Math Problem Statement
Evaluate the following integral in spherical coordinates. Integral from 0 to 2 pi Integral from pi divided by 6 to pi divided by 4 Integral from 0 to 3 cosecant phi rho squared sine phi d rho d phi d theta pi divided by 6 pi divided by 4 3
Solution
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Math Problem Analysis
Mathematical Concepts
Spherical Coordinates
Triple Integration
Integration in Multivariable Calculus
Formulas
Volume element in spherical coordinates: dV = ρ² sin(φ) dρ dφ dθ
Integral of ρ² over 0 to 3: ∫ρ² dρ
Integral of sin(φ) over bounds: ∫sin(φ) dφ
Angle bounds: ∫dθ from 0 to 2π
Theorems
Spherical Coordinate System
Integration of Trigonometric Functions
Suitable Grade Level
University Level (Calculus III or Multivariable Calculus)
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