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When deriving the formula for the area of a solid rotated about an axis, my textbook (Stewart calculus) uses portions of circular cones and integrates their lateral areas. I understand this process.

My question is why doesn't integrating the lateral areas of circular cylinders work as well. After all, we can use rectangles (and not trapezoids) to evaluate areas under curves, using cylinders as opposed to cone segments seems like the three dimensional equivalent.

I've been through both processes and I see why they don't give the same formula, but conceptually I don't understand why cylinders are wrong.

Thanks a lot

NSF
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    This has been discussed [here](http://math.stackexchange.com/questions/6979/areas-versus-volumes-of-revolution-why-does-the-area-require-approximation-by-a) – GaussTheBauss Dec 30 '15 at 07:27
  • [another great answer](http://math.stackexchange.com/questions/82788/why-cant-i-use-the-disk-method-to-compute-surface-area?lq=1) – GaussTheBauss Dec 30 '15 at 08:16

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