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 =
s0
—
Vzgt2,
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
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