Integration in angular momentum

[Rikudo]

Homework Statement

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Relevant Equations

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https://www.physicsforums.com/threa...f-a-translating-and-rotating-pancake.1005990/
So,I think I posted this in the wrong place. So, I will move it to here.
Here, in post #6, it is stated that $\int R dm = M R$. As far as I know, R change from time to time and it is not constant. Hence, isn't it incorrect to say that $\int R dm = M R$? Or, are there any techniques that are skipped?

 

[anuttarasammyak]

Though it is not an answer to your question but a matter of taste, I am not familiar with expression dm. I prefer to say it
$$\mathbf{L}=\int \mathbf{r} \times \mathbf{\pi}(\mathbf{r}) d^3\mathbf{r} =\int \mathbf{r} \times \rho(\mathbf{r}) \mathbf{v}(\mathbf{r}) d^3\mathbf{r}$$
where $\pi(r)$ is momentum density and $\rho(r)$ is mass density.
So now I know
$$dm = \rho(\mathbf{r}) d^3\mathbf{r}$$

 

[haruspex]

Here, in post #6, it is stated that $\int R dm = M R$. As far as I know, R change from time to time and it is not constant.

Right, but that integral has no moving parts. It is the integral over the mass at some instant.