HP 48g Graphing Calculator - Manual (Page 246)

20
The Accuracy Factor and the Uncertainty of Numerical
Integration
Numeric integration calculates the integral of a function f{x) by
computing a weighted average of the function’s values at many values
of X (sample points) within the interval of integration. The accuracy
of the result depends on the number of sample points considered:
generally, more sample points provide greater accuracy. There are two
reasons why you might want to limit the accuracy of the integral:
■ The length of time to calculate the integral increases as the number
of sample points increases.
m There are inherent inaccuracies in each calculated value of f{x):
□ Experimentally derived constants in f(x) may be inaccurate.
For example, if f(x) contains experimentally derived constants
that are accurate to only two decimal places, it is of little value
to calculate the integral to the full (T2-digit) precision of the
calculator.
□ If f(x) models a physical system, there may be inaccuracies in the
model.
n The calculator itself introduces round-off error into each
computation of f{x).
To indirectly limit the accuracy of the integral, you specify the
accuracy factor of the integrand f(x), defined as:
accuracy factor <
true value of f{x) — computed value of f(x)
computed value of f(x)
The accuracy factor is your estimation in decimal form of the error
in each computed value of/(*). You specify the accuracy factor by
setting the Display mode to n Fix. For example, if you set the display
mode to 2 Fix, the accuracy factor is 0.01, or 1%. If you set the
display mode to 5 Fix, the accuracy factor is 0.00001, or .001%.
The accuracy factor is related to the uncertainty of 'integration (a
measurement of the accuracy of the integral) by:
uncertainty of integration < accuracy factor x / \f{x)\d.x
20-6 Calculus and Symbolic Manipulation