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.

Wed Nov 1 13:27:42 EST 1995