More About Integration; How The Integral Is Evaluated - HP -32S Owner's Manual

D
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, JTN d x, calculates
the integral ofa function f(x) by computing a weighted average ofthe
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 con
sidered: 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 /(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 accu
racy 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 asis
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 off(x) would permit, the algo
rithm 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 justi
fied considering the inherent uncertainty in f(x).
D: More About Integration 273