An object occupies the solid region bounded by the upper nappe of the cone $$z^2=9x^2+y^2$$ and the plane $$z=9$$ Find the total mass of the object if the mass density at $\space (x,y,z) \space$ is equal to the distance from $\space (x,y,z) \space$ to the top.
I switched to spherical coordinates, I can't find the right bounds... The answer is $$243/4 \pi,$$ I tried many ways, nothing is working so I would appreciate if someone showed me the correct way to do this.
The density is $\rho(x,y,z) = 9-z$, the mass is $M = \iiint_V \rho(x,y,z) dxdydz$, where $V$ is the region occupied by the solid in your question.
$$ M = \int_0^9 dz(9-z)\iint_{\Omega_z} dx dy $$ where $\Omega_z = \{ (x,y): 0 \leq 9x^2+y^2\leq z^2\}$ and $|\Omega_z| = \iint_{\Omega_z} dx dy= \frac{\pi}{3}z^2$
Hence $M = \int_0^9 (9-z) \frac{\pi}{3}z^2 dz = \frac{729 \pi}{4}$