I learned how to do this:
Show[RevolutionPlot3D[{5 Cos[t], 5 Sin[t]},
{t, -Pi/2, Pi/2}, {\[Theta], 0, 3 Pi/2}],
RevolutionPlot3D[{Cos[t], Sin[t]},
{t, -Pi/2, Pi/2}, {\[Theta], 0, 3 Pi/2}]]
But I would like to show the region between the two surfaces. Can someone show me how?

ParametricRegion[]. – J. M.'s missing motivation May 23 '15 at 04:36