I am trying to convert
1
∫
0
1-
2
x
∫
-
1-
2
x
1
∫
2
x
+
2
y
(1)rxyz
into Cylindrical Coordinates using 4 different Wolfram Functions listed below and am not sure that any of them are giving me the correct answer. However, answer 1 looks more correct than any of the rest since it actually appears to have been converted from Cartesian to Cylindrical.
1. Using IntegrateChangeVariables[] function
In[]:=
Inactive[Integrate]1,{x,0,1},y,-
1-
2
x
,
1-
2
x
,z,
2
x
+
2
y
,1​​IntegrateChangeVariables[%,{r,θ,u},"Cartesian"->"Cylindrical"]
Out[]=
1
∫
0
1-
2
x
∫
-
1-
2
x
1
∫
2
x
+
2
y
1zyx
Out[]=
1
∫
0
π
2
∫
-
π
2
1
∫
r
ruθr
2. Using CoordinateTransform[] function
In[]:=
CoordinateTransform"Cartesian""Cylindrical",0,-
1-
2
x
,
2
x
+
2
y
​​CoordinateTransform"Cartesian""Cylindrical",1,
1-
2
x
,1
Out[]=

1-
2
x
,ArcTan0,-
1-
2
x
,
2
x
+
2
y

Out[]=

2-
2
x
,ArcTan1,
1-
2
x
,1
3. Using ToPolarCoordinates[] function
In[]:=
ToPolarCoordinates0,-
1-
2
x
,
2
x
+
2
y

Out[]=

1+
2
y
,
π
2
,ArcTan-
1-
2
x
,
2
x
+
2
y

In[]:=
ToPolarCoordinates1,
1-
2
x
,1
Out[]=

3-
2
x
,ArcCos
1
3-
2
x
,ArcTan
1-
2
x
,1
4. Using CoordinateData[] function
In[]:=
CoordinateTransformData"Cartesian""Cylindrical","Mapping",0,-
1-
2
x
,
2
x
+
2
y

Out[]=

1-
2
x
,ArcTan0,-
1-
2
x
,
2
x
+
2
y

In[]:=
CoordinateTransformData"Cartesian""Cylindrical","Mapping",1,
1-
2
x
,1
Out[]=

2-
2
x
,ArcTan1,
1-
2
x
,1