Tensor-product quadrature ($Q(f)={\sum }_{i_1,\dots ,i_d}{w}_{i_1}\cdots w_{i_d}f(x_{i_1},\dots ,x_{i_d})$)

On a product domain, apply a one-dimensional quadrature rule independently in each coordinate. The resulting quadrature nodes form a product grid and the quadrature weights multiply. With $m$ subdivisions per direction in fixed dimension $d$, the evaluation count is $N=O(m^d)$. If the one-dimensional rule has error $O(m^{-p})$, uniformly for the required coordinate slices, and its weight sums are bounded, telescoping the $d$ integration operators against their quadrature operators gives total error $O(m^{-p})=O(N^{-p/d})$. Thus composite Simpson's rule has $p=4$ for sufficiently smooth integrands, but loses efficiency as $d$ increases.