hi spec00!
 Quote by spec00
… i've seen a proof for the volume of a sphere using slices of it with infinitesimal height. Since the slices are not cylinders, but little cones, that deduction wouldn't get us in the wrong place too?
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we get this question quite often! …
the error for the volume is the difference between 2πr dh and (roughly) 2π(r + dr/2) dh …
a second-order error of π drdh
but the error for the area is the difference between 2π dh and 2π dh/cosθ …
a first-order error of 2π(1 - secθ) dh 
(in layman's terms, most of the
volume is in the middle, and the error is only an edge-effect, but for the area, it's
all edge!
)