The midpoint rule with n subintervals (designated as 
) usually
gives better accuracy than either the left endpoint rule ( 
) or
the right endpoint rule ( 
). This means that, for a given n,
is generally closer to A than either or . In
numericcal analysis texts it is shown that the error, 
, in using
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 
on
, 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.