Spherical coordinate question

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For this question I tried to use spherical coordinates but can't seem to find the interval for $\phi$. Also tried to use change of coordinates but no luck, any help is appreciated

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The first inequality defines a spherical shell.

The second really seems a typo, maybe it should be:

$$\gamma \leq \cos \frac{z}{\sqrt{x^2+y^2+z^2}} =\cos \frac{z}{\rho}=\cos \phi \leq \delta$$

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