HP -15C Advanced Functions Handbook page 50

48 Section 2: Working With [7F]
There are additional considerations, however, when you're
integrating functions relating to an actual physical situation.
Basically, with such functions you should ask yourself whether the
accuracy you would like in the integral is justified by the accuracy
in the function. For example, if the function contains empirical
constants that are specified to only, say, three significant digits, it
might not make sense to specify more than three digits in the
display format.
Another important consideration—and one which is more subtle
and therefore more easily overlooked—is that nearly every
function relating to a physical situation is inherently inaccurate to
a certain degree, because it is only a mathematical model of an
actual process or event. A mathematical model is itself an approxi-
mation that ignores the effects of known or unknown factors which
are insignificant to the degree that the results are still useful.
An example of a mathematical model is the normal distribution
function
L
dx
which has been found to be useful in deriving information
concerning physical measurements on living organisms, product
dimensions, average temperatures, etc. Such mathematical descrip-
tions typically are either derived from theoretical considerations or
inferred from experimental data. To be practially useful, they are
constructed with certain assumptions, such as ignoring the effects
of relatively insignificant factors. For example, the accuracy of
results obtained using the normal distribution function as a model
of the distribution of certain quantities depends on the size of the
population being studied. And the accuracy of results obtained
from the equation s = s
0
— Vzgt
2
, which gives the height of a falling
body, ignores the variation with altitude of g, the acceleration of
gfavity.
Thus, mathematical descriptions of the physical world can provide
results of only limited accuracy. If you calculated an integral with
an apparent accuracy beyond that with which the model describes