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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 really cant do this question. I know that $V=\rho /m$ but what is $m$?

I also don't understand how you can even find the volume with integrals because there is no limit of $z$ given so it would just be infinity wouldn't it.

Can someone start me off, I obviously don't need anyone to show me how to integrate. Just struggling on what the density and stuff is relevant.

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As a start cylindrical coordinates case is simpler perhaps by cylinders stacking summation

$$ dV = \rho \pi r^2 dz $$ $$ \rho = z, r = z $$ $$ V = \pi \int z^3 dz = z^4 \pi/4 $$ etc..

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  • $\begingroup$ where did you get your first expression from? $\endgroup$ – snowman Apr 28 '15 at 20:05
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I don't think using the given density is necessary because the triple integral of 1 using the cone's boundaries will give the volume of the cone. Although I don't know why the density would be mentioned if it's not part of the solution. Is this one of the questions from the UoB MVA module? If so, I'm taking the module too.. hi:D

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  • $\begingroup$ Yeah Hi. Yep not too nervous for it tbh. This q is so dodgy. $\endgroup$ – snowman Apr 30 '15 at 22:55

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