Find the exact value of fa²da. Find the error of approximation between the exact value and th value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.
Find the exact value of fa²da. Find the error of approximation between the exact value and th value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.
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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![### Problem Statement
Find the exact value of the integral \( \int_{2}^{6} x^2 \, dx \). Determine the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Illustrate this with a graph.
1. **The exact value is:** [__________]
2. **The error in the approximation is:** [__________]
Provide your answers accurate to four decimal places.
### Explanation
- **Integral Calculation**: This involves calculating the definite integral of the function \( x^2 \) from 2 to 6.
- **Trapezoidal Rule**: The approximation will use this method, which divides the interval into four subintervals and calculates areas of trapezoids under the curve.
- **Graph**: Would typically illustrate the comparison between the exact curve and the trapezoidal approximation, showing the areas of each trapezoid.
The results should highlight the small differences between the exact and approximated values, emphasizing numerical analysis techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F831589b4-5a0a-4072-b07c-b0f5f32736ed%2Fe43690d7-3f48-47a1-bc56-a9bac0a27024%2Fa61lg4r_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the exact value of the integral \( \int_{2}^{6} x^2 \, dx \). Determine the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Illustrate this with a graph.
1. **The exact value is:** [__________]
2. **The error in the approximation is:** [__________]
Provide your answers accurate to four decimal places.
### Explanation
- **Integral Calculation**: This involves calculating the definite integral of the function \( x^2 \) from 2 to 6.
- **Trapezoidal Rule**: The approximation will use this method, which divides the interval into four subintervals and calculates areas of trapezoids under the curve.
- **Graph**: Would typically illustrate the comparison between the exact curve and the trapezoidal approximation, showing the areas of each trapezoid.
The results should highlight the small differences between the exact and approximated values, emphasizing numerical analysis techniques.
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