Another Webwork Question

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For problem 13 on 12.7, Webwork does not like my phi limits. The problem
says the angle at the vertex is "pi/3" so wouldn't the limits be from pi/3
to 0? Having a hard time understand why this isn't the case.

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First of all, I don't think that that the statement of this problem very clearly specifies which angle is π/3 at the vertex, and yes I think it would be reasonable to assume that the angle they meant was the angle from the z-axis to the cone and this would indeed mean that the limits of integration in φ was from 0 to π/3.  However, it seems that the angle they mean is either the angle from the x-axis to the cone in the x-z plane or the angle from the one side of the cone to to the other in the x-z plane, which would make the  φ integration from 0 to π/6.