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More about Integration
E–7
File name 33s-English-Manual-040130-Publication(Edition 2).doc Page : 388
Printed Date : 2004/1/30 Size : 13.7 x 21.2 cm
In many cases you will be familiar enough with the function you want to integrate
that you will know whether the function has any quick wiggles relative to the
interval of integration. If you're not familiar with the function, and you suspect that
it may cause problems, you can quickly plot a few points by evaluating the
function using the equation or program you wrote for that purpose.
If, for any reason, after obtaining an approximation to an integral, you suspect its
validity, there's a simple procedure to verify it: subdivide the interval of integration
into two or more adjacent subintervals, integrate the function over each subinterval,
then add the resulting approximations. This causes the function to be sampled at a
brand new set of sample points, thereby more likely revealing any previously
hidden spikes. If the initial approximation was valid, it will equal the sum of the
approximations over the subintervals.
Conditions That Prolong Calculation Time
In the preceding example, the algorithm gave an incorrect answer because 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 as x approaches
∞
,
the contribution to the integral of the function at large values of
x is negligible.
Therefore, you can evaluate the integral by replacing
∞
, the upper limit of
integration, by a number not so large as 10
499
— say 10
3
.
Rerun the previous integration problem with this new limit of integration:
Keys: Display: Description:
0
Ï
}
3
_
New upper limit.
º
d
Selects Equation mode; displays
the equation.
º
"
X
∫
Integral. (The calculation takes a
minute or two.)