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.
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