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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 \varphi\cos(\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 $\frac{4}{3}\pi r^3$?

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10
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Employing most of your own code;

Integrate[
 f @@ CoordinatesToCartesian[#, Spherical @@ #] JacobianDeterminant[Spherical @@ #] &@{r, θ, ϕ}, 
 {r, 0, R}, {θ, 0, π}, {ϕ, -π, π}
 ]

(4 π R^3)/3

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

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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
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