"(b) Let $C$ be the solid cone with the boundary surfaces $x^2 +y^2 = z^2$ and $z = 0$. The density of the solid at point $(x, y,z)$ is $z$. Find the volume of the solid using the integrals in both the cylindrical coordinates and the spherical coordinates."

I have done this question and got an answer that is $\frac{\pi}{4z}$ in cylindrical and $\frac{\pi}{8z}$ in spherical. I took $\phi$ in spherical coordinates to be between $0$ and $\frac{\pi}{4}$. Is this correct?

  • $\begingroup$ Certainly the volume should be the same regardless of what coordinates you use. $\endgroup$ – Travis May 5 '15 at 15:26
  • $\begingroup$ Would the correct limits for phi be 0 and pi/4? $\endgroup$ – anon May 5 '15 at 15:32
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    $\begingroup$ I don't think the question even makes sense: The two surfaces doesn't actually bound a (finite) region. $\endgroup$ – Travis May 5 '15 at 15:39
  • $\begingroup$ It appears to work with limits of 0 and pi/2.. ? $\endgroup$ – anon May 5 '15 at 15:49
  • $\begingroup$ @JLL No, that is the start of the question, part a is unrelated... $\endgroup$ – anon May 5 '15 at 15:54

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