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.

Tue Nov 5 14:24:02 EST 1996