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Find the volume of the solid inside the surface defined by the equation $\rho=8\sin \phi$ in spherical coordinates

So far I've set up an integral in spherical coordinates with $\rho$ from $0$ to $\rho=8\sin \phi$ $\theta$ from $0$ to $2\pi$, $\phi$ from $0$ to $\pi$.

This is just a far off guess as I do not have a clue where to begin. I've been spending a lot of time just looking at the question and trying to make sense of it.

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Simply integrate the volume element: $$V= \int_0^{2\pi}\int_0^\pi\int_0^{8\sin\phi}\rho^2\sin\phi\,d\rho d\phi d\theta.$$

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Sometimes it helps to have a visual representation of the solid:enter image description here

It helps you convince yourself that indeed $\phi$ ranges from $0$ to $\pi$ and $\theta$ from $0$ to $2\pi$, as in Martín-Blas Pérez Pinilla 's answer.

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  • $\begingroup$ The OP knew the ranges from the beginning, the problem was the integrand. In any case, your idea is good. $\endgroup$ – Martín-Blas Pérez Pinilla Dec 17 '15 at 8:31

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