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Error bounds

The midpoint rule with n subintervals (designated as tex2html_wrap_inline251 ) usually gives better accuracy than either the left endpoint rule ( tex2html_wrap_inline253 ) or the right endpoint rule ( tex2html_wrap_inline255 ). This means that, for a given n, tex2html_wrap_inline251 is generally closer to A than either tex2html_wrap_inline253 or tex2html_wrap_inline255 . In numericcal analysis texts it is shown that the error, tex2html_wrap_inline263 , in using tex2html_wrap_inline251 to approximate the area under y = f(x) on [a,b] satisfies


where B is the absolute maximum of |f''(x)| on [a,b]. In practice, B is often approximated by a number K that is an upper bound for B, that is B < K. For instance, if tex2html_wrap_inline287 on tex2html_wrap_inline289 , K might be taken as 4. Do you see why? For more complicated functions, Maple can be used to get a value for K that is close to B. Note that the error bound formula gives a worst case estimate, the accuracy achieved for a given number of subintervals n may be much better than the guarantee given by the formula.

Sean O Anderson
Tue Nov 5 14:24:02 EST 1996