Simpson's Rule/Trapezoid RuleDate: 02/01/98 at 01:08:08 From: Allen Riddell Subject: Simpson's Rule/Trapezoidal Rule (Formula) for approximating integrals. I understand why the trapezoidal rule results in a better approximation of an integral. What I am uncertain about is how Simpson's rule is derived and why it is a better approximation of the integral than the midpoint (rule?) or trapezodial method. I'm finding it difficult to cope with my teacher's assertion that "it just is." I understand that Simpson's rule gets an approximation by using a parabola. What I'm completely confused on is why a parabola is on balance a better approximation than the trapazoid. Also - I don't see any x^2 or anything that suggests a curve in the actual equation that represents Simpson's rule. Jacob Krich is Swarthmore god! Date: 02/01/98 at 08:45:12 From: Doctor Jerry Subject: Re: Simpson's Rule/Trapezoidal Rule (Formula) for approximating integrals. Hi Allen, Perhaps an example will help. Let f(x)=e^x and consider int(0,1,e^x*dx). Divide the interval [0,1] into two parts with the subdivision {0,0.5,1}. The trapezoid rule can be seen as fitting a line to the curve between (0,0) and (0.5,e^0.5) and a second line between (0.5,e^0.5) and (1,e) and then, instead of integrating e^x over these two intervals, we integrate the equations of the two lines. Simpson's Rule can be seen as fitting a parabola to the three points (0,1), (0.5,e^0.5), and (1,e) and then, instead of integating e^x over these two intervals, we integrate the equation of the parabola. To my eye, the parabola does a better job of "snugging up" to the curve than the two lines. Here are some details: Integrating the two lines gives 1/4+sqrt(e)/2+e/4, which is 1.75393... This is the Trapezoid Rule. Integrating the parabola gives 1/6+2sqrt(e)/3+e/6, which is 1.71886... This is Simpson's Rule. The exact answer is e-1, which is 1.71828... -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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