# Finding the average value of a function over region in $\mathbb{R}^3$.

I want to know if I set this multiple integral up correctly (I always mess them up!). I want to find the average value of $z$ over the region (call it $M$) bounded by $x^2+y^2+z^2=16$ and $z=2\sqrt{x^2+y^2}$. I am using cylindrical coordinates with $0\leq r\leq \frac{4}{\sqrt{5}}$, $\theta\in[0,2\pi)$, and $2r\leq z \leq \sqrt{16-r^2}$. We then find the average value of $z$, denoted $A(z)$, by computing: $$A(z) = \left( \iiint\limits_M \,dx\,dy\,dz \right)^{-1}\iiint\limits_M z\,dx\,dy\,dz = \left(\int\limits_0^{2\pi} \int\limits_0^{4/\sqrt{5}} \int\limits_{2r}^{(\sqrt{16-r^2})} dz\,(r\,dr)\,d\theta \right)^{-1} \left( \int\limits_0^{2\pi} \int\limits_0^{4/\sqrt{5}} \int\limits_{2r}^{(\sqrt{16-r^2})} z\,dz\,(r\,dr)\,d\theta \right) \ .$$ Thank you for your help. Any critiques/suggestions are welcome.

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Looks good so far! –  Cameron Buie Apr 19 '13 at 1:09
Thank you! I have no problem in the calculations; I have just been struggling in setting up limits of integration for multiple integrals. –  user59083 Apr 19 '13 at 2:42