248 Appendix E: A Detailed Look at f

=∫aδ(x) dx

=∫ab [0.5×10−n+m(x) ] dx .b

This integral is calculated using the samples of δ(x) in roughly the same ways that the approximation to the integral of the function is calculated

using the samples of

ˆ

f (x) .

Because

is proportional to the factor 10-n, the uncertainty of an

approximation changes by about a factor of 10 for each digit specified in the display format. This will generally not be exact in i or ^ display format, however, because changing the number of digits specified may require that the function be evaluated at different sample points, so that δ(x) ~ 10m(x) would have different values.

Note that when an integral is approximated in •display format, m(x) = 0 and so the calculated uncertainty in the approximation turns out to be

=0.5×10-n(b – a).

Normally you do not have to determine precisely the uncertainty in the function. (To do so would frequently require a very complicated analysis.) Generally, it's more convenient to use i or ^display format if the uncertainty in the function's values can be more easily estimated as a relative uncertainty. On the other hand, it’s more convenient to use • display format if the uncertainty in the function’s values can be more easily estimated as an absolute uncertainly. • display format may be inappropriate to use (leading to peculiar results) when you are integrating a function whose magnitude and uncertainty have extremely small values within the interval of integration. Likewise, i display format may be inappropriate to use (also leading to peculiar results) if the magnitude of the function becomes much smaller than its uncertainty. If the results of calculating an integral seem strange, It may be more appropriate to calculate the integral in the alternate display format.