Triple Integration in Cylindrical and Spherical Coordinates

Triple Integration

Cylindrical Coordinates

• Also known as cylindrical polar coordinates
• Conversion from Cartesian coordinates:
  • x = r cos(θ)
  • y = r sin(θ)
  • z = z
• Volume element: dV = r dr dθ dz
• Integration order: typically dr dθ dz
• Useful for problems with cylindrical symmetry

Spherical Coordinates

• Also known as spherical polar coordinates
• Conversion from Cartesian coordinates:
  • x = ρ sin(φ) cos(θ)
  • y = ρ sin(φ) sin(θ)
  • z = ρ cos(φ)
• Volume element: dV = ρ^2 sin(φ) dρ dφ dθ
• Integration order: typically dρ dφ dθ
• Useful for problems with spherical symmetry

Cartesian Coordinates

• Rectangular coordinates (x, y, z)
• Volume element: dV = dx dy dz
• Integration order: typically dx dy dz or dy dz dx or dz dx dy
• Useful for problems with rectangular symmetry or no particular symmetry

General Triple Integration

• Iterated integrals: ∫∫∫ f(x, y, z) dx dy dz
• Fubini's theorem: allows iterated integrals to be computed in any order
• Importance: used in physics, engineering, and computer science to model and solve problems involving volume and surface integrals.

Triple Integration

Coordinate Systems

• Cylindrical Coordinates
  • Also known as cylindrical polar coordinates
  • Conversion from Cartesian coordinates: x = r cos(θ), y = r sin(θ), z = z
  • Volume element: dV = r dr dθ dz
  • Integration order: typically dr dθ dz
  • Useful for problems with cylindrical symmetry
• Spherical Coordinates
  • Also known as spherical polar coordinates
  • Conversion from Cartesian coordinates: x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ)
  • Volume element: dV = ρ^2 sin(φ) dρ dφ dθ
  • Integration order: typically dρ dφ dθ
  • Useful for problems with spherical symmetry
• Cartesian Coordinates
  • Rectangular coordinates (x, y, z)
  • Volume element: dV = dx dy dz
  • Integration order: typically dx dy dz or dy dz dx or dz dx dy
  • Useful for problems with rectangular symmetry or no particular symmetry

Key Concepts

• Iterated Integrals: ∫∫∫ f(x, y, z) dx dy dz
• Fubini's Theorem: allows iterated integrals to be computed in any order
• Importance: used in physics, engineering, and computer science to model and solve problems involving volume and surface integrals

Understand the concepts of cylindrical and spherical coordinates, conversions from Cartesian coordinates, and integration methods. Useful for problems with cylindrical and spherical symmetry.