c. Use the graph to find the Right-endpoint Riemann sum for f(z) on the interval [0, 8] with the partition P = {0, 1, 4, 6, 8}: Σf(ck) - Azk d. If the function, y = f(z) graphed above represents flow rate of water in gallons per minute where is in minutes Loves
c. Use the graph to find the Right-endpoint Riemann sum for f(z) on the interval [0, 8] with the partition P = {0, 1, 4, 6, 8}: Σf(ck) - Azk d. If the function, y = f(z) graphed above represents flow rate of water in gallons per minute where is in minutes Loves
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I am not sure how to complete the last two parts of the attached question. Please help.
![### Calculus: Integral and Riemann Sum Applications
#### Given Information:
You have the graph of \( y = f(x) \) and the information:
\[ \int_{0}^{8} f(x) \, dx = \frac{40}{3} \]
#### Graph Description:
The graph presented depicts the function \( y = f(x) \) over the interval [0, 18]. Key characteristics of the graph are:
- The function starts from the point (0, -2) and exhibits a peak at (6, 7).
- It descends steeply to the point (12, -7) and then flattens out to the point (15, -7), eventually ending at (18, -7).
#### Tasks:
1. **Compute the integral: \(\int_{0}^{18} f(x) \, dx\)**
- Answer: \( \boxed{14} \)
2. **Compute the integral: \(\int_{4}^{0} f(x) \, dx\)**
- Answer: \( \boxed{-\frac{8}{3}} \)
3. **Use the graph to find the Right-endpoint Riemann sum for \( f(x) \) on the interval [0, 8] with the partition \( P = \{0, 1, 4, 6, 8\} \)**
- Answer:
\[
\sum_{k=1}^{4} f(c_k) \Delta x_k = \boxed{}
\]
4. **Interpretation of the integral result:**
- If the function \( y = f(x) \) represents the flow rate of water in gallons per minute where \( x \) represents minutes, write a sentence to interpret the value of your answer in part a.
#### Detailed Diagram Explanation:
- The x-axis ranges from 0 to 18, with intervals of 1 unit.
- The y-axis ranges from -10 to 10, with intervals of 1 unit.
- The function starts negative, rises sharply to a positive peak, then descends to a lower negative value, and finally flattens out.
By analyzing the given graph and formulas, these exercises guide you in applying integral calculus concepts to calculate areas under curves and perform Riemann sums, as well as in interpreting results within real-world contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f31f885-3acd-4362-8d52-f6c21adcaba6%2F183ea1eb-ace3-48da-b0db-6dfe8227dcdc%2F3scrv4n_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus: Integral and Riemann Sum Applications
#### Given Information:
You have the graph of \( y = f(x) \) and the information:
\[ \int_{0}^{8} f(x) \, dx = \frac{40}{3} \]
#### Graph Description:
The graph presented depicts the function \( y = f(x) \) over the interval [0, 18]. Key characteristics of the graph are:
- The function starts from the point (0, -2) and exhibits a peak at (6, 7).
- It descends steeply to the point (12, -7) and then flattens out to the point (15, -7), eventually ending at (18, -7).
#### Tasks:
1. **Compute the integral: \(\int_{0}^{18} f(x) \, dx\)**
- Answer: \( \boxed{14} \)
2. **Compute the integral: \(\int_{4}^{0} f(x) \, dx\)**
- Answer: \( \boxed{-\frac{8}{3}} \)
3. **Use the graph to find the Right-endpoint Riemann sum for \( f(x) \) on the interval [0, 8] with the partition \( P = \{0, 1, 4, 6, 8\} \)**
- Answer:
\[
\sum_{k=1}^{4} f(c_k) \Delta x_k = \boxed{}
\]
4. **Interpretation of the integral result:**
- If the function \( y = f(x) \) represents the flow rate of water in gallons per minute where \( x \) represents minutes, write a sentence to interpret the value of your answer in part a.
#### Detailed Diagram Explanation:
- The x-axis ranges from 0 to 18, with intervals of 1 unit.
- The y-axis ranges from -10 to 10, with intervals of 1 unit.
- The function starts negative, rises sharply to a positive peak, then descends to a lower negative value, and finally flattens out.
By analyzing the given graph and formulas, these exercises guide you in applying integral calculus concepts to calculate areas under curves and perform Riemann sums, as well as in interpreting results within real-world contexts.
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