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D–8 More about Integration
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24 Size : 17.7 x 25.2 cm
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
`
3
_
New upper limit.
{
G
%º%1.%2
Selects Equation mode; displays
the equation.
{
)
X
!!
∫
/)
Integral. (The calculation takes a
minute or two.)