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Formula for the Trapezoid Rule


Date: 07/31/99 at 10:10:54
From: Lester J. Snyder
Subject: The Trapezoid Rule

I have searched on the Web, but cannot find the formula for the 
trapezoid rule.  I also need to know how to use it.  (I am writing a 
program to find the moment area of any shape.)


Date: 08/01/99 at 22:58:12
From: Doctor Jaffee
Subject: Re: The Trapezoid Rule

To find an approximation for the integral of some function that is 
integrable from x(0) to x(n), divide the x-axis into equal parts and 
draw vertical segments from the endpoints of each part to a point on 
the curve. Connect those points and you will have n trapezoids whose 
total accumulation approximates the integral of the function for that 
interval. That is the idea behind the trapezoid rule.

To use geometry to find the area of each trapezoid, add the total 
areas (keeping in mind that areas below the x-axis will have negative 
values). The result will be the width of the partition divided by 
2 times f(x(0)) + 2f(x(1)) +2f(x(2)) + 2f(x(3)) + ... + 2f(x(n-1)) + 
f(x(n)), which is the formula for the trapezoid rule.

An interesting Web page that illustrates this procedure can be found 
at:

   http://www.cs.utah.edu/~zachary/isp/applets/Integrate/Integrate.html   

Just follow the instructions (there is a useful help screen).

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus
High School Geometry
High School Triangles and Other Polygons

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