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So I am looking at the following question

Find the volume of the ice-cream cone shape given by the region bounded between the upper half of the sphere $x^{2}+y^{2}+z^{2}= 16$ and the cone $z=\frac {1}{\sqrt{3}}\sqrt{x^{2}+y^{2}}$.

I can easily understand that we need to convert this into spherical coordinates, and can see that I will have $0 < p < 4$ and $0 < \theta < 2\pi$. But how do I know the bounds of integration for $\phi$?

In my notes, it is simply noted that the intersection when squaring the cone equation and replacing $z^{2}$ with the subsequent value leads to $x^{2} + y^{2} = 12$, which occurs in the plane $z=2$. What does this mean? Why does it occur in this plane?

Further, it then notes that the cone equation forms an angle of $\frac{\pi}{3}$ with the z-axis, and thus we have $0 < \phi < \frac{\pi}{3}$. I don't want to have to draw out or plot my cone to see this, I would much rather know how to solve for my bounds on $z$ and subsequently $\phi$, but I'm struggling to understand where these values are coming from.

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@muaddib has answered one part of your question so I'll answer the other part. Actually, you've practically answered it yourself.

At z = 1 and y = 0 what is the value of x on the cone? This will allow you to draw a triangle from which you can determine $\phi$ using basic trig. You'll find that it is the $\pi/3$ value that your notes tell you it should be.

Now to integrate over the whole volume of the shape you need to integrate $\phi$ from the z-axis out to the surface of the cone. Given the above, do you see what this range is?

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  • $\begingroup$ Thanks! So I just set y = 0 and draw the resulting triangle in z-x coords? And yes past this I can solve the integral. $\endgroup$ – HavelTheGreat Aug 11 '15 at 3:53
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We can solve for $z$ as follows. On one hand you have $$x^2 + y^2 + z^2 = 16$$ You can square both sides of your cone equation and then multiply by 3 to get $$3z^2 = x^2 + y^2$$ So substituting the second equation into the first we obtain $$3z^2 + z^2 = 16$$ This has two possible solutions $z = -2, 2$. However, looking at the origin equation for the cone, we see that $z$ has to be positive so $z = 2$. Now substituting back $z = 2$ into the first equation we obtain $$x^2 + y^2 + 4 = 16$$

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  • $\begingroup$ Nice. How does it form that angle with the z-axis though? How can I determine that? $\endgroup$ – HavelTheGreat Aug 11 '15 at 3:38
  • $\begingroup$ @HavelTheGreat - Please consider voting this up if "this answer is useful". $\endgroup$ – muaddib Aug 11 '15 at 15:27
  • $\begingroup$ have vote up from me. $\endgroup$ – gleedadswell Aug 12 '15 at 20:38
  • $\begingroup$ @gleedadswell - thanks much :) $\endgroup$ – muaddib Aug 12 '15 at 20:46

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