Conditions
That
Prolong
Calculation
Time
In the preceding
example,
the algorithm gavean
incorrect
answer be
cause it never detected the spike in the function. This happened
because the variation in the function was too quick relative to the
width of the interval of integration. If the width of the interval were
smaller,
you
would
get the
correct
answer;
but it
would
take a very
long time if the interval
were
still too
wide.
Consider
an
integral
where
the
interval
of
integration
is
wide
enough
to
require
excessive
calculation
time,
but not so
wide
that it
would
be
calculated
incorrectly.
Note that
because
f(x)
=
xe~x
approaches
zero
very
quickly
asx
approaches
oo,
the
contribution
to
the
integral
of
the
function
at
large
values
ofxis
negligible.
Therefore,
you
can
evaluate
the
integral
by
replacing
oo,
the
upper
limit
of
integration,
bya
num
ber
not
so
large
as
10499—say
10*
Re-run
the
previous
integration
problem
with
this
new
limit
of
inte
gration.
If
you
have
notrun any
other
integrations
in
the
meantime,
you do not have to
re-specify
FN=
F.
Keys:
Display:
0 I
ENTER
I
ID
3
1E3_
HI SOLVE/J |
{/FN}
X
/=1.000E0
l**y|
1.824E-4
Description:
New upper limit.
Integral. (The calcula
tion
takes
a
while.)
Uncertainty of
approximation.
Thisis the correct
answer,
but it took a verylong time.
To
understand
why,
compare
the
graph
of
the
function
between
x= 0
and
x=
103,
which looks about the same as that shown on page 276, with the
graph of the
function
between
x = 0 and x =
10:
D:
More
About
Integration
279
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