28 52²₂ Use Simpson's Rule and all the data in the following table to estimate the value of the integral X y 22 23 8 3 24 25 26 27 28 -3 -4 1 6 7 Round your answer to 4 decimal places ydx.
28 52²₂ Use Simpson's Rule and all the data in the following table to estimate the value of the integral X y 22 23 8 3 24 25 26 27 28 -3 -4 1 6 7 Round your answer to 4 decimal places ydx.
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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![**Estimating Integrals Using Simpson's Rule**
To estimate the value of the integral \( \int_{22}^{28} y \, dx \) using Simpson's Rule, we have the following data in the table:
| x | 22 | 23 | 24 | 25 | 26 | 27 | 28 |
|----|----|----|----|----|----|----|----|
| y | 8 | 3 | -3 | -4 | 1 | 6 | 7 |
**Instructions:**
1. Use all the data provided in the table.
2. Apply Simpson's Rule to estimate the integral.
3. Round your final answer to four decimal places.
### Explanation of Simpson's Rule
Simpson's Rule is a numerical method for approximating the value of a definite integral. It works by dividing the interval of integration into an even number of subintervals and approximating the integrand using quadratic polynomials.
- For the integral \(\int_{a}^{b} f(x) \, dx\), divide the interval \([a, b]\) into \(n\) subintervals of equal width \(h = \frac{b-a}{n}\).
- Simpson's Rule is given by:
\[
\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right]
\]
**Important Notes:**
- Ensure \(n\) is even.
- Apply the rule by substituting corresponding \(x\) and \(y\) values from the table into the formula.
Once the calculation is done, input the estimated value in the provided space, rounded to four decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54cb8c7a-f5c8-4f70-beb9-4c302d85da57%2Fa4196d67-505f-4588-afd5-df66af13179d%2Frb8ql6l_processed.png&w=3840&q=75)
Transcribed Image Text:**Estimating Integrals Using Simpson's Rule**
To estimate the value of the integral \( \int_{22}^{28} y \, dx \) using Simpson's Rule, we have the following data in the table:
| x | 22 | 23 | 24 | 25 | 26 | 27 | 28 |
|----|----|----|----|----|----|----|----|
| y | 8 | 3 | -3 | -4 | 1 | 6 | 7 |
**Instructions:**
1. Use all the data provided in the table.
2. Apply Simpson's Rule to estimate the integral.
3. Round your final answer to four decimal places.
### Explanation of Simpson's Rule
Simpson's Rule is a numerical method for approximating the value of a definite integral. It works by dividing the interval of integration into an even number of subintervals and approximating the integrand using quadratic polynomials.
- For the integral \(\int_{a}^{b} f(x) \, dx\), divide the interval \([a, b]\) into \(n\) subintervals of equal width \(h = \frac{b-a}{n}\).
- Simpson's Rule is given by:
\[
\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right]
\]
**Important Notes:**
- Ensure \(n\) is even.
- Apply the rule by substituting corresponding \(x\) and \(y\) values from the table into the formula.
Once the calculation is done, input the estimated value in the provided space, rounded to four decimal places.
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