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'.(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Evaluating Volume of a Sphere in Cartesian Framework

**Physics Forums | Science Articles, Homework Help, Discussion**