Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral 15 1₁³²x³ Exact Trapezoidal Simpson's x³ dx, n = 6
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral 15 1₁³²x³ Exact Trapezoidal Simpson's x³ dx, n = 6
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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![**Using Numerical Methods to Approximate Definite Integrals**
In this exercise, we're focusing on approximating the value of a definite integral using the Trapezoidal Rule and Simpson's Rule.
We are given the definite integral:
\[
\int_{1}^{5} x^3 \, dx, \quad n = 6
\]
Here, \( n = 6 \) indicates the number of subintervals to be used in the approximation.
**Objective:**
- Calculate the exact value of the integral.
- Approximate the integral using the Trapezoidal Rule.
- Approximate the integral using Simpson’s Rule.
**Results Table:**
- **Exact**: [Calculate and insert the exact value here]
- **Trapezoidal**: [Calculate and insert the approximate value using the Trapezoidal Rule]
- **Simpson's**: [Calculate and insert the approximate value using Simpson's Rule]
**Explanation of Methods:**
1. **Trapezoidal Rule**:
This numerical method approximates the region under the graph of a function as a series of trapezoids and computes the area of these trapezoids to estimate the integral.
2. **Simpson's Rule**:
This method uses parabolic arcs instead of line segments to approximate the function. It generally provides a more accurate approximation than the Trapezoidal Rule, especially when dealing with a smooth curve.
By comparing the approximations from these methods with the exact value, students can gauge the effectiveness of each technique and understand how increasing the number of subintervals (\(n\)) can improve accuracy.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3738e16a-10fd-4378-b4a3-b8733d2d5bb5%2F0998ad40-6593-494d-b1ca-9a73733bbdb3%2Foh6oufp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Numerical Methods to Approximate Definite Integrals**
In this exercise, we're focusing on approximating the value of a definite integral using the Trapezoidal Rule and Simpson's Rule.
We are given the definite integral:
\[
\int_{1}^{5} x^3 \, dx, \quad n = 6
\]
Here, \( n = 6 \) indicates the number of subintervals to be used in the approximation.
**Objective:**
- Calculate the exact value of the integral.
- Approximate the integral using the Trapezoidal Rule.
- Approximate the integral using Simpson’s Rule.
**Results Table:**
- **Exact**: [Calculate and insert the exact value here]
- **Trapezoidal**: [Calculate and insert the approximate value using the Trapezoidal Rule]
- **Simpson's**: [Calculate and insert the approximate value using Simpson's Rule]
**Explanation of Methods:**
1. **Trapezoidal Rule**:
This numerical method approximates the region under the graph of a function as a series of trapezoids and computes the area of these trapezoids to estimate the integral.
2. **Simpson's Rule**:
This method uses parabolic arcs instead of line segments to approximate the function. It generally provides a more accurate approximation than the Trapezoidal Rule, especially when dealing with a smooth curve.
By comparing the approximations from these methods with the exact value, students can gauge the effectiveness of each technique and understand how increasing the number of subintervals (\(n\)) can improve accuracy.
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