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.)

math expression

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

math expression

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 math expression 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

math expression

 

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



get MathTutorDVDs

Easy to understand math lessons on DVD. Try before you commit.
MathTutorDVD.com


Book mark this page in Del.icio.us, Furl, Digg, StumbleUpon, whatever...


Didn't find what you are looking for? Try search:


Need a break? Play a math game. Well, they all involve math... No, really!

dumbolf memoTST bola shadow factory mindfields trick-hoops-challenge crystal clear