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Volume of a sphere Watch

1. I'm stuck!

Surface area of a sphere is 4Pi R^2 = Pi D^2 D=R/2 so D^2 = R^2 / 4

The integral of the surface area is the volume:

So 4/3 Pi R^3 which is right which goes to Pi D^3 / 6 which is also correct.

But integrating the original expression for area with diameter gives Pi D^3 / 3 which is wrong... why?

Can you only integrate this using the radius?
2. Roughly speaking you need to think of how the solid sphere (say of radius a) is built up of infinitesimal spherical shells of radii 0<r<a.

In which case you're working out

You could work out the same this using diameter, but you'd need to evaluate

which does give the same answer.

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Updated: December 19, 2010
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