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bezimeni [28]
2 years ago
6

Use spherical coordinates to evaluate the

Mathematics
1 answer:
kotykmax [81]2 years ago
5 0

Answer:

4096π / 5

Step-by-step explanation:

∫∫∫ (x² + y² + z²) dV

In spherical coordinates, x² + y² + z² = r², and dV = r² sin φ dr dθ dφ.

E is the range 0 ≤ r ≤ 4, 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π.

∫₀ᵖⁱ∫₀²ᵖⁱ∫₀⁴ (r²) (r² sin φ dr dθ dφ)

∫₀ᵖⁱ∫₀²ᵖⁱ∫₀⁴ (r⁴ sin φ) dr dθ dφ

Evaluate the first integral.

∫₀ᵖⁱ∫₀²ᵖⁱ (⅕ r⁵ sin φ)|₀⁴ dθ dφ

∫₀ᵖⁱ∫₀²ᵖⁱ (¹⁰²⁴/₅ sin φ) dθ dφ

¹⁰²⁴/₅ ∫₀ᵖⁱ∫₀²ᵖⁱ (sin φ) dθ dφ

Evaluate the second integral.

¹⁰²⁴/₅ ∫₀ᵖⁱ (θ sin φ)|₀²ᵖⁱ dφ

¹⁰²⁴/₅ ∫₀ᵖⁱ (2π sin φ) dφ

²⁰⁴⁸/₅ π ∫₀ᵖⁱ sin φ dφ

Evaluate the third integral.

²⁰⁴⁸/₅ π (-cos φ)|₀ᵖⁱ

²⁰⁴⁸/₅ π (-cos π + cos 0)

²⁰⁴⁸/₅ π (1 + 1)

⁴⁰⁹⁶/₅ π

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