MHB Mass of a Torus in Spherical Coordinates

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To determine the mass of a torus defined by the equation r=2sin(φ) in spherical coordinates, the volume can be calculated using the integral formula for a torus. The mass is then found by integrating the density function ρ=φ over the defined volume. The appropriate triple integral for mass involves integrating the density multiplied by r²sin(θ) across the specified limits for θ and φ. The discussion emphasizes the need for a clear approach to setting up these integrals to find the mass accurately. Understanding the relationship between the torus's geometry and the density function is crucial for solving the problem.
pamsandhu
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consider a torus whose equation in terms of spherical coordinates(r,\theta,\phi) is r=2sin\phi for 0\le\phi\le2\Pi. determine the mass of the region bounded by the torus if the density is given by \rho=\phi.
 
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pamsandhu said:
consider a torus whose equation in terms of spherical coordinates(r,\theta,\phi) is r=2sin\phi for 0\le\phi\le2\Pi. determine the mass of the region bounded by the torus if the density is given by \rho=\phi.
Since you chose not to show any attempt to do this yourself, it is difficult to know what advice would help. Do you, at least, know that the volume of a torus given by r= f(\theta, \phi) is \int_{\theta= 0}^{2\pi}\int_{\phi= 0}^{2\pi}\int_0^{f(\theta, \phi)} r^2 sin(\theta)drd\theta d\phi and so the mass of such an object with density given by \rho(r,\theta, \phi) is \int_{\theta= 0}^{2\pi}\int_{\phi= 0}^{2\pi}\int_0^{f(\theta, \phi)}\rho(r,\theta,\phi) r^2 sin(\theta)drd\theta d\phi?
 
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