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.

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ParametricPlot3D[ CoordinateTransform[ "Cylindrical" -> "Cartesian", {Sin[z], t, z}] // Evaluate, {z, 0, Pi}, {t, 0, 2 Pi}, PlotTheme -> "Business"]

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Use Area to find the surface area directly in cylindrical coordinates.

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Area[{Sin[z], t, z}, {z, 0, Pi}, {t, 0, 2 Pi}, "Cylindrical"]

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Volume can also be computed in non-Cartesian coordinates.

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Volume[{r Sin[z], t, z}, {z, 0, Pi}, {t, 0, 2 Pi}, {r, 0, 1}, "Cylindrical"]

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These computations can be carried out in any number of dimensions with RegionMeasure.

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RegionMeasure[{Sin[z], t, z}, {{z, 0, Pi}, {t, 0, 2 Pi}}, "Cylindrical"]

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RegionMeasure[{r Sin[z], t, z}, {{z, 0, Pi}, {t, 0, 2 Pi}, {r, 0, 1}}, "Cylindrical"]

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