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Evaluate the integral ∬ₑ y² dV by using spherical coordinates, where is the solid hemisphere x²+y²+z² ≤ 9, y ≥ 0.

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Final answer:

To evaluate the integral ∬ₑ y² dV using spherical coordinates, express the integral in terms of spherical coordinates. Substitute the expressions for ρ and y² into the integral and integrate the term ρ² sin³(φ) with respect to ρ, θ, and φ. Perform the integration and obtain the final result.

Step-by-step explanation:

To evaluate the integral ∬ₑ y² dV using spherical coordinates, we need to express the integral in terms of spherical coordinates. In spherical coordinates, the volume element dV is given by dV = ρ² sin(φ) dρ dθ dφ, where ρ, θ, and φ represent the radial distance, the azimuthal angle, and the polar angle, respectively.

The equation of the solid hemisphere can be expressed in spherical coordinates as ρ ≤ 3, 0 ≤ θ ≤ 2π, and 0 ≤ φ ≤ π/2. Since y = ρ sin(φ) in spherical coordinates, we have y² = ρ² sin²(φ).

Substituting these expressions into the integral, we get ∬ₑ y² dV = ∫ᵠ₀ ∫²π₀ ∫³₀ ρ² sin³(φ) dρ dθ dφ.

Now, we can integrate the term ρ² sin³(φ) with respect to ρ, θ, and φ, using the given limits of integration. After performing the integration, the final result will be obtained.

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