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## Main Question or Discussion Point

I'm trying to sharpen up my maths before I go back to university to start my PhD and am working through Roel Snieder's excellent book 'A Guided Tour of Mathematical Methods for the Physical Sciences'.

The problem I am working on is how to evaluate the volume of a sphere in Cartesian co-ordinates.

I was never really that advanced in my mathematics, but always managed to pick up the bits I needed. I have made some progress on my own but have got stuck.

So far I have formulated the problem in terms of a volume integral:

[tex]\int^{R}_{-R}\int^{\sqrt{R^{2}-x^{2}}}_{-\sqrt{R^{2}-x^{2}}}\int^{\sqrt{R^{2}-x^{2}-y^{2}}}_{-\sqrt{R^{2}-x^{2}-y^{2}}}dzdydx[/tex]

I have performed the integration as far as this:

[tex]\int^{R}_{-R}2\sqrt{R^{2}-x^{2}} sin^{-1}\sqrt{R^{2}-x^{2}}dx[/tex]

Which may well be wrong, because I have had to do quite a bit of looking around and take leaps of faith to get even just that far!

But now I am completely stumped. I have reformulated this integral by trigonometric substitution to give this:

[tex]2R\int^{R}_{-R}cos^{2}\theta sin^{-1}(Rcos\theta) d\theta[/tex]

Which looks a bit easier to solve, but I really need some help here, I am weak and cannot progress.

Thanks

The problem I am working on is how to evaluate the volume of a sphere in Cartesian co-ordinates.

I was never really that advanced in my mathematics, but always managed to pick up the bits I needed. I have made some progress on my own but have got stuck.

So far I have formulated the problem in terms of a volume integral:

[tex]\int^{R}_{-R}\int^{\sqrt{R^{2}-x^{2}}}_{-\sqrt{R^{2}-x^{2}}}\int^{\sqrt{R^{2}-x^{2}-y^{2}}}_{-\sqrt{R^{2}-x^{2}-y^{2}}}dzdydx[/tex]

I have performed the integration as far as this:

[tex]\int^{R}_{-R}2\sqrt{R^{2}-x^{2}} sin^{-1}\sqrt{R^{2}-x^{2}}dx[/tex]

Which may well be wrong, because I have had to do quite a bit of looking around and take leaps of faith to get even just that far!

But now I am completely stumped. I have reformulated this integral by trigonometric substitution to give this:

[tex]2R\int^{R}_{-R}cos^{2}\theta sin^{-1}(Rcos\theta) d\theta[/tex]

Which looks a bit easier to solve, but I really need some help here, I am weak and cannot progress.

Thanks