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Simpson’s 3/8 Rule By: Mufan Yang

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**What is Simpson’s 3/8 Rule**

Simpson’s 3/8 is very similar to the Simpson’s Method that we already learned in class. The different is Simpson’s 3/8 method uses a third degree polynomial (cubic) to estimate the curve you are trying to find the integral of while Simpson’s Method (also called Simpson’s 1/3 Method) uses a second degree polynomial (quadratic)

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To get the estimate for an integral in this case, we will write out the 3rd degree polynomial using the general equation below and then integrate it.

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**How do we get the formula:**

We begin by integrating the 3rd degree Polynomial p(t)=c3t3+c2t2+c1t+c0.

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**Substitute 𝑐 0 , 𝑐 2 , 𝑐 3 , and 𝑐 1 values back into the equation.**

h = (𝑏−𝑎)/N , where N is a positive multiple of 3 so another way to look at this is h = (𝑏−𝑎)/3𝑛 , where n is the number of partitions being used.

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x y a y0=f(a) a+h y1=f(a+h) a+2h y2=f(a+2h) a+3h y3=f(b) Here we are using a single partition. 1 Partition = 𝑃 1 a a + h a + 2h b=a+3h

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Use 2 partitions for this formula: 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 ≈ 3ℎ 8 ( 𝑦 0 +3 𝑦 1 +3 𝑦 2 + 𝑦 3 ) + 3ℎ 8 ( 𝑦 3 +3 𝑦 4 +3 𝑦 5 + 𝑦 6 ) = 3 8 ( 𝑏−𝑎 3∗2 )( 𝑦 0 +3 𝑦 1 +3 𝑦 2 + 2𝑦 3 +3 𝑦 4 +3 𝑦 5 + 𝑦 6 ) = 3 8 ( 𝑏−𝑎 6 )(f(a)+3f(a+h)+3f(a+2h)+2f(a+3h)+3f(a+4h)+3f(a+5h)+f(b)) 𝑃 1 𝑃 2 a+3h a+4h a+5h b = a+6h

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Conclusion: So we can see that the equation for the Simpson’s 3/8 Rule is: Each of the function values will be multiplied by 3 or 2 except for the first and the last. The number of subinterval points will need to be multiples of 3’s.

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Example: 0 4 𝑥 3 𝑑𝑥 m = 1: h = 1: 0 4 𝑥 3 𝑑𝑥≈ 3 8 ∗1∗[𝑦 0 +3𝑦 1 +3𝑦 2 +𝑦(3)] = 3 8 ∗1∗78=29.25 m = 2: h = 1 2 : 0 4 𝑥 3 𝑑𝑥≈ 3 8 ∗ 1 2 ∗[𝑦 0 +3𝑦 𝑦 1 +2𝑦 𝑦 2 +3𝑦 5 2 +𝑦(3)] = 3 16 ∗108=20.25 m = 4: h = 1 4 : 0 4 𝑥 3 𝑑𝑥≈ 3 8 ∗ 1 4 ∗[𝑦 0 +3𝑦 𝑦 𝑦 𝑦 1 +3𝑦 𝑦 𝑦 𝑦 2 +2𝑦 𝑦 𝑦 𝑦(3)] = 3 32 * 216 =20.25

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**Simpson’s 1/3 Rule VS Simpson’s 3/8 Rule**

Advantages: Can use an odd or even number of subintervals rather than just an even number of subintervals as used in the Simpson’s 1/3 Rule. Can converge to estimated integral values much quicker than Simpson’s 1/3 Rule Has the same order of accuracy as the Simpson’s 1/3 Rule when calculating the absolute error, but the Simpson’s 3/8 Rule is actually slightly more accurate. But the fraction used in this error will be smaller and then will provide a more accurate error. Disadvantages: This rule requires multiple of 3 segments only. When calculating the value by hand this rule is more complicated. 1/3 Rule: and

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References Module for Simpson’s 3/8 Rule for Numerical Integration. (2012). Retrieved from Nguyen, Duc. Chapter 07.08: Simpson 3/8 Rule For Integration. University of South Florida. Wikipedia. Simpson’s Rule. Retrieved from

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