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### Trapezoid Rule

Suppose we take just two terms in the sum from Eq.(2) including just with and . Then we have

What behavior of the function is consistent with taking just these two terms? We claim that if we include just the functions 1 and x and demand that keeping just two terms in the Taylor series, we can get and . Also, since we need just two weights for this lowest approximation, it makes sense that we would need behavior summarized in two pieces of information about the function's variation in the interval [a,b]. Another way of putting this is to say that taking the functions 1 and x amounts to a restriction to values of x which are close to zero, or thus necessarily values of a and b which are not too different.

So, let's demand that the integral of 1 and x over the interval be exactly equal to the sum of two weights times two function values. That is we have

These represent two equations in two unknowns. When solved for and we find that

The resulting approximation to the integral is called the trapezoid rule. The formula is:

where . Although we shall not show it here, it turns out to be the case that the error made when using the trapezoid rule is , so that if h is small, then the error can become small. However, when h is not small, this approximation is just too crude to be of general utility. The rule is called the trapezoid rule because the formula amounts to approximating the integral assuming that the area under a trapezoid formed using the values of the function at the ends of the interval.

Comer Duncan
Tue Sep 23 17:09:49 EDT 1997