I'm currently in second year calculus and have come across a problem that I'm struggling badly to try and understand. The question is as follows:
Sketch the region of integration of the following triple integral and eveluate the integral:
$$ \int_0^3 \int_0^{\sqrt{9-x^2}} \int_0^{\sqrt{9-x^2-y^2}} z^2\sqrt{x^2+y^2+z^2}\,dz\,dy\,dx $$
I understand that this will likely require a shift to a different coordinate system, and I'm fairly sure that the $\sqrt{x^2 + y^2 + z^2}$ indicates that a shift to spherical would be most convenient. Based off of the limits, I've also come up with the equations $x^2 + y^2 = 9$ and $x^2 + y^2 + z^2 = 9$. However, beyond this, I can't seem to go any further, and have gone through about 6 pieces of paper and 3 hours of trying to make sense of this problem. Can anyone help me solve this, or at least put me on the right track for what to do? I cannot for the life of me seem to get a conversion that creates anything other than a tangled mess that's harder to solve than the original equation. Thank you so much to anyone that replies.
EDIT: I see that my formatting has been fixed, thank you for that!
Using spherical coordinates, we obtain
$\displaystyle\int_0^{2\pi}\int_0^{\frac{\pi}{2}}\int_0^3(\rho\cos\phi)^2\cdot\rho\cdot\rho^2\sin\phi \;d\rho\; d\phi \;d\theta=\int_0^{2\pi}\int_0^{\frac{\pi}{2}}\int_0^3 \rho^5\cos^2\phi\sin\phi\;d\rho\; d\phi \;d\theta$,
which should be straightforward to evaluate.