Integral of an affine function over a simplex (${\int }_{S}f(x)dx=\frac{\mathrm{vol}(S)}{d+1}{\sum }_{i=0}^{d}f(v_i)$)

For a nondegenerate $d$-dimensional simplex $S$ with vertices $v_0,\dots ,v_d$ and an affine function $f$, the integral equals the volume times the value at the centroid $c=(v_0+\cdots +v_d)/(d+1)$. To prove this, write $x={\sum }_i{\lambda }_i(x)v_i$ in a barycentric coordinate system. On the standard simplex, symmetry permuting the vertices gives equal integrals of all ${\lambda }_i$; since ${\sum }_i{\lambda }_i=1$, each is $\mathrm{vol}(S)/(d+1)$. An invertible affine map transfers this result to any simplex. Affineness gives $f(x)={\sum }_i{\lambda }_i(x)f(v_i)$, so linearity of integration proves the formula. For a tetrahedron the factor is one quarter. The rule is exact for affine integrands; a nonlinear integrand generally needs additional moments.