Calculus triple integral

Question

Find the mass of the solid bounded below by the circular cone  and above by the hemisphere  if the density .Write your answer in terms of .

Answer 1

Using Spherical Coordinates

The radius of the sphere is √13/2 .

The projection of the intersection of the cone and the sphere on the xy-plane is a circle with radius √(13/8).

That is sinφ = [√(13/8)] / [ √13/2 ] = √2/2 Hence φ = π/4.

Then the mass m= ∫_θ=0^θ=2π ∫_φ=0 ^φ=π/4 ∫_ρ=0 ^ρ=√13/2 ρ⋅ρ²sinφ dρ dφ dθ〉

m = ∫_θ=0^θ=2π ∫_φ=0 ^φ=π/4 ∫_ρ=0 ^ρ=√13/2 ρ³sinφ dρ dφ dθ = 1/4 ∫_θ=0^θ=2π ∫_φ=0 ^φ=π/4 [ρ⁴] |₀ √13/2 sinφ dφ dθ=

169/64 ∫_θ=0^θ=2π ∫_φ=0 ^φ=π/4 sinφ dφ dθ = 169/64 ∫_θ=0^θ=2π[ - cosφ ] |₀^π/4d θ = [ 169( 2 - √2 )π]/64