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6. Simpson's Rule

by M. Bourne

The Trapezoidal Rule is an improvement over using rectangles because we have much less "missing" from our calculations. We used straight lines to model the curve in trapezoidal Rule.

We seek an even better approximation. In Simpson's Rule, we use parabolas to approximate each part of the curve. This proves to be very efficient. (See more about Parabolas.)

We can show (by integrating the area under each parabola and adding these areas) that the approximate area is given by Simpson's Rule:

Note: In Simpson's Rule, n must be EVEN.

In the LiveMath document, note how the approximation is very good, even with a small number of divisions.

LIVEMath

Exercise: Approximate using Simpson's Rule with n = 4.

Answer

Δx = (3 - 2)/4 = 0.25

y₀ = f(a) = f(2) = 1/(2 + 1) = 0.3333333

y₁ = f(a + Δx) = f(2.25) = 1/(2.25+1) = 0.3076923

y₂ = f(a + 2Δx) = f(2.5) = 1/(2.5+1) = 0.2857142

y₃ = f(a + 3Δx) = f(2.75) = 1/(2.75+1) = 0.2666667

y₄ = f(b) = f(3) = 1/(3+1) = 0.25

 

[The actual answer to this problem is 0.287682 so our Simpson's Rule approximation has an error of only 3.6 × 10^-4%.]

 

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Chapter Contents

• Integration
• 1. The Differential
• 2. Antiderivatives and The Indefinite Integral
• 3. The Area Under a Curve
• 4. The Definite Integral
• 5. Trapezoidal Rule
• 6. Simpson’s Rule
• Riemann Sums
• Integration Mini-lectures
• Integration Problem Solver
• Millionaire Calculus Game

• Comments, Questions?

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