Triple integrals in spherical coordinates

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Transforming triple integrals into spherical coordinates involves understanding the limits for the angle phi, which measures the angle from the positive z-axis. Phi typically ranges from 0 to Pi/2 for the upper octants and from Pi/2 to Pi for the lower octants. If the integration requires angles beyond these ranges, additional considerations are necessary. Properly defining these limits is crucial for accurate calculations in spherical coordinates. Understanding these conventions helps in effectively setting up triple integrals in spherical coordinates.
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i have a question concerning transforming triple integrals into spherical coordinates. the problem is, i do not know how to find the limits of phi. Can anyone help me? Thanks...
 
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What's the problem?
 
Phi is the variable indicating the angle subtended from the positive z-axis to the negative z-axis. It is similar to theta which is usually defined from positive x, but all the way around to positive x again.

Phi is usually integrated from 0 to Pi/2 for the top four octants, and Pi/2 to Pi for the bottom four octants. Anything more than that and we'd need a problem.
 

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