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Explanation:
We have the triple integral:
∫∫∫E x^2 + y^2 + z^2 dV
where E is the ball x^2 + y^2 + z^2 ≤ 81.
In spherical coordinates, the volume element is given by:
dV = ρ^2 sin φ dρ dθ dφ
where ρ is the radial distance, φ is the polar angle (measured from the positive z-axis), and θ is the azimuthal angle (measured from the positive x-axis).
Substituting into the integral, we get:
∫φ=0 to φ=π ∫θ=0 to θ=2π ∫ρ=0 to ρ=9 ρ^2 sin φ (ρ^2) dρ dθ dφ
= ∫φ=0 to φ=π ∫θ=0 to θ=2π ∫ρ=0 to ρ=9 ρ^4 sin φ dρ dθ dφ
= ∫φ=0 to φ=π ∫θ=0 to θ=2π (1/5)(9^5) sin φ dθ dφ
= (2π/5)(9^5)(-cos φ)|φ=0 to φ=π
= (2π/5)(9^5)(-(-1 - 1))
= (4π/5)(9^5)
≈ 114413.73
Therefore, the value of the triple integral is approximately 114413.73.