Find the centroid of a solid hemisphere of radius R

Find the centroid of a solid hemisphere of radius $R$, centered at the origin, lying above the $xOy$ plane.

Step 1: Volume

\[V=\frac23\pi R^3\]

Step 2: Moment about the $xy$-plane

In spherical coordinates ($z=\rho\cos\phi$, $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$, $\phi\in[0,\pi/2]$):

\[M_{xy}=\int_0^{2\pi}d\theta\int_0^{\pi/2}\cos\phi\sin\phi\,d\phi\int_0^R\rho^3\,d\rho = 2\pi\cdot\frac12\cdot\frac{R^4}4=\frac{\pi R^4}4\]

Step 3: Centroid

\[\bar z=\frac{M_{xy}}V=\frac{\pi R^4/4}{2\pi R^3/3}=\frac{3R}8\]

\[\boxed{\left(0,0,\frac{3R}8\right)}\]

Step 4: Comparison

The hemisphere's centroid ($3R/8=0.375R$ from the base) sits between the cone's ($h/4=0.25h$) and the cylinder's ($h/2$), consistent with the hemisphere's cross-section shrinking more gradually than a cone's but not staying constant like a cylinder's.