Volume integral (${\int }_{V}fdV$)

A volume integral integrates a scalar or componentwise vector field over a three-dimensional region with its volume form. In spherical polar coordinates the Euclidean volume form is $r^2\sin \theta drd\theta d\varphi$. The divergence theorem relates a volume integral of a divergence to a boundary flux integral.

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 givesThus 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.