The question: A solid bounded by the (y,z)-plane, the (x,y)-plane, the cone $x^2 + y^2 = z^2$, and the surface $x^2 + y^2 + z^2 - 2y = 0$. Suppose a 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.
So far I have $$\int_0^{\pi/4} \int_0^{\pi/2} \int_0^{2\sin\phi \sin\theta} \rho^3\sin\phi d\rho d\theta d\phi $$ but I'm unsure about the range for $\phi$ and $\theta$?
The question should use inequality sign for the given equations to clearly identify the region. The integral you have is correct if the region is bound above $z = 0$ and by the surfaces of the cone, the sphere and yz-plane i.e. the solid is defined by,
$x^2 + y^2 + z^2 \leq 2y, x^2 + y^2 \leq z^2, x \geq 0, z \geq 0$
But if I strictly go by that yz and xy planes are both parts of the surface of the solid then the solid the question is referring to is,
$x^2 + y^2 \ge z^2, ~x^2 + y^2 + z^2 \leq 2y, x \ge 0, z \geq 0$
In that case, the integral would be
$ \displaystyle \int_{\pi/4}^{\pi/2} \int_0^{\pi/2} \int_0^{2\sin\phi \sin\theta} \rho^3\sin\phi ~ d\rho~ d\theta ~d\phi$