I'm trying to wrap my head around the second moment of area and I can understand its meaning in engineering calculations (it tells the strength of the shape in the sense of its cross-sectional area, http://www.learneasy.info/MDME/MEMmods/MEM30006A/Area_Moment/Area_Moment.html).
But I cannot grasp how exactly does the mathematical formulation produce this kind of measure. Or whether this measure is even the original meaning of the second moment of area.
Also how is this really conceptually different from the first moment of area?
This may not be the answer you're looking for but here's how I see it. We know by definition that the moment if inertia for a point mass is $I=mr^2$ where $r $ is the distance from the axis of rotation. So for the moment of area, we are just considering a region with constant density of 1 so that the area is essentially the same as the "mass". Then to get the moment of area/inertia of that region we would just sum up all the contributions of each point mass in the region. Since the area, $dA $ is the "mass" because density is 1 we just say $$\int\int _R r^2dA$$