Edit: I haven't read the last comments made while I typed - sorry!
My club mates are probably not going to appreciate this, but let me give this a try. My collection of basic aero texts is mostly older ones that do not focus on compressibility and transonics/supersonics. Five of my seven texts develop the idea of MAC, one assumes its use, and the last seems to over-simplify it. Most texts fail to answer your question, stopping at the idea of it being the chord length of an "equivalent" rectangular wing or similar idea. Futhermore, I've found that some internet "authorities" have it wrong as far as curved wings are concerned, opting for invalid short cuts. I'm going to try here - 'hope I don't #8/?@&!#& this up.
First, every MAC does have a length and spanwise position, BUT the MAC is not necessarily the same length as the actual wing chord at that position. So some expectations should be discarded.
The MAC definitions are all based on the simplifying assumption that every point on a wing exerts the same amount of lift for a given aoa. So it is an approximation, and we can expect any answer to give a chord actually located outboard of what aerodynamic effects like tip losses would otherwise dictate. By this approximation, the lift at any point along a semispan is simply proportional to the chord at that position. That reduces the position of the MAC to a geometrical determination - no tip losses, camber changes, etc. are considered. For a tapered or elliptical wing then, we expect the MAC to be located closer to the root than the tip, because the larger chords are there. For twisted wings, varying lift coefficients have to be factored in later. For the straight tapered wing, it turns out that the MAC is the same size as the actual chord at its position, but for an elliptical wing, the MAC is really shorter than the actual chord at that position (a lot of internet folks get this wrong by short cutting). So those are our limitations and expectations.
The MAC is not the "average" chord. For its location, we would get a better figure if we found the point where the chord divides the half-span area into two equal parts, but even that is not quite the spanwise location. The MAC position computation actually takes moments about the root so that not just chord length, but chord distance from the root are taken into account. MAC is actually located at this idealized center of lift, even though its length is not necessarily equal to the actual wing chord at that point. Think of it as located at the center of gravity of a wing cut-out of uniform material.
The integral definitions for MAC and its location account for this. The MAC equation just takes an infinite number of thin chord-wise rectangular areas making up the right (+) wing half, multiplies each of these area incriments by its local chord (we're finding a 'weighted average' of the chords), and divide their sum by the entire right-wing area. Think of this as multiplying each chord by it's own fraction of the wing area (da/A) and adding them together.
To find the MAC's distance from the root, we just multiply each of the infinite number of local chord-wise rectangular areas by its distance from the wing center (root) and again divide by the total area - same as multiplying the spanwise distances by their fractions of the entire area and adding the results. This just sums up the lift moments about the root and divides them by the entire lift (lift is proportional to the area in the simplified assumption) to leave only a span-wise distance.
So I think these pass through the centroid, although that often confuses this old-er mind. Without showing pictures and math, this is probably pretty obstruse, and I hope I have not misused the language. A couple years ago, I worked all this out for tapered and elliptical wings of varying sweep and dimensions. I checked my work many ways, and the derivations and sample answers are correct. If anyone wants to examine this 7-pg. document, typed out using the Microsoft equation application, I'll be happy to e-mail it to you; my scanner doesn't like that document. I've gotta stop now!
SK