I am trying to plot a certain piecewise function using RevolutionPlot3D, so it revolves around an axis.

u[x_] := Piecewise[{{.5, {0 <= x <= pi/6, 5 pi/6 < x <= 13 pi/6,

17 pi/6 < x <= 10}}, {Sin[x], {pi/6 < x <= 5 pi/6,

13 pi/6 < x <= 17 pi/6}}}]

(It's the upper portion of

this graph)

However, whatever I try, the plot is always blank. I can't even plot it in 2d; it is still blank. Here are some things I've tried:

RevolutionPlot3D[u[x], {x, 0, 2}]

RevolutionPlot3D[Evaluate[u[x]], {x, 0, 2},

RevolutionAxis -> {1, 0, 0}]

RevolutionPlot3D[Piecewise[{{.5, {0 <= x <= pi/6, 5 pi/6 < x <= 13 pi/6,

17 pi/6 < x <= 10}}, {Sin[x], {pi/6 < x <= 5 pi/6,

13 pi/6 < x <= 17 pi/6}}}], {x, 1, 2},

RevolutionAxis -> {1, 0, 0}]

Plot[Evaluate[u[x]], {x, 0, 2}]

Plot[u[x], {x, 0, 2}]

Why won't Mathematica plot my function?