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After these preparations, we can finally do the last integral to arbitrary precision numerically

After these preparations, we can finally do the last integral to arbitrary precision numerically:

After these preparations, we can finally do the last integral to arbitrary precision numerically:

After these preparations, we can finally do the last integral to arbitrary precision numerically

EDIT It was asked how I integrated numerically. Here is the answer.

r11 = FullSimplify[r[[11]]];
a11 = Integrate[Boole[r11], {z, 0, 2}];
b11 = FullSimplify[a11 && 0 < y < 1];
c11 = Integrate[b11, {y, 0, 1}];

r7 = FullSimplify[r[[7]]];
a7 = Integrate[Boole[r7], {z, 0, 2}];
b7 = FullSimplify[a7 && 0 < y < 1];
c7 = Integrate[b7, {y, 0, 1}];

r9 = FullSimplify[r[[9]]];
a9 = Integrate[Boole[r9], {z, 0, 2}];
b9 = FullSimplify[a9 && 0 < y < 1];
c9 = Integrate[b9, {y, 0, 1}];

After these preparations, we can finally do the last integral to arbitrary precision numerically:

vol = NIntegrate[8 (c7 + c9 + c11), {x, 0, 1}, WorkingPrecision -> 200]
RootApproximant[vol] // FullSimplify