Integrity vs. Integration - an etymological perspective

It's interesting to see that we have educators who believe that they can maintain the 'integrity' of their programs by sticking to traditional, didactic methodologies, whilst they still want us to respect the traditional roots of language and the 'moral codes' of yore.

I present a simple argument based upon the etymological roots of 'integrity' and 'integration' (as borrowed from www.etymonline):

integrity
c.1450, "wholeness, perfect condition," from O.Fr. integrité, from L. integritatem (nom. integritas) "soundness, wholeness," from integer "whole" (see integer). Sense of "uncorrupted virtue" is from 1548.

integrate (v.)
1638, "to render (something) whole," from L. integratus, pp. of integrare "make whole," from integer "whole" (see integer). Meaning "to put together parts or elements and combine them into a whole" is from 1802.

Thus, aside from the obvious modern similarities between the words 'integrity' and 'integrate,' we find that the roots of both of these words are strikingly similar, in the sense that we are are striving toward the "whole," rather than just a disparate collection of parts.

Hence, is it 'reasonable' to educate in parallel domains without looking toward both the integrity and the integration of these parts? Is it reasonable for an educator to teach in isolation from history, sociology, technology, natural sciences, linguistics, and all of the other 'parts' that we've gradually transformed into 'specialities'?

Why do we look back at The Enlightenment and The Renaissance with their capitalized articles? Parallel programs do not make 'integrated' programs...and 'integrity' has unfortunately be nominalized to the point where it means little more than 'security' or 'lack of corruption' in modern colloquial terms.