Evaluating a triple integral in spherical coordinates

 Integration in spherical coordinates

Let f (⍴, θ, ψ) be a function continuous over a bounded spherical box defined by:

Let's divide each interval in l, m. n subintervals such that 

Let's consider any sample point (ρᵢⱼₖ, θᵢⱼₖ, ψᵢⱼₖ) in the subbox Bᵢⱼₖ. The volume element ΔV of the subbox B can be written in spherical coordinates by:

as shown in the following figure:

Let's take the Rieman sum of the expression:

The limit of this expression when l, m, n approach infinity is the triple integral of the function in spherical coordinates as defined above

Definition of a triple integral in spherical coordinates

The properties already examined for previous integrals work for triple integrals in cylindrical coordinates as well as iterated integrals. As always, Fubini's theorem allows us to evaluate a triple a integral by setting it up as an iterated integral. The theorem is stated below:

Theorem

Example

Solution

The variables being independent of each other, we can integrate each piece and multiply: