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I know the formula of sphere and have seen number of derivations using with and without calculus.

I was using a common sense approach of using a slice and rotating this slice 360deg. Number of such slices would be 2(pi)(r) which is length of base circle of hemi-sphere. Then we can multiply the volume by 2 to get volume of entire sphere. I am using same logic as used for volume of a cylinder which is using a small slice and using number of such slices as height of the cylinder which is also its length.

volume of sphere calculation

What is wrong with my approach ?

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  • $\begingroup$ What is the question? $\endgroup$ – copper.hat Nov 18 '14 at 16:26
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    $\begingroup$ The "thickness" of your "slice" is not uniform (thinner close to the center). This is not an issue with cylinders since slicing the cylinder you get "uniform thickness" for the slices. $\endgroup$ – Milly Nov 18 '14 at 16:29
  • $\begingroup$ @Milly I dont think so. I am assuming thickness to be unity. $\endgroup$ – AnjumSKhan Nov 18 '14 at 16:33
  • $\begingroup$ @copper.hat question is : what is wrong with my approach. $\endgroup$ – AnjumSKhan Nov 18 '14 at 16:33
  • $\begingroup$ @AnjumSKhan You can't. Assuming thickness to be uniform is exactly what breaks your argument. $\endgroup$ – Milly Nov 18 '14 at 18:06
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Because of the way you slice the sphere.

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As I see your slices cannot be considered as cylinders because they all meet at the center. So their width decrease to zero at the center. Just imagine that you have an internal sphere into you sphere. All slices will be convereted in the corresponding subslices with the smaller initial width on the surface.

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  • $\begingroup$ with respect to your knowledge, can i ask for more clarification using images or otherwise. $\endgroup$ – AnjumSKhan Nov 18 '14 at 17:24
  • $\begingroup$ Sorry, I am not so good in posting images here. Just think about what I tried to explain. The Sphere can not built through rotating of cylinder of the infinitesimal width. $\endgroup$ – Alexander Vigodner Nov 18 '14 at 17:49
  • $\begingroup$ I am sorry again. Are you pointing to some property of sphere which I can't figure out ? Can't we take the slice shown in the image to be part of a perfect circle ? $\endgroup$ – AnjumSKhan Nov 18 '14 at 18:43
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    $\begingroup$ Ok let me give an analogy for 2d circle. An infinitestimal 2D slice is not a rectangle but triangle. Similarly 3d dimenstional slice with infinitestimal width is not a cylinder. $\endgroup$ – Alexander Vigodner Nov 18 '14 at 19:07
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Milly's comment :

The "thickness" of your "slice" is not uniform (thinner close to the center). This is not an issue with cylinders since slicing the cylinder you get "uniform thickness" for the slices.

I think this answers my question, so I am posting it as a self-answer. Because if we try to cut a slice from the exact center. The slice tapers into a corner resulting into obvious squeezing of thickness at the corner.

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