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
Related questions
Question
100%
![### 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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
