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Consider the following region shown in green:

enter image description here

It is a part of the annular cylinder with the inner radius $R$ and the outer radius $R+r$, which is cut off from the left and right. Is it possible to get this figure as a Region in Mathematica? For the test parameters, we may consider its longitudinal length dz = 10, the radii R = 10, r = 2, and the transverse length of the remaining shell dx = 2.

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You could use RegionProduct of an annulus and a line:

reg = RegionProduct[Annulus[{0,0},{10,12},{-ArcSin[.2],ArcSin[.2]}], Line[{{0},{10}}]];

Visualization:

Region @ reg

enter image description here

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