How to Solve Integrals on a Sphere Using Pathria's Stat. Mech. Approach?

[QuantumDefect]

Homework Statement

Problem 2.9

Solve the integral:
\begin{equation}
\int \dddot \int dx_{1}dx_{2}\dddot dx_{N}
\end{equation}

where
\begin{equation}
0 \geq \sum_{i=1}^{N}x_{i} \leq R\]
\end{equation}

Homework Equations

The Attempt at a Solution

My issue is not really this problem but the general how-to. Pathria, it seems, does not like explaining the how and the why and instead thinks that a single example in Appendix C suffices. My attempt at a solution includes defining

\begin{equation}
x_{i} = u_{i}^{2}
\end{equation}

in order to get the hypervolume defined on a sphere as the condition becomes

\begin{equation}
0 \geq \sum_{i=1}^{N}u_{i}^{2} \leq R
\end{equation}

this is it however since the general theory is not explained. I have never felt so lost before...if anyone could point me to a good resource(I have yet to find any example or explanation online) or could give me a push in the right direction, it would be appreciated.

 

[QuantumDefect]

Bah, my TeX is apparently incompatible, sorry.

 

[nicksauce]

Instead of begin equation use
[ tex] and [ /tex] to open and close your tex environment (without the spaces).