Qfyilyi

I used the code in the second answer on this page to switch from Cartesian to Spherical coordinates and integrate a function over the sphere. I would like to use this code to calculate the volume of a sphere. However if I set $ f(x, y, z) = 1 $, I get the following output:

$ -frac{1}{3}r^3 varphicos(theta) $

Here is the code I used:

Needs["VectorAnalysis`"]



JacobianDeterminant[Spherical[r, θ, ϕ]]



f[x_, y_, z_] := 1



Integrate[

 f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[

     Spherical @@ #] &@{r, θ, ϕ},

 r, θ, ϕ]

How can I make the code return the volume of a sphere which is $frac{4}{3}pi r^3$?

Try this

transform[f : (_Function | _Symbol), coordi_: {x, y, z}, coordf_: {r, θ, ϕ}] := Module[{rules, jdet},

          rules = Thread[coordi -> CoordinateTransformData["Spherical" -> "Cartesian", "Mapping", coordf]];

          jdet = CoordinateTransformData["Spherical" -> "Cartesian", "MappingJacobianDeterminant", coordf];

          jdet f @@ coordi /. rules

         ]



f[x_, y_, z_] := 1



Integrate[transform[f], r, {θ, 0, π}, {ϕ, 0, 2 π}]

(4 π r^3)/3

Employing most of your own code;

Integrate[

 f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[Spherical @@ #] &@{r, θ, ϕ}, 

 {r, 0, R}, {θ, 0,Pi}, {ϕ, -Pi, Pi}

 ]

That is, you just have to add the integral boundaries.