Mathematica Mathematica - what is wrong with ImplicitRegion[]

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The discussion centers on evaluating the integral of z over the lateral surface area of a cylinder defined by the equation y^2 + z^2 = 4, constrained by the planes x = y - 3 and x = 6 - z. The user successfully computes the integral using a parametrization of the surface, yielding a result of -8π. However, when attempting to use Mathematica for the same calculation, the software returns a result of 8π, which contradicts the manual computation. The user initially suspects an error in their approach but later concludes that the discrepancy is due to a bug in Mathematica, confirming that their manual calculations are correct.
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Trying to understand how to use an implicit region in Integrate.
I'm working the integral
$$\displaystyle \int_S z\,dS$$

Where S is the lateral (surface) area of the cylinder y^2+z^2=4 cut off by two planes, x=y-3 and x=6-z.

Using the parametrization r(x,\theta)=(x,2\cos(\theta), 2\sin(\theta)) this is pretty straight forward

$$\int 2\sin(\theta)|\frac{\partial r}{\partial x}\times\frac{\partial r}{\partial \theta}|\,dxd\theta=-8\pi$$

However, when I use Mathematica I get the following:

Code:
myReg2 = ImplicitRegion[y^2 + z^2 == 4, {{x, y - 3, 6 - z}, y, z}];
Integrate[z, {x, y, z} \[Element] myReg2]

>8 \[Pi]

Even using my exact parametrization:

Code:
myRegion = 
  ParametricRegion[{x, 2*Cos[\[Theta]], 
    2*Sin[\[Theta]]}, {{x, 2*Cos[\[Theta]] - 3, 
     6 - 2*Sin[\[Theta]]}, {\[Theta], 0, 2 \[Pi]}}];
Integrate[z, {x, y, z} \[Element] myRegion]

>8 \[Pi]

Obviously I'm missing something, but I can't figure it out and it drives me nuts. Any ideas would be welcome!
 
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Quick update*; so I was not missing something. Apparently, this is a bug in Mathematica.
 
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