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More about Integration E-1
E
More about Integration
This appendix provides information about integration beyond that given in chapter
8.
How the Integral Is Evaluated
The algorithm used by the integration operation, ∫ , calculates the integral of
a function f(x) by computing a weighted average of the function's values at many
values of x (known as sample points) within the interval of integration. The accuracy
of the result of any such sampling process depends on the number of sample points
considered: generally, the more sample points, the greater the accuracy. If f(x) could
be evaluated at an infinite number of sample points, the algorithm could —
neglecting the limitation imposed by the inaccuracy in the calculated function f(x) —
always provide an exact answer.
Evaluating the function at an infinite number of sample points would take forever.
However, this is not necessary since the maximum accuracy of the calculated
integral is limited by the accuracy of the calculated function values. Using only a
finite number of sample points, the algorithm can calculate an integral that is as
accurate as is justified considering the inherent uncertainty in f(x).
The integration algorithm at first considers only a few sample points, yielding
relatively inaccurate approximations. If these approximations are not yet as accurate
as the accuracy of f(x) would permit, the algorithm is iterated (repeated) with a
larger number of sample points. These iterations continue, using about twice as
many sample points each time, until the resulting approximation is as accurate as is
justified considering the inherent uncertainty in f(x).