= Spherical wedge
A <spherical wedge> is the portion of a solid <sphere> between two meridional half-planes meeting at a dihedral angle $\beta$, with $0<\beta<2\pi$. Put the common axis along $z$ and its azimuthal bisector along $x$. For radius $a$, use <spherical coordinates> $0\leq r\leq a$, $0\leq\theta\leq\pi$, $-\beta/2\leq\varphi\leq\beta/2$. Its <volume> is $2\beta a^3/3$.
For constant <mass density>, reflection symmetries put the <centre of mass> on the bisector. The first moment is
$$
\int_Vx\,dV=\left(\int_0^a r^3\,dr\right)
\left(\int_0^\pi\sin^2\theta\,d\theta\right)
\left(\int_{-\beta/2}^{\beta/2}\cos\varphi\,d\varphi\right)
=\frac{\pi a^4}{4}\sin\frac\beta2.
$$
Dividing by <volume> gives $\bar x=3\pi a\sin(\beta/2)/(8\beta)$. It decreases from the limiting value $3\pi a/16$ for an infinitesimal wedge to zero for a full sphere.
The curved boundary is a <spherical lune>. Its outward <vector area> is $\pi a^2\sin(\beta/2)\mathbf e_x$. Indeed, its $x$ component has the same angular integrals as the first moment, multiplied by $a^2$; the other components cancel by symmetry. Its scalar <surface area> is $2\beta a^2$, which is generally different from the magnitude of its <vector area>.
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