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I'm trying to solve this:

$\frac {1}{4 \pi} \int_{Β (x,t)} \frac{(\vert y \vert)^2}{\vert y-x \vert} dV\;\;where\;Β (x,t) \subset \mathbb R^3$

I thought to use spherical coordinates such as , for $y=(y_1,y_2,y_3)\;and\;x=(x_1,x_2,x_3) \\y_1=x_1+rcosθsinφ\;\\y_2=x_2+rsinθsinφ\;\\y_3=x_3+rcosφ\;$

But then I conclude to this :

$\frac{r^2+2r(x_1cosθsinφ+x_2sinθsinφ+x_3cosφ)+{x_1}^2 +{x_2}^2 +{x_3}^2}{r} r^2sinφ\;$ inside the integral. I think I'm doing this in the wrong way.. Shouldn't I have vanished $x$?

I know it's a bit elementary what I'm asking, but I would appreciate if someone could make that clear to me.

Thanks in advance!!

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    Are you integrating over a spherical shell, or a solid ball? Either way, $dy$ by itself is wrong. – 2017-01-20
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    @Arthur the integral is over a solid ball.. why is it wrong? – 2017-01-20
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    You use $dy$ by itself when whatever you're integrating over is one-dimensional. For three-dimensional objects, either $dV$ or $dx\,dy\,dz$ are the most common ones in use. If you use spherical coordinates, then $dr\,d\theta\,d\varphi$. – 2017-01-20
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    @Arthur well, yes... In class we use this notation... Thank you, I'll edit it – 2017-01-20
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    Not entirely sure what you mean by "Shouldn't I have vanished $x$", but if you write out the triple integral (including limits of integration), it's easy to see the terms in the numerator of the fraction that contain trig functions integrate to zero; the remaining terms are easily evaluated. – 2017-01-20
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    @AndrewD.Hwang I 've been so silly... this integral was a part of a solution for a pde.. I don't need to vanish $x$.. Thank you for your comment. I just computed the integral.. You helped me a lot with this really! – 2017-01-20

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