My question is simple, when setting up the bounds for this problem, I believed that the bounds would be from $\frac \pi 3$ to $\frac \pi 6$. however its wrong and its actually the other way around, from $\frac \pi 6$ to $\frac \pi 3$. This doesn't make sense to me as the cone made by $\frac \pi 6$ sits on top of the cone made by $\frac \pi 3$. What am I not understanding? Thanks.
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asked Jul 7, 2020 at 22:40
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$\begingroup$ $\frac{\pi}{6}<\frac{\pi}{3}$ or do I misunderstood your question? $\endgroup$– babemcnuggetsJul 7, 2020 at 22:48
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$\begingroup$ that is true but in spherical coordinates ϕ is the angle made by the z-axis. So since pi/6 is smaller that would mean the angle is close to the z axis, and pi/3 is futher. There fore pi/6 lies on top of pi/3 right? $\endgroup$– Not Friedrich gaussJul 7, 2020 at 23:01
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1 Answer
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In spherical coordinates, when you're integrating with respect to $\phi$ you integrate from the z-axis down, so if you only want the top half of $\mathbb{R}^3$ then you integrate $\phi=0$ to $\phi=\frac{\pi}{2}$. So in this case you would integrate from $\phi=\frac{\pi}{6}$ to $\phi=\frac{\pi}{3}$
