Subdividing The Interval Of Integration - HP -15C Advanced Functions Handbook

50 Section 2: Working With [TT]
Subdividing the Interval of Integration
In regions where the slope of f(x) is varying appreciably, a high
density of sample points is necessary to provide an approximation
that changes insignificantly from one iteration to the next.
However, in regions where the slope of the function stays nearly
constant, a high density of sample points is not necessary. This is
because evaluating the function at additional sample points would
not yield much new information about the function, so it would not
dramatically affect the disparity between successive approxima-
tions. Consequently, in such regions an approximation of
comparable accuracy could be achieved with substantially fewer
sample points: so much of the time spent evaluating the function in
these regions is wasted. When integrating such functions, you can
save time by using the following procedure:
1. Divide the interval of integration into subintervals over
which the function is interesting and subintervals over
which the function is uninteresting.
2. Over the subintervals where the function is interesting,
calculate the integral in the display format corresponding to
the accuracy you would like overall.
3. Over the subintervals where the function either is not
interesting or contributes negligibly to the integral, calculate
the integral with less accuracy, that is, in a display format
specifying fewer digits.
4. To get the integral over the entire interval of integration, add
together the approximations and their uncertainties from
the integrals calculated over each subinterval. You can do
this easily using the | £ + | key.
Before subdividing the integration, check whether the calculator
underflows when evaluating the function around the upper (or
lower) limit of integration.* Since there is no reason to evaluate the
function at values of x for which the calculator underflows, in some
cases the upper limit of integration can be reduced, saving
considerable calculation time.
* When the calculation of any quantity would result in a number less than 10 , the
result is replaced by zero. This condition is known as underflow.