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I'd like to calculate the center of mass of $S=${$x\in\mathbb{R}^n : |x| \leq 1, x_n >0$}. I know that the COM is calculated with $C(M) = \frac{1}{\lambda(M)}\int_M x d\lambda(x)$ with component-wise integral. I also intuitively know that just the last coordinate of the resulting vector can be different from 0. Thanks in advance.

mathemagiker
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    Your $S$ is a half-ball, not a hemisphere. Similar to the hemisphere case, the volume element will be $r^{n - 1} \sin^{n - 2} \theta_1 \cdots \sin \theta_{n - 2} , dr d\theta_1 \cdots d\theta_{n - 1}$, the integrals over $\theta_2, \ldots, \theta_{n - 1}$ will cancel out. – Maxim Feb 13 '21 at 17:29
  • see https://math.stackexchange.com/q/870253 – Jean Marie Feb 13 '21 at 19:11

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