Write the equation z√3=√(x^2+y^2) in spherical coordinates.?
√3*(rho)*(cos(ph))=√(rho^2sin^2(phi)) is the simplification I got, but it is incorrect.
Also, express the the average distance from a point in a ball of radius 2 to its center as a triple integral.
I got:
∫ θ=0 to θ=2π, ∫ Φ=0 to Φ=π, ∫ p=0 to p=2, ______________ dp dΦ dθ
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1) z√3 = √(x² + y²)
==> (ρ cos φ)√3 = √((ρ cos θ sin φ)² + (ρ sin θ sin φ)²)
==> (ρ cos φ)√3 = √(ρ² (cos²θ + sin²θ) sin²φ)
==> (ρ cos φ)√3 = √(ρ² * 1 * sin²φ)
==> (ρ cos φ)√3 = ρ sin φ
==> sin φ/cos φ = √3
==> tan φ = √3
==> φ = π/3.
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2) The average distance equals
(1/Volume of the ball) * ∫∫∫ √(x² + y² + z²) dV
= [1/((4π/3) * 2³)] * ∫∫∫ √(x² + y² + z²) dV
= (3/(32π)) * ∫∫∫ √(x² + y² + z²) dV
Converting to spherical coordinates:
(3/(32π)) * ∫(θ = 0 to 2π) ∫(φ = 0 to π) ∫(ρ = 0 to 2) ρ * (ρ² sin φ dρ dφ dθ)
= ∫(θ = 0 to 2π) ∫(φ = 0 to π) ∫(ρ = 0 to 2) (3/(32π)) ρ³ sin φ dρ dφ dθ.
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I hope this helps!