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I just want to know if the region among the cones $$z=\sqrt{x^{2}+y^{2}}$$ and $$z=\sqrt{\frac{x^{2}+y^{2}}{3}}$$ and the sphere $$x^{2}+y^{2}+z^{2}=3z$$ in spherical coordinates is $$0\leq\theta\leq2\pi$$ $$0\leq\rho\leq\frac{3}{2}$$ $$\frac{\pi}{4}\leq\phi\leq\frac{\pi}{3}$$

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    $\begingroup$ I think the boundary for $\rho$ is wrong. Note that $\rho$ measures the distance from the origin, while the sphere you wrote is not centered in the origin. $\endgroup$ – Ernie060 Sep 27 '18 at 21:16

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