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Finding density using multiple integrals

For this question, I used the formula:
M=∫ρ dV where M=mass and ρ=density=cos²θ. I converted this into spherical co ordinates and integrated to obtain M=4πR³/9 . How do I find the density from this? The answer is ρ=9M/4πR³ but I am not sure how this was obtained
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Original post by bobbricks
For this question, I used the formula:
M=∫ρ dV where M=mass and ρ=density=cos²θ. I converted this into spherical co ordinates and integrated to obtain M=4πR³/9 . How do I find the density from this? The answer is ρ=9M/4πR³ but I am not sure how this was obtained


Question says ρ∝cos⁡2θ\rho\propto \cos^2\theta, not equal to.

Lets say ρ=kcos⁡2θ\rho=k \cos^2\theta

So, using M=∫ρ  dVM=\int\rho \;dV we can determine the constant k.

At Z, theta=0, so density at Z is simply k, whatever that works out to be.
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bobbricks
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Original post by ghostwalker
Question says ρ∝cos⁡2θ\rho\propto \cos^2\theta, not equal to.

Lets say ρ=kcos⁡2θ\rho=k \cos^2\theta

So, using M=∫ρ  dVM=\int\rho \;dV we can determine the constant k.

At Z, theta=0, so density at Z is simply k, whatever that works out to be.


So k=9M/4πR³ and density=kcos^2(theta) and at z=0, theta=0 so density=k. Makes sense, thanks! :smile:
Original post by bobbricks
So k=9M/4πR³ and density=kcos^2(theta) and at z=0, theta=0 so density=k. Makes sense, thanks! :smile:


Yep. Though it's not the case z=0, rather, at Z we have theta=0....

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