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.