A Method for Checking Round Or Periodic Integrals of Functions

Most basically, when checking a round integral, re-do the integral with double the span. If the integral was over 2 Pi, for instance, then integrate over 4 Pi and divide the result by two. This second answer should be the same as the first answer. If not, there is a discontinuity in the integral, hence it is not completely round. Do not make any assumptions about symmetry! In a math book, the student is often instructed to 'notice the symmetry' and use this to simplify the integration. For example, instead of integrating from zero to 2 Pi, the student is told to notice the symmetry and integrate from 0 to Pi, then double the answer. Unfortunately, the only way to 'notice the symmetry' is to do the whole integral!

Changing the limits of integration based on some presumed symmetry should be a no-no, as this amounts to changing the limits of the integral during the integration. More exactingly, an arbitrarily small number called Delta is presumed. An integral from zero to (2 Pi minus Delta) should be approximately equal to an integral from zero to (2 Pi plus Delta) as long as Delta is quite small when compared with 2 Pi. In the limit where Delta is vanishingly small, both of these answers should be exactly the same as the original integral from zero to 2 Pi. If the answers are not the same as the original integral, then some sort of discontinuity exists in the original function. As such, the function HAS NO SYMMETRY. This same check should be performed at both ends of the integral, for example, integrate from (Zero minus Delta) through 2 PI, then integrate from (Zero plus Delta) through 2 PI. In the limit where delta becomes vanishingly small, both of these answers should be exactly the same as the original integral from zero to 2 Pi.