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Use spherical coordinates to evaluate the triple integral ∫∫∫Ex2+y2+z2dV

, where E is the ball: x2+y2+z2≤81
.

User TyMarc
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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.

User JohnnyDH
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