HP HP-32S - Page 282

You canseethat this function is'interesting* only at small values ofx.
At
greater values ofx,the function isnotinteresting, since it decreases
smoothly and gradually in a predictable manner.
The algorithm samples the function with higher densities of sample
points untilthe disparity between successive approximations becomes
sufficientlysmall. For a narrow interval in an area where the function
is interesting, it takes less time to reach this critical density.
To achieve the same density of sample points, the total number of
sample points required over the larger interval is much greater than
the number required over the smaller interval. Consequently, several
more iterations are required over the larger interval to achieve anap
proximation with the same accuracy, and therefore calculating the
integral requires considerably more time.
Because the calculation time depends on how soon a certain density
of sample points is achieved in the region where the function isinter
esting, the calculation of the integral of any function will be
prolonged if the interval of integration includes mostiy regions where
the function isnot interesting. Fortunately, if you must calculate such
an integral,youcan modifythe problemso that the calculation time is
considerably reduced. Two such techniques are subdividing the inter
val of integration and transformation of variables. These methods
enable you to change the function or the limits of integration sothat
the integrand is better behaved over the interval(s) of integration.
280 D: More About Integration