How can I draw solid when rotate this rigion around Ox axis?

I have this picture, knowing that $$AC=2$$. The point $$B$$ is midpoint of $$AC$$, $$D$$ is intersection point of the semicirle and arc center at $$B$$ and radius 1. I want to rotate the region with gray fill aroun the line $$AC$$.

I put

coordinate of $$D$$ is found

SolveValues[{y == Sqrt[1 - x^2], (x + 1)^2 + y^2 == 1}, {x, y}, Reals]


{{-(1/2), Sqrt[3]/2}}

I put

Clear["Global*"];
b = {0, 0};
a = {-1, 0};
c = {1, 0};
d = {-(1/2), Sqrt[3]/2};
f[x_] := Sqrt[1 - x^2];
g[x_] = Sqrt[1 - (x + 1)^2 ];


Clear["Global*"];
cir1[t_] := {Cos[t], Sin[t]};
cir2[s_] := {-1, 0} + {Cos[s], Sin[s]};
{s0, t0} = {s, t} /.
First@FindInstance[{cir2[s] == cir1[t], 0 <= s <= π,
0 <= t <= π}, {s, t}]
Show[ParametricPlot[cir1[t], {t, 0, t0}],
ParametricPlot[cir2[s], {s, 0, s0}]]


Show[RevolutionPlot3D[cir1[t] // Evaluate, {t, 0, t0},
RevolutionAxis -> {1, 0, 0}],
RevolutionPlot3D[cir2[s] // Evaluate, {s, 0, s0},
RevolutionAxis -> {1, 0, 0}], PlotRange -> All, Boxed -> False,
Axes -> False]


b1 = Ball[{0, 0, 0}, 1];
b2 = Ball[{-1, 0, 0}, 1];
rd = RegionDifference[b1, b2];

RegionImage@rd