On my drive home I convinced myself that my method is correct (even if my explanation may not have been very good
I convinced myself with the following analogy. Imagine that the object in question--lets say it's a turned chair leg with lots of nice curves and a constantly changing diameter--is made not out of wood but out of something like an uncompressible yet malleable clay.
Constraining its longitudinal dimension, it then stands to reason that that chair leg can be reformed into one and only one solid cylinder. The longitudinal cross section of that cylinder will then have an area which is, of course, symmetric around the chair leg's long axis. Obviously this area must then be the same as the area of the chair leg's longitudinal cross section. Therefore, if I can measure (or estimate) the area of the chair leg's longitudinal cross section-because maybe, all I have is a 2D image of the chair leg-I can then come up with a corresponding cylinder (again, holding length constant) which has the same longitudinal cross sectional area and hence, derive a volume for the leg.
Which of course gets me nowhere
because I want to ultimately figure out the surface area of the leg, not the volume.
Oh well, at least my drive home wasn't boring