Spherical Coordinates Integration

Integration in spherical coordinate system can be defined as Integration over a volume defined in Spherical Coordinates. Coordinate system is a system which uses either one or more Numbers or one or more coordinates.
Spherical coordinate system can be explained by three quantities.
Radial distance or radial coordinate (ρ), Polar angle or inclination angle (θ) and azimuth angle (φ). So ρ, θ, and φ are spherical coordinates
If we want to convert an integral of Cartesian coordinates into spherical coordinates then triple Integral is used and to apply triple integral in spherical coordinates following steps are used:
(1) Suitable form of limits and the integrands must be used.
(2) Multiply with correct volume element to convert an integral of Cartesian coordinates into spherical coordinates.
To determine the integration using spherical coordinates (ρ, θ and φ), solid is first divided into small pieces which has dimensions Δρ, Δθ, and Δφ.
Lets consider a function which is dependent on spherical coordinates as shown below:
p (ρ, θ, φ),
When this function p (ρ, θ, φ) will be integrated over a Solid 'V' in spherical coordinates then spherical coordinates integration will be:
∫ ∫ ∫V p (ρ, θ, φ) ρ² sin ( φ ) dρ dθ dφ,
Order of integration can be changed if solid 'V' is a basic solid. Above integration can also be written as:
∫ ∫ ∫V  p ( ρ, θ, φ) ρ² sin ( φ ) dρ dφ dθ,
∫ ∫ ∫V  p ( ρ, θ, φ) ρ² sin ( φ ) dθ dρ dφ,
∫ ∫ ∫V  p ( ρ, θ, φ) ρ² sin ( φ ) dθ dφ dρ,
∫ ∫ ∫V  p ( ρ, θ, φ) ρ² sin ( φ ) dφ dθ dρ,
∫ ∫ ∫V  p ( ρ, θ, φ) ρ² sin ( φ ) dφ dρ dθ.