What are the conditions for the volume formula V = (H/6)(B + 4M + T) to work?

If H = height, B = area of bottom face, M = area of cross section halfway up, and T = area of top face, then the volume formula V = (H/6)(B + 4M + T) works for a variety of solids, such as a cone, a box, a fustrum, a pyramid, even a sphere or ellipsoid. When does it work or does not work? What are the conditions... show more If H = height, B = area of bottom face, M = area of cross section halfway up, and T = area of top face, then the volume formula V = (H/6)(B + 4M + T) works for a variety of solids, such as a cone, a box, a fustrum, a pyramid, even a sphere or ellipsoid. When does it work or does not work? What are the conditions necessary for it to work?
Update: Dr D, yes, it's based on Simpson's rule, which is an APPROXIMATION. Now, how about those exactly correct volume formulas for certain solids?
Update 2: Right, it won't work for all solids, even those solids of revolution. The question is, what are the necessary conditions for it to work?
Update 3: Thanks, everybody, for your interesting answers to this problem. I'm still trying to find exceptions to the rule, where the cross section is not necessarily a cubic function of height.
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