4.9 Simpson and other approximations to integrals 277function F for which f f (x) dx = F(x) orf(x) = F' (x) and can then evaluate
b
f(x) dx by the calculationfa(4.91) f 'f(x) dx= F(x)Ja =F(b) - F(a).
As was pointed out in Section 3.6, derivatives of elementary functions
are always elementary functions that can be calculated by use of appro-
priate rules. It must not be presumed, however, that if f is an elementary
function, then there must exist an elementary function F for which
f(x) = F'(x). While proofs of such things do not grow in ordinary
gardens, it is nevertheless known that if f(x) is one or another of
-%/1-I-x4, x , VI±sin'x,
1/4 - sine xthen there is no elementary function F for which f(x) = F'(x).
This section is devoted to methods by which we can obtain useful
decimal approximations to(4.92) Jab f (x) dxin cases where it is impossible or difficult to obtain a useful formula for a
function F such that (4.91) holds. Some pedestrian methods are worthy
of brief mention. When a reasonably accurate graph of f is drawn on
graph paper as in Figure 4.93, we can ob-
tain an informative approximation by
counting the squares and estimating the
partial squares that lie within the appro-
priate region. Chemists and others who
have access to scissors and appropriate
scales can cut out the region and weigh the
paper. Another method involves use of a
planimeter, an instrument which will reveal
a useful approximation to the area of a
region after it has been suitably adjusted
and a needle point on a movable arm has
traced the boundary of the region. In
some situations, the simplest and most
direct method is illustrated by Figure 4.931.
The interval a < x <- b is cut into n equala b x
Figure 4.93a b x
Figure 4.931subintervals of length h, where h = (b - a)/n, and a point xk* is selected
in the kth subinterval. Then(4.932) f ab f(x) dx= E + h
f(xx ),sin x 1 eywhere a is an error term and the sum is a particular Riemann sum. Of