Compute Areas and Volumes in Non-Cartesian Coordinates
The "nut" defined by revolving the curve about the
axis can be easily parameterized in cylindrical coordinates.
In[1]:=
![Click for copyable input](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/In_151.png)
ParametricPlot3D[
CoordinateTransform[ "Cylindrical" -> "Cartesian", {Sin[z], t, z}] //
Evaluate, {z, 0, Pi}, {t, 0, 2 Pi}, PlotTheme -> "Business"]
Out[1]=
![](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/O_85.png)
Use Area to find the surface area directly in cylindrical coordinates.
In[2]:=
![Click for copyable input](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/In_152.png)
Area[{Sin[z], t, z}, {z, 0, Pi}, {t, 0, 2 Pi}, "Cylindrical"]
Out[2]=
![](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/O_86.png)
Volume can also be computed in non-Cartesian coordinates.
In[3]:=
![Click for copyable input](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/In_153.png)
Volume[{r Sin[z], t, z}, {z, 0, Pi}, {t, 0, 2 Pi}, {r, 0,
1}, "Cylindrical"]
Out[3]=
![](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/O_87.png)
These computations can be carried out in any number of dimensions with RegionMeasure.
In[4]:=
![Click for copyable input](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/In_154.png)
RegionMeasure[{Sin[z], t,
z}, {{z, 0, Pi}, {t, 0, 2 Pi}}, "Cylindrical"]
Out[4]=
![](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/O_88.png)
In[5]:=
![Click for copyable input](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/In_155.png)
RegionMeasure[{r Sin[z], t,
z}, {{z, 0, Pi}, {t, 0, 2 Pi}, {r, 0, 1}}, "Cylindrical"]
Out[5]=
![](assets.en/compute-areas-and-volumes-in-non-cartesian-coordin/O_89.png)