HP 35s Scientific Calculator - Manual (Page 338)

E-2 More about Integration
As explained in chapter 8, the uncertainty of the final approximation is a number
derived from the display format, which specifies the uncertainty for the function. At
the end of each iteration, the algorithm compares the approximation calculated
during that iteration with the approximations calculated during two previous
iterations. If the difference between any of these three approximations and the other
two is less than the uncertainty tolerable in the final approximation, the calculation
ends, leaving the current approximation in the X–register and its uncertainty in the
Y–register.
It is extremely unlikely that the errors in each of three successive approximations —
that is, the differences between the actual integral and the approximations — would
all be larger than the disparity among the approximations themselves.
Consequently, the error in the final approximation will be less than its uncertainty
(provided that f(x) does not vary rapidly). Although we can't know the error in the
final approximation, the error is extremely unlikely to exceed the displayed
uncertainty of the approximation. In other words, the uncertainty estimate in the Y–
register is an almost certain "upper bound" on the difference between the
approximation and the actual integral.
Conditions That Could Cause Incorrect Results
Although the integration algorithm in the HP 35s is one of the best available, in
certain situations it — like all other algorithms for numerical integration — might
give you an incorrect answer. The possibility of this occurring is extremely remote.
The 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.
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 below, showing (over a
portion of the interval of integration) three functions whose graphs include the many
sample points in common.