As ane fule kno the volume of a ball (ie., the interior of a sphere) is .
But I have just had one of those “why’s that then” moments and sought to prove to myself, using integration, why this would be so.
But my logic is flawed and I get the wrong result – so risking looking stupid – I am asking for someone to correct the error in my logic.
Starting from the start and a familiar (I hope!) lemma:
is the ratio of a circle’s circumference () to its diameter (), hence or, more familiarly, where is the radius of the circle.
Now the area (a) of a circle can then be found by integration:
, giving the familiar
So, my reasoning runs, the volume of a ball would then be:
or , which is precisely half the figure it should be – so where have I gone wrong?
Update: with thanks to Hugh in the comments – of course what I have described is a (double) cone with a height equal to the radius of the base. Cones and circles are not the same, obviously.
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