Integrasi Numerik_Metode Simpson 1/3

Summary

This video delves into numerical integration methods, focusing on Simpson’s 1/3 and 3/8 rules. It explains how higher-order polynomials, such as parabolas and cubic functions, provide more accurate approximations for integrals compared to the trapezoidal method. The video demonstrates how Simpson’s rules reduce error through increasing the number of subintervals, with detailed examples and error analysis. The viewer learns how to apply these methods effectively, emphasizing the benefits of refining the approximation for greater precision in practical scenarios like physics and engineering.

Takeaways

• 😀 **Numerical Integration** is a technique used to approximate the area under a curve using various methods such as the Trapezoidal Rule and Simpson’s Rule.
• 😀 **Trapezoidal Rule** is a basic method where the area is approximated by dividing the interval into smaller sections and applying linear approximations.
• 😀 **Simpson’s 1/3 Rule** uses a quadratic polynomial (parabola) to fit through three points: the two endpoints and the midpoint of the interval, offering better accuracy than the Trapezoidal Rule.
• 😀 **Simpson’s 3/8 Rule** involves dividing the interval into four points and using a cubic polynomial for better approximation over larger subintervals.
• 😀 **Subdivision of Intervals**: To increase the accuracy of Simpson's methods, the interval can be divided into multiple smaller intervals. More intervals lead to lower error.
• 😀 **Simpson’s 1/3 Rule Formula**: The formula for Simpson’s 1/3 Rule is: ∫_a^b f(x) dx ≈ (b - a) / 3 [f(a) + 4f((a+b)/2) + f(b)]
• 😀 **Simpson’s 3/8 Rule Formula**: The formula for Simpson’s 3/8 Rule is: ∫_a^b f(x) dx ≈ 3(b - a) / 8 [f(a) + 3f((a+2b)/3) + 3f((2a+b)/3) + f(b)]
• 😀 **Accuracy Comparison**: Simpson’s 1/3 and 3/8 Rules provide more accurate results than the Trapezoidal Rule, especially as the number of subintervals increases.
• 😀 **Error Analysis**: The error can be reduced by increasing the number of subintervals. For example, the error from Simpson’s 1/3 Rule decreases from 5.9% to 0.5% by using more intervals.
• 😀 **Application Example**: For a function like f(x) = e^x between 0 and 4, Simpson’s 1/3 Rule and Simpson’s 3/8 Rule yield estimates that are compared to the exact solution to determine accuracy and error.

The video is abnormal, and we are working hard to fix it.
Please replace the link and try again.