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I have the following integral:

[tex]

I = \int_0^\infty dx_1\ldots \int_0^\infty dx_{3N} \int_0^\infty dy_1\ldots \int_0^\infty dy_{3N}\Theta\left(1-\sum_{j=1}^{3N}\left(|x_j|^2+|y_j|^2\right)\right)

[/tex]

Now I want to change to polar coordinates by the following substitution:

[tex]

\begin{equation}

x_j &=& r_j \cos \varphi_j;

y_j &=& r_j \sin\varphi_j

\end{equation}

[/tex]

Then the integral becomes

[tex]

\left[\int_0^{2\pi}d\varphi\right]^{3N}\int_0^\infty dr_1\ldots \int_0^\infty dr_{3N}r_1r_2\ldots r_{3N} \Theta\left(1-\sum_{j=1}^{3N}|r_j|^2\right)

[/tex]

Now this factors [itex]x_j[/itex] in front of the theta function are bothering me, because if they weren't there, the integral would be simply the volume of sphere with radius 1 in 3N dimensions, and this is a known formula.

But, they are there so this is a problem...

Or not, I don't know, is there another maybe easier way to solve this integral?

Is my substitution perhaps not wel chosen?

(there are some typo's in the formula, but not important, seems like the preview doesn't renew the latex code)

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# Integration by change to polar coordinates

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