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- TL;DR Summary
- Trying to understand how to use an implicit region in Integrate.
I'm working the integral
$$\displaystyle \int_S z\,dS$$
Where [itex]S[/itex] is the lateral (surface) area of the cylinder [itex]y^2+z^2=4[/itex] cut off by two planes, [itex]x=y-3[/itex] and [itex]x=6-z[/itex].
Using the parametrization [itex]r(x,\theta)=(x,2\cos(\theta), 2\sin(\theta))[/itex] 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:
Even using my exact parametrization:
Obviously I'm missing something, but I can't figure it out and it drives me nuts. Any ideas would be welcome!
$$\displaystyle \int_S z\,dS$$
Where [itex]S[/itex] is the lateral (surface) area of the cylinder [itex]y^2+z^2=4[/itex] cut off by two planes, [itex]x=y-3[/itex] and [itex]x=6-z[/itex].
Using the parametrization [itex]r(x,\theta)=(x,2\cos(\theta), 2\sin(\theta))[/itex] 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!