Approximate the area under the graph of the function f(x) = 5x' from 3 to 11 for n = 4 and n = 8 subintervals by using lower and upper sums. (Use symbolic notation and fractions where needed.) (a) By using lower sums s, (rectangles that lie below the graph of f(x)). lower sum s4 = lower sum s8 = (b) By using upper sums S, (rectangles that lie above the graph of f(x)).

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Chapter2: Second-order Linear Odes
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**Title: Approximating the Area Under a Curve Using Lower and Upper Sums**

**Objective:**
Learn how to approximate the area under the graph of the function \( f(x) = 5x^3 \) from \( x = 3 \) to \( x = 11 \) using lower and upper sums with \( n = 4 \) and \( n = 8 \) subintervals.

**Instructions:**
Use symbolic notation and fractions where needed to calculate the following:

**(a) Lower Sums (\(s_n\)):**
Compute the area using rectangles that lie below the graph of \( f(x) \).

- **For \( n = 4 \) subintervals:**
  - Lower sum \( s_4 = \) [Enter Answer Here]

- **For \( n = 8 \) subintervals:**
  - Lower sum \( s_8 = \) [Enter Answer Here]

**(b) Upper Sums (\(S_n\)):**
Compute the area using rectangles that lie above the graph of \( f(x) \).

- **For \( n = 4 \) subintervals:**
  - Upper sum \( S_4 = \) [Enter Answer Here]

- **For \( n = 8 \) subintervals:**
  - Upper sum \( S_8 = \) [Enter Answer Here]

This exercise helps understand the concept of Riemann sums and the method of approximating integrals by dividing the area under a curve into a series of rectangles.
Transcribed Image Text:**Title: Approximating the Area Under a Curve Using Lower and Upper Sums** **Objective:** Learn how to approximate the area under the graph of the function \( f(x) = 5x^3 \) from \( x = 3 \) to \( x = 11 \) using lower and upper sums with \( n = 4 \) and \( n = 8 \) subintervals. **Instructions:** Use symbolic notation and fractions where needed to calculate the following: **(a) Lower Sums (\(s_n\)):** Compute the area using rectangles that lie below the graph of \( f(x) \). - **For \( n = 4 \) subintervals:** - Lower sum \( s_4 = \) [Enter Answer Here] - **For \( n = 8 \) subintervals:** - Lower sum \( s_8 = \) [Enter Answer Here] **(b) Upper Sums (\(S_n\)):** Compute the area using rectangles that lie above the graph of \( f(x) \). - **For \( n = 4 \) subintervals:** - Upper sum \( S_4 = \) [Enter Answer Here] - **For \( n = 8 \) subintervals:** - Upper sum \( S_8 = \) [Enter Answer Here] This exercise helps understand the concept of Riemann sums and the method of approximating integrals by dividing the area under a curve into a series of rectangles.
(b) By using upper sums \( S_n \) (rectangles that lie above the graph of \( f(x) \)).

- upper sum \( S_4 = \) [text box]

- upper sum \( S_8 = \) [text box]
Transcribed Image Text:(b) By using upper sums \( S_n \) (rectangles that lie above the graph of \( f(x) \)). - upper sum \( S_4 = \) [text box] - upper sum \( S_8 = \) [text box]
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