Find the mass of the object, which is in the form of a sphere of radius √5cm, centred at the origin. The density at any point is given to be the constant 2.
Since we have the density of the sphere (which is constant and given in g/cm3) and the radius we will have to use the information given to find the mass in grams:
We also know that the radius is and then we can establish that x2+y2+z2 = r2 = 5 (this also because the sphere was centered at the origin). The volume differential dV will be changed to polar coordinates to evaluate the integral easily:
The volume element in spherical coordinates is and we also have to considerate the limits for the integration: (0⪕θ⪕π), (0⪕ϕ ⪕2π) and (0⪕r⪕ ). With this information we proceed to evaluate the integral for the volume and after we multiply it for the density we'll have the mass of the object:
(these are the integrals that have to be evaluated to find the mass)
In conclusion, the mass of the sphere is found as m = ρ*∫dV ≈ 93.664 g
References:
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