Section 14: Numerical Integration 201
Because the accuracy of any integral is limited by the accuracy of the
function (as indicated in the display format), the calculator cannot compute
the value of an integral exactly, but rather only approximates it. The
HP-15C places the uncertainty
*
of an integral's approximation in the Y-
register at the same time it places the approximation in the X-register. To
determine the accuracy of an approximation, check its uncertainty by
pressing ®.
Example: With the display format set to i 2, calculate the integral in
the expression for J
1
(1) (from the example on page 197).
The integral is 1.38 ± 0.00188. Since the uncertainty would not affect the
approximation until its third decimal place, you can consider all the
displayed digits in this approximation to be accurate. In general, though, it
is difficult to anticipate how many digits in an approximation will be
unaffected by its uncertainty. This depends on the particular function being
integrated, the limits of integration, and the display format.
*
No algorithm for numerical integration can compute the exact difference between its approximation and
the actual integral. But the algorithm in the HP-
is the uncertainty of the approximation. For example, if the integral Si (2) is 1.6054 ± 0.0001, the
approximation to the integral is 1.6054 and its uncertainty is 0.0001. This means that while we don't know
the exact difference between the actual integral and its approximation, we do know that it is highly
unlikely that the difference is bigger than 0.0001. (Note the first footnote on page 200.)
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