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