I am trying to evaluate the following integral: $\int_{0}^{6}\int_{-\sqrt{36 - x^2}}^{\sqrt{36 - x^2}}\int_{-\sqrt{36 - x^2 - z^2}}^{\sqrt{36 - x^2 - z^2}} \dfrac{1}{\sqrt{x^2 + y^2 + z^2}} \ dy dz dx$.
My guess is that I need to change to coordinates but I'm unsure how to approach the problem. So I am thinking that I could use spherical coordinates and re-arrange this problem as:
$\int_{0}^{2\pi}\int_{-6}^{6} \int_{0}^{6} \rho sin(\phi) \ dr d\phi d\theta$
however, I get the result as $0$ which isn't right. I'm know that I"m wrong here somehow but I'm not sure how. Any help is appreciated. Thank you in advance.
You have $\phi$ running from -6 to 6 there, my friend.