# Intersection of volumes defined by inequalities

I have 5 inequalities with some parameters like dh, din and thetawhich I intend to vary.

In Mathematica, I am trying to:

1. Check if the inequalities have a solution.
2. Find the common volume if they do.

I tried this:

theta = 85* Pi/180;
dh = 0.5;
din = 1.016;
dout = 1.27;
zz = z*Cos[theta] - x*Sin[theta];
xx = z*Sin[theta] + x*Cos[theta];
yy = y;
region1 = z^2 >= (dh/2 + 0.0254)^2 - (dout/2)^2*x^2/(x^2 + y^2);
region2 = (din/2)^2 < x^2 + y^2 < (dout/2)^2;
region3 = -1 <= z <= 1;
region4 = zz^2 <= (dh/2)^2 - (dout/2)^2*xx^2/(xx^2 + yy^2);
region5 = (din/2)^2 <= xx^2 + yy^2 <= (dout/2)^2;
region = region1 && region2 && region3 && region4 && region5;
AbsoluteTiming[Resolve[Exists[{x, y, z}, region], Reals]]


whose execution is not completing.

Plotting with

AbsoluteTiming[RegionPlot3D[region1 && region2 && region3 && region4 && region5,
{x , -1.27/2, 1.27/2}, {y, -1.27/2, 1.27/2}, {z, -1, 1}, PlotRange
-> {{-0.7, 0.7}, {-0.7, 0.7}, {-0.7, 0.7}}, AspectRatio -> 1, PlotPoints -> 500,
MaxRecursion -> 15]]


gives an inaccurate, broken volume after 15 minutes of running.

Numerical integration gives 0 (possibly due to some volumes being negative?)

How can I quickly and accurately find if the regions have a common volume and if they do, the magnitude of the common volume?

Show[RegionPlot3D[ region1 && region2 && region5,