= Volume of a solid with affine thickness
{title2=$V=A\,h(\overline{\mathbf x})$}
Let a bounded planar region $D$ have positive <area> $A$ and <centroid> $\overline{\mathbf x}=A^{-1}\int_D\mathbf x\,dA$. If a solid has nonnegative thickness $h(\mathbf x)=c+\mathbf v\cdot\mathbf x$ above $D$, its <volume integral> gives
$$
V=\int_D h(\mathbf x)\,dA=A\left(c+\mathbf v\cdot\overline{\mathbf x}\right)=A\,h(\overline{\mathbf x}).
$$
Thus the average of an <affine function> over a region is its value at the <centroid>. The proof is the <linearity of integration>. For a <disk> the <centroid> is its centre, making the formula particularly useful for a cylindrical bore cut through a solid whose width varies linearly with height.
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