TITLE: Multivariate Calculus: finding mass
QUESTION [2 upvotes]: Consider a solid $E$ bounded by the $yz$-plane, the $xy$-plane, the cone $z^2 = x^2 + y^2$ and the surface $x^2 + y^2 + z^2 - 2y=0$.
(There are four such solids, but it doesn't matter which one you use). Suppose the density of a chunk of metal of the shape of this solid at the point $(x,y,z)$ is $\sqrt{x^2 + y^2 + z^2}$.

Find the mass of the chunk of metal.

What I did so far:
$$\iiint_Ef(x, y, z) \,\mathrm dV =\iiint_E \sqrt{x^2 + y^2 + z^2}\,\mathrm dV$$ with
$$E = \{(ρ\sinϕ\cosθ, ρ\sinϕ\sinθ, ρ\cosϕ): 0≤θ≤{π\over 2}, 0≤ϕ≤{π\over 4}, 0≤ρ≤2\sinϕ\sinθ\}.$$
Would appreciate any help on how to solve. Thanks.

REPLY [0 votes]: You want to convert from Cartesian to Spherical Coordinates
$${x = \rho\sin\phi\cos\theta\\y = \rho\sin\phi\sin\theta\\z = \rho\cos\phi\\~\\\rho^2=x^2+y^2+z^2}$$
Use  $\iiint\ldots\mathrm d V=\iiint\ldots\rho^2\sin\phi\,\mathrm d\phi\,\mathrm d\theta\,\mathrm d\rho$
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