper explanation of any step?

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Here are 5 related questions that might help expand your understanding:

1. How would you calculate \( L_8 \) and \( R_8 \) with 8 subintervals instead of 4 for the same function?
2. What does the difference between \( L_4 \) and \( R_4 \) suggest about the behavior of \( f(x) = 7x^2 \) over \([0, 7]\)?
3. How can we improve the accuracy of the Riemann sum estimates?
4. Can we use the midpoint rule for this function over \([0, 7]\), and how would it compare to \( L_4 \) and \( R_4 \)?
5. What is the exact integral of \( f(x) = 7x^2 \) over \([0, 7]\), and how does it compare to \( L_4 \) and \( R_4 \)?

**Tip:** Using more subintervals in Riemann sums generally improves the approximation to the actual area under the curve.

Calculate L4 and R4 over [0, 7] for the function f(x) = 7x^2. (Use decimal notation. Give your answers to three decimal places.)

Learn how to calculate L4 and R4 for the function f(x) = 7x^2 over the interval [0, 7]. This problem demonstrates the use of left and right Riemann sums to approximate the area under a curve. Detailed step-by-step calculations are provided, suitable for students in Grades 11-12.

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