Find the volume enclosed by the surface ρ = cos φ
10 points for the best explanation.
Thanks.
Z
10 points for the best explanation.
Thanks.
Z
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V = ∫∫∫ 1 dV
....= ∫(θ = 0 to 2π) ∫(φ = 0 to π) ∫(ρ = 0 to cos φ) 1 * (ρ^2 sin φ dρ dφ dθ), via spherical coord's
....= 2π ∫(φ = 0 to π) ∫(ρ = 0 to cos φ) ρ^2 sin φ dρ dφ
....= 2π ∫(φ = 0 to π) (1/3)ρ^3 sin φ {for ρ = 0 to cos φ} dφ
....= 2π ∫(φ = 0 to π) (1/3) cos^3(φ) sin φ dφ
....= (2π/3) * (-1/4)cos^4(φ) {for φ = 0 to π}
....= π/3.
I hope this helps!
....= ∫(θ = 0 to 2π) ∫(φ = 0 to π) ∫(ρ = 0 to cos φ) 1 * (ρ^2 sin φ dρ dφ dθ), via spherical coord's
....= 2π ∫(φ = 0 to π) ∫(ρ = 0 to cos φ) ρ^2 sin φ dρ dφ
....= 2π ∫(φ = 0 to π) (1/3)ρ^3 sin φ {for ρ = 0 to cos φ} dφ
....= 2π ∫(φ = 0 to π) (1/3) cos^3(φ) sin φ dφ
....= (2π/3) * (-1/4)cos^4(φ) {for φ = 0 to π}
....= π/3.
I hope this helps!