HP 15c - Manual (Page 249)

Appendix E: A Detailed Look at
f
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Conditions That Could Cause Incorrect Results
Although the f algorithm in the HP-15C is one of the best available, in
certain situations it – like nearly all algorithms for numerical integration –
might give you an incorrect answer. The possibility of this occurring is
extremely remote. The f algorithm has been designed to give accurate
results with almost any smooth function. Only for functions that exhibit
extremely erratic behavior is there any substantial risk of obtaining an
inaccurate answer. Such functions rarely occur in problems related to actual
physical situations; when they do, they usually can be recognized and dealt
with in a straightforward manner.
As discussed on page 240, the f algorithm samples the function f(x) at
various values of x within the interval of integration. By calculating a
weighted average of the function's values at the sample points, the
algorithm approximates the integral of f(x).
Unfortunately, since all that the algorithm knows about f(x) are its values at
the sample points, it cannot distinguish between f(x) and any other function
that agrees with f(x) at all the sample points. This situation is depicted in the
illustration on the next page, which shows (over a portion of the interval of
integration) three of the infinitely many functions whose graphs include the
finitely many sample points.