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Find the volume of region outside the cone $\varphi = \frac{\pi}{4}$ and inside the sphere $\rho =4cos(\varphi)$.

Solution Attempt: I can visualize the surfaces and see that the volume is two spherical caps at the edges of the cone but am not sure how to set up the integral.

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In spherical coordinates, you are looking for the volume of the following region: $$ E=\{(\rho,\theta,\phi)\;|\;0\le \rho\le 4\cos\phi,0\le \theta \le2\pi, 0\le \phi\le\frac{\pi}{4}\} $$ So your volume equals $$ V(E)=\iiint_E \rho^2\sin\phi\; d\rho d\theta d\phi $$

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