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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Picture what you are integrating: the area of a disc as it grows to the full radius. But that isn’t a set of slices that makes up a ball. It’s just a set of overlapping slices.

You want to integrate the surface area of a sphere that grows from 0 to the radius. Or you want to integrate a set of slices along a diameter, with radius appropriate (not linear).

Thanks – I’ve updated the article to reflect this!