Volume of a ball

As ane fule kno the volume of a ball (ie., the interior of a sphere) is \frac{4}{3}\pi r^3.

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:

\pi is the ratio of a circle’s circumference (c ) to its diameter (d ), hence c = \pi d or, more familiarly, c = 2 \pi r where r is the radius of the circle.

Now the area (a) of a circle can then be found by integration:

a = \int_0^r 2\pi x dx , giving the familiar \pi r^2

So, my reasoning runs, the volume of a ball would then be:

2 \int_0^r \pi x^2 dx or \frac{2}{3}\pi r^3 , 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.


2 responses to “Volume of a ball”

  1. 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).

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

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