I'm trying to solve this problem in different coordinates, but I get different answers. Here's what I've managed to set up: In Cartesian coordinates: $$ \int_{0}^{8} \int_{0}^{\sqrt{64-z^2}} \int_{0}^{\sqrt{64-y^2-z^2}} z(x^2+y^2)\, dx dy dz$$
In cylindrical coordinates: $$ \int_{0}^{\frac{\pi}{2}}\int_{0}^{8 }\int_{0}^{\sqrt{64-z^2}} z\rho^{3}\,d\rho dz d\theta $$
In spherical: $$ \int_{0}^{\frac{\pi}{2}} \int_{0}^{\frac{\pi}{2}} \int_{0}^{8 } r^{5}\,\sin^{3}(\phi)\cos{\phi}\,dr d\theta d\phi $$ My calculator gives 4096π for the Cartesian coordinates, I get $(4/3) 4096π$ when I use cylinderical or spherical coordinates and my calculator agrees with me on those. Did I set them up wrong? I read the chapters on these topics like thrice and still can't figure out what I'm doing wrong, this is fairly simple problem. Pls help ;-; (I didn't do Cartesian one with hand, I'll try it and see what happens). Okay I miscalculated, it's solved now. Thank you math lover.