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Actually, this isn't too far off. See http://mathworld.wolfram.com/PappussCentroidTheorem.html, where they talk about the 2nd Theorem of Pappus. If you determine the centroid of the region that's being revolved, and then determine the distance the centroid moves in rotation, the volume of the solid of revolution is the product of the distance the centroid moves, and the area of the region being revolved.why can't it be like this?
AHHH! that's it. wedge shape like cutting cakes. i now see why my formula can't work haha! yes it has to vary from the axis of symmetry! thank you for adding more insight into it :DYour intuition comes from the human faculty for pattern recognition. It works well as long as you understand where the patterns come from.
You got the formula you did by following the general pattern ... but the setup was generally fine: you can rotate a function like that, so long at the volume integrated is wedge shaped slices - like cutting a cake. It comes in handy if the shape being cut varies in distance from the symmetry axis ... i.e. what if the cylinder in the example above had an oval hole in it instead of a circular one?
If you explore this area you'll run ahead of your course ;)
i've always wondered if this was how it's supposed to be in geometry class. thank you for bringing that theorem up sir!Actually, this isn't too far off. See http://mathworld.wolfram.com/PappussCentroidTheorem.html, where they talk about the 2nd Theorem of Pappus. If you determine the centroid of the region that's being revolved, and then determine the distance the centroid moves in rotation, the volume of the solid of revolution is the product of the distance the centroid moves, and the area of the region being revolved.