Math Problem Statement

Find the specified expressions and integrals in spherical and cylindrical coordinates from the exercises shown in the uploaded image.

Solution

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Spherical Coordinates
Cylindrical Coordinates
Triple Integration
Moment of Inertia
Mass Distribution

Formulas

Vector cross product: (∂r/∂r × ∂r/∂θ) · ∂r/∂ϕ
Mass: M = ∫∫∫ ρ dV
Moment of Inertia: I = ∫∫∫ ρ r^2 sin^2(θ) dV

Theorems

Divergence Theorem
Triple Integral Transformation

Suitable Grade Level

Undergraduate (Calculus III or Advanced Calculus)