Approximating the volume integral by a surface integral via the divergence theorem

By utilising the divergence theorem for $n$-dimensional integrals, we provide some error estimates for approximating the integral on a body $B,$ a bounded closed subset of $\mathbb {R}^{n}$ $(n\geq 2)$ with smooth (or piecewise smooth) boundary $\partial B,$ by an integral on the surface $\partial B$ and some other simple terms. Some examples in the $3$-dimensional case are also given.