Verified answer

We are given an upper hemispherical shell in the first three quadrants in the xy-plane

with radii 4 and 7.

So, converting to spherical coordinates yields

∫(θ = 0 to 3π/2) ∫(φ = 0 to π/2) ∫(ρ = 4 to 7) (3ρ cos θ sin φ + 2ρ sin θ sin φ) * (ρ^2 sin φ dρ dφ dθ)

Evaluating this yields

∫(θ = 0 to 3π/2) (3 cos θ + 2 sin θ) dθ * ∫(φ = 0 to π/2) sin^2(φ) dφ * ∫(ρ = 4 to 7) ρ^3 dρ

= [(3 sin θ - 2 cos θ) {for θ = 0 to 3π/2}] * ∫(φ = 0 to π/2) (1/2)(1 - cos(2φ) dφ]

* [ρ^4/4 {for ρ = 4 to 7}]

= -1 * (π/4) [(7^4 - 4^4)/4]

= -2145π/16.

I hope this helps!

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