HP -32S Owner's Manual page 279

The graph is a spike very close to the origin. Because no sample point
happened to discover the spike, the algorithm assumed that f(x) was
identically equal to zero throughout the interval of integration. Even if
you increased the number ofsample points by calculating the integral
in SCI 11 or ALL format, none of the additional sample points would
discover the spike when this particular function is integrated over this
particular interval. (For better approaches to problems such as this,
see the next topic, 'Conditions That Prolong Calculation Time.*)
Fortunately, functions exhibiting such aberrations (afluctuation thatis
uncharacteristic of the behavior of the function elsewhere) are un
usual enough that you are unlikely to have to integrate one unknow
ingly. Afunction that could lead to incorrect results can be identified
in simple terms by how rapidly it and its low-order derivatives vary
across the interval of integration. Basically, the more rapid the varia
tion in the function or its derivatives, and the lower the order of such
rapidly varying derivatives, the less quickly will the calculation finish,
and the less reliable will be the resulting approximation.
Note that the rapidity of variation in the function (or its low-order
derivatives) must be determined with respect to the width of the in
terval of integration. With a given number of sample points, a
function f(x) that has three fluctuations can be better characterized by
its samples when these variations are spread out over most of the in
terval of integration than if they are confined to only a small fraction
of the interval. (These two situations are shown in the following two
illustrations.) Considering the variations or fluctuation as a type of
oscillation in the function, the criterion of interest is the ratio of the
period of the oscillations to the width of the interval of integration:
the larger this ratio, the more quickly the calculation will finish, and
the more reliable will be the resulting approximation.
D: More About Integration 277